tag:blogger.com,1999:blog-82441316731480792242024-02-21T17:43:18.208+00:00MatifutbolMatifutbol: Cool Maths, logic games, and some football. Here you'll find posts full of funny puzzles, charts, Math games, logic riddles, geometry tricks, arithmetical problems, statistical paradoxes and probabiliy challenges. WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.comBlogger39125tag:blogger.com,1999:blog-8244131673148079224.post-47401848435707554022016-08-30T16:24:00.000+01:002016-08-31T22:14:33.786+01:00Index<div align="center">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCjjXaPIFLocJRpyykmLrkRVGeR8q6F6jAo4HRMwWfDihhKRIaqP6v9mIW8W2IBvqq-zIp5LIdK2p449tAd76wxNMLP_58If8e64-qOF92TTLN0So2NREULq6vObgNLjTW0ooIZOSOswA/s1600/150+Pepe+Vitruvio+-+Joe+Vitruvius+-+Pep+Vitruvi.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCjjXaPIFLocJRpyykmLrkRVGeR8q6F6jAo4HRMwWfDihhKRIaqP6v9mIW8W2IBvqq-zIp5LIdK2p449tAd76wxNMLP_58If8e64-qOF92TTLN0So2NREULq6vObgNLjTW0ooIZOSOswA/s200/150+Pepe+Vitruvio+-+Joe+Vitruvius+-+Pep+Vitruvi.jpg" width="98" /></a>Many <b><i><span style="color: #351c75; font-family: "georgia" , "times new roman" , serif; font-size: 140%;">thanks </span></i></b>for visiting this website!<br />
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Here you'll find some <b><span style="color: red; font-size: 140%;">math </span></b>& <span style="color: #274e13; font-family: "times" , "times new roman" , serif; font-size: 140%;"><b>football </b></span>problems. Surely you'll have fun and entertainment with them.</div>
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Which one will you pick up first?</div>
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<h3>
Last entries
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<ul>
<li><a href="http://mathsball.blogspot.com/2015/06/the-awkward-question.html" target="_blank">The awkward question</a></li>
<li><a href="http://mathsball.blogspot.com/2015/04/power-factor-number.html" target="_blank">A galactic prize</a></li>
<li><a href="http://mathsball.blogspot.com/2015/01/flavius-josephus-problem.html" target="_blank">Flavius Josephus and the puisi-nanoq game</a></li>
<li><a href="http://mathsball.blogspot.com/2014/12/mathematical-greetings-card.html" target="_blank">A very special greetings card</a></li>
<li><a href="http://mathsball.blogspot.com.es/2014/10/mathematical-halloween-vampire-numbers.html" target="_blank">Mathematical Halloween</a></li>
<li><a href="http://mathsball.blogspot.com.es/2014/06/mathematical-murderer-combinatorics-pascal-triangle.html" target="_blank" title="The mathematical murderer. Combinatorics, Pascal triangle">The mathematical murderer</a></li>
<li><a href="http://mathsball.blogspot.com/2014/06/mathematical-binary-trees-estructure.html" target="_blank" title="Trees and googols at World Cup 2014. Binary trees, tree structure, Wedderburn-Etherington">Trees and googols at World Cup 2014.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/03/probability-two-envelopes-paradox-exchange.html" target="_blank" title="The probability and the ocean. Probability, optimal stopping theory, taking decisions, two-envelopes paradox">Probability and sunrise.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/01/fibonacci-also-plays-football.html" target="_blank" title="Fibonacci also plays football. Fibonacci sequence.">Fibonacci also plays football.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/12/magic-2014.html" target="_blank" title="Magic 2014. Theory of numbers.">Magic 2014</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/11/the-amazing-case-of-extra-car.html" target="_blank" title="The amazing case of the extra car. Alabama paradox, voting, apportionment.">The amazing case of the extra car.</a></li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBq6rJsC-VMhs4QI2oSMqn9c1JDSeq2flX1TtR0HJ87VupQZ80OWXTNmdXOGxdIbiYIsCzc_M6gvdZAOTqpNrhwzXfViE5K3NKYCYftx_rLyRBTXKQjSw0H3_FulZr6k7svWWdVr76hyE/s1600/eye-151331_640.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="110" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBq6rJsC-VMhs4QI2oSMqn9c1JDSeq2flX1TtR0HJ87VupQZ80OWXTNmdXOGxdIbiYIsCzc_M6gvdZAOTqpNrhwzXfViE5K3NKYCYftx_rLyRBTXKQjSw0H3_FulZr6k7svWWdVr76hyE/s1600/eye-151331_640.png" width="200" /></a><br />
<h3>
Most visited
</h3>
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<ul>
<li><a href="http://mathsball.blogspot.com/2014/01/fibonacci-also-plays-football.html" target="_blank" title="Fibonacci also plays football. Fibonacci sequence.">Fibonacci also plays football.</a></li>
<li><a href="http://mathsball.blogspot.com/2015/04/power-factor-number.html" target="_blank">A galactic prize</a></li>
<li><a href="http://mathsball.blogspot.com/2015/01/flavius-josephus-problem.html" target="_blank">Flavius Josephus and the puisi-nanoq game</a></li>
<li><a href="http://mathsball.blogspot.com.es/2014/10/mathematical-halloween-vampire-numbers.html" target="_blank">Mathematical Halloween</a></li>
<li><a href="http://mathsball.blogspot.com.es/2014/06/mathematical-murderer-combinatorics-pascal-triangle.html" target="_blank" title="The mathematical murderer. Combinatorics, Pascal triangle">The mathematical murderer</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/12/magic-2014.html" target="_blank" title="Magic 2014. Theory of numbers.">Magic 2014</a></li>
<li><a href="http://www.matifutbol.com/en/triangle.eng.html" target="_blank" title="Özil and the Fermat point. Notable points of the triangle. Geometry.">Özil and the Fermat point.</a></li>
<li><a href="http://www.matifutbol.com/en/penalty.html" target="_blank" title="And now, who should kick de penalty? Statistics, Simpson's paradox, Gerrymandering, Yule-Simpson efect.">And now, who should kick the penalty?</a></li>
<li><a href="http://www.matifutbol.com/en/glove.html" target="_blank" title="The Golden Goalkeeper. Arrow's paradox, Voting systems">The Golden Goalkeeper.</a></li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSl82Cd_8Fd-C-UNFlWsd1A40rEy-fvTirk1yUlkS9uZcoZWpD-Z5UiLH6cXq1us17NhZc_lNu69iOJn927K3Q2YZlwn6iTGxukRMe-eJbLG64ohCvalfJrcFrmfv9B8WTAOtH6Xi70hk/s1600/dice-147157_640.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="121" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSl82Cd_8Fd-C-UNFlWsd1A40rEy-fvTirk1yUlkS9uZcoZWpD-Z5UiLH6cXq1us17NhZc_lNu69iOJn927K3Q2YZlwn6iTGxukRMe-eJbLG64ohCvalfJrcFrmfv9B8WTAOtH6Xi70hk/s1600/dice-147157_640.png" width="200" /></a><br />
<h3>
Probability
</h3>
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<ul>
<li><a href="http://www.matifutbol.com/en/reserve.html" target="_blank" title="Beckham and the reserve player's chance. The problem of Monty Hall, Probability">Beckham and the reserve player's chance.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/08/team-win-champions-league.html" target="_blank" title="Which team will win the Champions League? Probability, Inverse probability.">Which team will win the Champions League?</a></li>
<li><a href="http://www.matifutbol.com/en/flags.html" target="_blank" title="The importance of a flag. Conditioned probability.">The importance of a flag.</a></li>
<li><a href="http://www.matifutbol.com/en/eurobirthdays.html" target="_blank" title="Eurobirthdays 2012. Birthday paradox, Probability.">Eurobirthdays 2012.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/10/breakfast-at-les-deux-magots.html" target="_blank" title="Breakfast at Les Deux Magots. Friendship paradox, probability, mean and variance.">Breakfast at Les Deux Magots.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/03/probability-two-envelopes-paradox-exchange.html" target="_blank" title="The probability and the ocean. Probability, optimal stopping theory, taking decisions, two-envelopes paradox">Probability and sunrise.</a></li>
<li><a href="http://mathsball.blogspot.com/2015/06/the-awkward-question.html" target="_blank">The awkward question</a></li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCVnc05bo0hhwJ98wEQbfaPLhOwFsPx2rkP02jFfTaAX5ZP-I86kpd62V4ZJUgYpihFhUBVb1t1HzjHAoCUEVAQ6X-yyqcfC4WKCCWyt4Hh0hqsPemB-uJJcFg8jIj3cukQOTb6OKtOD4/s1600/cubes-161964_640.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="121" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCVnc05bo0hhwJ98wEQbfaPLhOwFsPx2rkP02jFfTaAX5ZP-I86kpd62V4ZJUgYpihFhUBVb1t1HzjHAoCUEVAQ6X-yyqcfC4WKCCWyt4Hh0hqsPemB-uJJcFg8jIj3cukQOTb6OKtOD4/s1600/cubes-161964_640.png" width="200" /></a><br />
<h3>
Geometry
</h3>
<br />
<ul>
<li><a href="http://www.matifutbol.com/en/triangle.eng.html" target="_blank" title="Özil and the Fermat point. Notable points of the triangle. Geometry.">Özil and the Fermat point.</a></li>
<li><a href="http://www.matifutbol.com/en/balls.html" target="_blank" title="Let's give away some balls. Compact packing. Geometry.">Let's give away some balls.</a></li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwg4QiXNxJ2Z8YL8wL8Sc-Cc-_0da83FrznifSOH53ht-lWjkI-9i0hOfOvjr2OB1THVYBMwt5Aeq414X_mxPHGwdDjnrgI22P3PozQDP87-or6yDaVqnOhsWZVAA6ksfNujqW9BoNKq8/s1600/rubiks-cube-41382_640.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwg4QiXNxJ2Z8YL8wL8Sc-Cc-_0da83FrznifSOH53ht-lWjkI-9i0hOfOvjr2OB1THVYBMwt5Aeq414X_mxPHGwdDjnrgI22P3PozQDP87-or6yDaVqnOhsWZVAA6ksfNujqW9BoNKq8/s1600/rubiks-cube-41382_640.png" width="200" /></a><br />
<h3>
Logic
</h3>
<br />
<ul>
<li><a href="http://www.matifutbol.com/en/liars.html" target="_blank" title="Does anybody tell the truth? Logical thinking, The paradox of the caterpillar and the lizard.">Does anybody tell the truth?</a></li>
<li><a href="http://www.matifutbol.com/en/decision.eng.html" target="_blank" title="Llorente's hard decision. Optimal stopping theorem, taking decisions, logical thinking and strategy.">Llorente's hard decision.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/09/herds-of-goats-and-group-decisions.html" target="_blank" title="Herds of goats and group decisions. Logic. Group thinking. Voting. Abilene paradox.">Herds of goats and group decisions.</a></li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVz-l-42oLrALAR-Y1l5v_jnafz5XrxNIbbRiaAtgAL0gKyES8zXkSwLlFLPPLyUoRbTStJClCfhyrx35QzHQJf1oCv7sKf_8uRxd0-QWpoxwCB5MgQD6-Fvoa2gAWcqrg4WqmVyqiUEc/s1600/abacus-33275_640.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="141" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVz-l-42oLrALAR-Y1l5v_jnafz5XrxNIbbRiaAtgAL0gKyES8zXkSwLlFLPPLyUoRbTStJClCfhyrx35QzHQJf1oCv7sKf_8uRxd0-QWpoxwCB5MgQD6-Fvoa2gAWcqrg4WqmVyqiUEc/s1600/abacus-33275_640.png" width="200" /></a><br />
<h3>
Arithmetics
</h3>
<br />
<ul>
<li><a href="http://www.matifutbol.com/en/grass.html" target="_blank" title="Simple Arithmetics, Composed rule of three, The problem of water lilies.">Juventus and the Chinese grass.</a></li>
<li><a href="http://www.matifutbol.com/en/gates.html" target="_blank" title="The troubled stadium gates. Arithmetic and geometric progressions.">The troubled stadium gates.</a></li>
<li><a href="http://www.matifutbol.com/en/blackhole.html" target="_blank" title="A black (and red) hole. Mathematical black holes.">A black (and red) hole.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/07/football-and-happiness.html" target="_blank" title="Football and happiness. Happy numbers.">Football and happiness.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/10/breakfast-at-les-deux-magots.html" target="_blank" title="Breakfast at Les Deux Magots. Friendship paradox, probability, mean and variance.">Breakfast at Les Deux Magots.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/01/fibonacci-also-plays-football.html" target="_blank" title="Fibonacci also plays football. Fibonacci sequence.">Fibonacci also plays football.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/06/mathematical-binary-trees-estructure.html" target="_blank" title="Trees and googols at World Cup 2014. Binary trees, tree structure, Wedderburn-Etherington">Trees and googols at World Cup 2014.</a></li>
<li><a href="http://mathsball.blogspot.com/2015/01/flavius-josephus-problem.html" target="_blank">Flavius Josephus and the puisi-nanoq game</a></li>
<li><a href="http://mathsball.blogspot.com/2015/04/power-factor-number.html" target="_blank">A galactic prize</a></li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibplnm52q1lHKsmTWN792qoenyfYUHReexkkBy3rb6zLo6Ey63yAlY96nT34MXgpDBsYG9WhIOt_Qwnt-1ZiomXLUPs1LFJla8D7ApklDILzVQfW0_NDm2kYETw-JXm8WZ6puY4Z4IzqQ/s1600/pensamiento+lateral.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibplnm52q1lHKsmTWN792qoenyfYUHReexkkBy3rb6zLo6Ey63yAlY96nT34MXgpDBsYG9WhIOt_Qwnt-1ZiomXLUPs1LFJla8D7ApklDILzVQfW0_NDm2kYETw-JXm8WZ6puY4Z4IzqQ/s1600/pensamiento+lateral.jpg" width="200" /></a><br />
<h3>
Lateral thinking
</h3>
<br />
<ul>
<li><a href="http://www.matifutbol.com/en/goal.html" target="_blank" title="Ribéry's strange training. Lateral thinking, Even and odd.">Ribéry's strange training.</a></li>
<li><a href="http://www.matifutbol.com/en/crosses.html" target="_blank" title="Do you dare to count? A forward-and-back problem. Logical thinking. The problem of the ships">Do you dare to count? A forward-and-back problem.</a></li>
<li><a href="http://www.matifutbol.com/en/fairdivision.html" target="_blank" title="The problem of the fair division. Lateral thinking.">The problem of the fair division.</a></li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibJeOC-FsJVEHPtp1gjZU70kB74wjUhZCym517i3q9N6rTMUSCz5mZ9L53pLelxdHixZumRwJcXz0Ieoje1NCYLPtI1kse7YmrxmC5aalBgRR6Y12yW9qmbuUOVHkSjqVj7IuHnqXDRck/s1600/2694239747_21bbdeb560_z.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibJeOC-FsJVEHPtp1gjZU70kB74wjUhZCym517i3q9N6rTMUSCz5mZ9L53pLelxdHixZumRwJcXz0Ieoje1NCYLPtI1kse7YmrxmC5aalBgRR6Y12yW9qmbuUOVHkSjqVj7IuHnqXDRck/s1600/2694239747_21bbdeb560_z.jpg" width="200" /></a><br />
<h3>
Numbers
</h3>
<br />
<ul>
<li><a href="http://mathsball.blogspot.com/2014/06/mathematical-binary-trees-estructure.html" target="_blank" title="Trees and googols at World Cup 2014. Binary trees, tree structure, Wedderburn-Etherington">Trees and googols at World Cup 2014.</a></li>
<li><a href="http://www.matifutbol.com/en/pencil.html" target="_blank" title="The fantastic pencil of Sir Alex Ferguson. Champernowne constant, Statistics.">The fantastic pencil of Sir Alex Ferguson.</a></li>
<li><a href="http://www.matifutbol.com/en/blackhole.html" target="_blank" title="A black (and red) hole. Mathematical black holes.">A black (and red) hole.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/07/football-and-happiness.html" target="_blank" title="Football and happiness. Happy numbers.">Football and happiness.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/12/magic-2014.html" target="_blank" title="Magic 2014. Theory of numbers.">Magic 2014.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/01/fibonacci-also-plays-football.html" target="_blank" title="Fibonacci also plays football. Fibonacci sequence.">Fibonacci also plays football.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2014/10/mathematical-halloween-vampire-numbers.html" target="_blank">Mathematical Halloween</a></li>
<li><a href="http://mathsball.blogspot.com/2014/12/mathematical-greetings-card.html" target="_blank">A very special greetings card</a></li>
<li><a href="http://mathsball.blogspot.com/2015/04/power-factor-number.html" target="_blank">A galactic prize</a></li>
</ul>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSAWcVqTTUjwcFOjWjLTKz-u6wGNWGS-0zv6dQXZOgSaTKp49OODRZm-CZMzM9cwns6QBGZ6vZQO5yDTqsUM7yX7UWMnMccWP93eXvb7G8eaMMB3ondpY3mcUDjpTUxqrGG1BpeexbaYw/s1600/como-utilizar-google-trends.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="171" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSAWcVqTTUjwcFOjWjLTKz-u6wGNWGS-0zv6dQXZOgSaTKp49OODRZm-CZMzM9cwns6QBGZ6vZQO5yDTqsUM7yX7UWMnMccWP93eXvb7G8eaMMB3ondpY3mcUDjpTUxqrGG1BpeexbaYw/s1600/como-utilizar-google-trends.png" width="200" /></a><br />
<h3>
Statistics
</h3>
<br />
<ul>
<li><a href="http://www.matifutbol.com/en/penalty.html" target="_blank" title="And now, who should kick de penalty? Statistics, Simpson's paradox, Gerrymandering, Yule-Simpson efect.">And now, who should kick the penalty?</a></li>
<li><a href="http://www.matifutbol.com/en/glove.html" target="_blank" title="The Golden Goalkeeper. Arrow's paradox, Voting systems">The Golden Goalkeeper.</a></li>
<li><a href="http://www.matifutbol.com/en/bus.html" target="_blank" title="The prodigious memory of Van Bommel. Statistic, Bolzano's theorem, The intermediate value theorem">The prodigious memory of Van Bommel.</a></li>
<li><a href="http://www.matifutbol.com/en/benefit.html" target="_blank" title="A benefit match. A reflection on arithmetic means.">A benefit match.</a></li>
<li><a href="http://mathsball.blogspot.com/2013/06/press-stats-and-football-charts.html" target="_blank" title="Press, stats and football charts. Statistical paradoxes with index numbers.">Press, stats and football charts.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/10/breakfast-at-les-deux-magots.html" target="_blank" title="Breakfast at Les Deux Magots. Friendship paradox, probability, mean and variance.">Breakfast at Les Deux Magots.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/11/the-amazing-case-of-extra-car.html" target="_blank" title="The amazing case of the extra car. Alabama paradox, voting, apportionment.">The amazing case of the extra car.</a></li>
</ul>
</div>
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<div align="center">
<div style="background-color: #eaffdd; border: 1px double #000000; color: black; padding: 10px; text-align: justify; width: 630px;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6Hnj55su6HlEx3Qs0TW_OduHK7kQVmPIxDDtP16DkaFGtLGYn7cOKxIg0n1zUou9KIuZXfLsqMBNDg4tPEs40tqyf3i8iEGR-qzUQbPCnlg887Szq0uAfshmRj4g-WoxSdXb8YYsqXbE/s1600/mathematics-299384_640.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6Hnj55su6HlEx3Qs0TW_OduHK7kQVmPIxDDtP16DkaFGtLGYn7cOKxIg0n1zUou9KIuZXfLsqMBNDg4tPEs40tqyf3i8iEGR-qzUQbPCnlg887Szq0uAfshmRj4g-WoxSdXb8YYsqXbE/s1600/mathematics-299384_640.jpg" width="200" /></a><br />
<h3>
Other subjects
</h3>
<br />
<ul>
<li><a href="http://mathsball.blogspot.com/2014/06/mathematical-murderer-combinatorics-pascal-triangle.html" target="_blank" title="The mathematical murderer. Combinatorics, Pascal triangle">The mathematical murderer.</a></li>
<li><a href="http://www.matifutbol.com/en/kicks.html" target="_blank" title="Guardiola: the last training session. Lateral thinking, The problem of the bridge and the lantern">Guardiola: the last training session.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/06/mathematical-binary-trees-estructure.html" target="_blank" title="Trees and googols at World Cup 2014. Binary trees, tree structure, Wedderburn-Etherington">Trees and googols at World Cup 2014.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/12/mathematical-greetings-card.html" target="_blank">A very special greetings card</a></li>
<li><a href="http://mathsball.blogspot.com/2015/01/flavius-josephus-problem.html" target="_blank">Flavius Josephus and the puisi-nanoq game</a></li>
<li><a href="http://mathsball.blogspot.com/2015/06/the-awkward-question.html" target="_blank">The awkward question</a></li>
</ul>
</div>
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<div align="center">
<div style="background-color: #eaffdd; border: 1px double #000000; color: black; padding: 10px; text-align: justify; width: 630px;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-LbhZl-tE9MSxSGq2II2ZWWvTVeRVIqLxJWAw2h0nxMLnh3UluQOgRMRLTNaIqyz8YuGY8GLC1OWzOtOPPhXlaJghV-7wi008M5r8ghTg2y3OkhWhANk6xe9prBjlC_0WRg3k3Kx7edU/s1600/fractals-65831_640.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-LbhZl-tE9MSxSGq2II2ZWWvTVeRVIqLxJWAw2h0nxMLnh3UluQOgRMRLTNaIqyz8YuGY8GLC1OWzOtOPPhXlaJghV-7wi008M5r8ghTg2y3OkhWhANk6xe9prBjlC_0WRg3k3Kx7edU/s1600/fractals-65831_640.jpg" width="200" /></a><br />
<h3>
All the published puzzles
</h3>
<br />
<ul>
<li><a href="http://www.matifutbol.com/en/triangle.eng.html" target="_blank" title="Özil and the Fermat point. Notable points of the triangle. Geometry.">Özil and the Fermat point.</a></li>
<li><a href="http://www.matifutbol.com/en/penalty.html" target="_blank" title="And now, who should kick de penalty? Statistics, Simpson's paradox, Gerrymandering, Yule-Simpson efect.">And now, who should kick the penalty?</a></li>
<li><a href="http://www.matifutbol.com/en/glove.html" target="_blank" title="The Golden Goalkeeper. Arrow's paradox, Voting systems">The Golden Goalkeeper.</a></li>
<li><a href="http://www.matifutbol.com/en/liars.html" target="_blank" title="Does anybody tell the truth? Logical thinking, The paradox of the caterpillar and the lizard.">Does anybody tell the truth?</a></li>
<li><a href="http://www.matifutbol.com/en/decision.eng.html" target="_blank" title="Llorente's hard decision. Optimal stopping theorem, taking decisions, logical thinking and strategy.">Llorente's hard decision.</a></li>
<li><a href="http://www.matifutbol.com/en/reserve.html" target="_blank" title="Beckham and the reserve player's chance. The problem of Monty Hall, Probability">Beckham and the reserve player's chance.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2013/08/team-win-champions-league.html" target="_blank" title="Which team will win the Champions League? Probability, Inverse probability.">Which team will win the Champions League?</a></li>
<li><a href="http://www.matifutbol.com/en/flags.html" target="_blank" title="The importance of a flag. Conditioned probability.">The importance of a flag.</a></li>
<li><a href="http://www.matifutbol.com/en/eurobirthdays.html" target="_blank" title="Eurobirthdays 2012. Birthday paradox, Probability.">Eurobirthdays 2012.</a></li>
<li><a href="http://www.matifutbol.com/en/grass.html" target="_blank" title="Simple Arithmetics, Composed rule of three, The problem of water lilies.">Juventus and the Chinese grass.</a></li>
<li><a href="http://www.matifutbol.com/en/gates.html" target="_blank" title="The troubled stadium gates. Arithmetic and geometric progressions.">The troubled stadium gates.</a></li>
<li><a href="http://www.matifutbol.com/en/bus.html" target="_blank" title="The prodigious memory of Van Bommel. Statistic, Bolzano's theorem, The intermediate value theorem">The prodigious memory of Van Bommel.</a></li>
<li><a href="http://www.matifutbol.com/en/benefit.html" target="_blank" title="A benefit match. A reflection on arithmetic means.">A benefit match.</a></li>
<li><a href="http://www.matifutbol.com/en/goal.html" target="_blank" title="Ribéry's strange training. Lateral thinking, Even and odd.">Ribéry's strange training.</a></li>
<li><a href="http://www.matifutbol.com/en/crosses.html" target="_blank" title="Do you dare to count? A forward-and-back problem. Logical thinking. The problem of the ships">Do you dare to count?</a></li>
<li><a href="http://www.matifutbol.com/en/pencil.html" target="_blank" title="The fantastic pencil of Sir Alex Ferguson. Champernowne constant, Statistics.">The fantastic pencil of Sir Alex Ferguson.</a></li>
<li><a href="http://www.matifutbol.com/en/kicks.html" target="_blank" title="Guardiola: the last training session. Lateral thinking, The problem of the bridge and the lantern">Guardiola: the last training session.</a></li>
<li><a href="http://www.matifutbol.com/en/fairdivision.html" target="_blank" title="The problem of the fair division. Lateral thinking.">The problem of the fair division.</a></li>
<li><a href="http://www.matifutbol.com/en/balls.html" target="_blank" title="Let's give away some balls. Compact packing. Geometry.">Let's give away some balls.</a></li>
<li><a href="http://www.matifutbol.com/en/blackhole.html" target="_blank" title="A black (and red) hole. Mathematical black holes.">A black (and red) hole.</a></li>
<li><a href="http://mathsball.blogspot.com/2013/06/press-stats-and-football-charts.html" target="_blank" title="Press, stats and football charts. Statistical paradoxes with index numbers.">Press, stats and football charts.</a></li>
<li><a href="http://mathsball.blogspot.com/2013/07/football-and-happiness.html" target="_blank" title="Football and happiness. Happy numbers.">Football and happiness.</a></li>
<li><a href="http://mathsball.blogspot.com/2013/08/herds-of-goats-and-group-decisions.html" target="_blank" title="Herds of goats and group decisions. Logic. Groupo thinking. Voting. Abilene paradox.">Herds of goats and group decisions.</a></li>
<li><a href="http://mathsball.blogspot.com/2013/10/breakfast-at-les-deux-magots.html" target="_blank" title="Breakfast at Les Deux Magots. Friendship paradox, probability, mean and variance.">Breakfast at Les Deux Magots.</a></li>
<li><a href="http://mathsball.blogspot.com/2013/11/the-amazing-case-of-extra-car.html" target="_blank" title="The amazing case of the extra car. Alabama paradox, voting, apportionment.">The amazing case of the extra car.</a></li>
<li><a href="http://mathsball.blogspot.com/2013/12/magic-2014.html" target="_blank" title="Magic 2014. Theory of numbers.">Magic 2014.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/01/fibonacci-also-plays-football.html" target="_blank" title="Fibonacci also plays football. Fibonacci sequence.">Fibonacci also plays football.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/03/probability-two-envelopes-paradox-exchange.html" target="_blank" title="The probability and the ocean. Probability, optimal stopping theory, taking decisions, two-envelopes paradox">Probability and sunrise.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/06/mathematical-binary-trees-estructure.html" target="_blank" title="Trees and googols at World Cup 2014. Binary trees, tree structure, Wedderburn-Etherington">Trees and googols at World Cup 2014.</a></li>
<li><a href="http://mathsball.blogspot.com/2014/06/mathematical-murderer-combinatorics-pascal-triangle.html" target="_blank" title="The mathematical murderer. Combinatorics, Pascal triangle">The mathematical murderer.</a></li>
<li><a href="http://mathsball.blogspot.com.es/2014/10/mathematical-halloween-vampire-numbers.html" target="_blank">Mathematical Halloween</a></li>
<li><a href="http://mathsball.blogspot.com/2015/01/flavius-josephus-problem.html" target="_blank">Flavius Josephus and the puisi-nanoq game</a></li>
<li><a href="http://mathsball.blogspot.com/2015/04/power-factor-number.html" target="_blank">A galactic prize</a></li>
<li><a href="http://mathsball.blogspot.com/2015/06/the-awkward-question.html" target="_blank">The awkward question</a></li>
</ul>
</div>
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
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<div align="center">
<div style="background-color: #eaffdd; border: 1px double #000000; color: black; padding: 10px; text-align: justify; width: 630px;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwgy64QCp8IYtldPhWE2BEuB3zyqkiu7uxJ95gIsch1-aSmfwIyuOjqnay_ZCL2Blrt0DBkzc1lrDy9ZVXM9IiyVvVPy38NUNNewwr0k17kzJtxI9C6HaDN8W0KyJlSPzCwolfVjj52BQ/s1600/ball-142738_640.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="141" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwgy64QCp8IYtldPhWE2BEuB3zyqkiu7uxJ95gIsch1-aSmfwIyuOjqnay_ZCL2Blrt0DBkzc1lrDy9ZVXM9IiyVvVPy38NUNNewwr0k17kzJtxI9C6HaDN8W0KyJlSPzCwolfVjj52BQ/s1600/ball-142738_640.jpg" width="200" /></a><br />
<h3>
Other websites you may like to visit
</h3>
<br />
<ul>
<li><a href="http://www.science4all.org/" target="_blank" title="Quality popular science.">Science4All.org</a></li>
<li><a href="http://cardcolm-maa.blogspot.com.es/" target="_blank">Card Colm</a></li>
<li><a href="http://www.theguardian.com/science/mathematics" target="_blank">The Guardian / </a><a href="http://www.theguardian.com/science/mathematics" target="_blank">Mathematics</a></li>
<li><a href="http://www.math-math.com/" target="_blank" title="Mathematics explained">Math-math</a></li>
<li><a href="https://www.facebook.com/pages/Math-Explosioncom/161708027242841" target="_blank" title="Mathematics connection with the world around us">Math-explosion.com</a></li>
<li><a href="http://hoodamath.com/" target="_blank" title="Math games, apps, movies and tutorials">Hooda math</a></li>
<li><a href="http://www.cut-the-knot.org/" target="_blank" title="Cut the Knot. Learn to enjoy!">Cut the Knot</a></li>
<li><a href="http://aperiodical.com/" target="_blank">The aperiodical</a></li>
<li><a href="http://www.puzzletweeter.com/" target="_blank" title="Puzzle Tweeter">Puzzle Tweeter</a></li>
<li><a href="http://nrich.maths.org/frontpage" target="_blank">Enriching Mathematics</a></li>
<li><a href="http://www.wildaboutmath.com/" target="_blank" title="Making Math fun and accessible">Wild about Math!</a></li>
<li><a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank" title="Carnaval de Matemáticas">Carnival of Mathematics</a></li>
<li><a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank" title="Páginas web interesantes para visitar, relacionadas con las Matemáticas">The world and the Mathematics</a></li>
</ul>
</div>
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<div align="center">
<div style="background-color: #eaffdd; border: 1px double #000000; color: black; padding: 10px; text-align: justify; width: 630px;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0yY7ujW_1YspNUuNfftc-DbolwkhsWTUbNSITlNJ5JVxSgpajdM8A4b88tjXwPHDgZvvJT9vTRy1h1v_3V_U8Xq3QasOUtmYW3GkqvwxvxzE9l8dl3gFsgXaFiqfrHgLGYjLYd-4j0Cc/s1600/beach-hut-237489_640.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="133" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0yY7ujW_1YspNUuNfftc-DbolwkhsWTUbNSITlNJ5JVxSgpajdM8A4b88tjXwPHDgZvvJT9vTRy1h1v_3V_U8Xq3QasOUtmYW3GkqvwxvxzE9l8dl3gFsgXaFiqfrHgLGYjLYd-4j0Cc/s1600/beach-hut-237489_640.jpg" width="200" /></a><br />
<h3>
Websites you can see when you're tired about Mathematics
</h3>
<br />
<ul>
<li><a href="http://pinterest.com/matifutbol/the-world-is-not-just-math-el-mundo-no-es-tan-solo/" target="_blank" title="El mundo no es solo Matemáticas">The world is not just Math</a></li>
<li><a href="http://photoxperiences.blogspot.com.es/" target="_blank" title="Una galería de reportajes fotográficos, para amantes de la fotografía y de los viajes.">Photoxperiences</a></li>
<li><a href="http://manualnavidades.blogspot.com.es/" target="_blank" title="Manualidades sencillas basadas en la papiroflexia, para decorar nuestra casa en Navidad.">ManualNavidades</a> (Spanish)</li>
<li><a href="http://4vium.blogspot.com.es/" target="_blank">4vium</a> (Spanish)</li>
<li><a href="http://lavueltaalondresensietedias.blogspot.com.es/" target="_blank">La vuelta a Londres en siete días</a> (Spanish)</li>
</ul>
</div>
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
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<div align="center">
<div style="background-color: #eaffdd; border: 1px double #000000; color: black; padding: 10px; text-align: justify; width: 630px;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmCxGgExEz9g4BBGadlHHVfWeslRfg5id-RI7cj6-YuPcCcKg11t6Aqs1_ucAEe-hM6S0MqN-qdF4BJiZtj9PC_jh0_1YAGBsp1wzw5lOakY5bXONgdoaO-tX_9UWH1xzeA5GLkf7hVcQ/s1600/cosmic_gc3_new_med.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmCxGgExEz9g4BBGadlHHVfWeslRfg5id-RI7cj6-YuPcCcKg11t6Aqs1_ucAEe-hM6S0MqN-qdF4BJiZtj9PC_jh0_1YAGBsp1wzw5lOakY5bXONgdoaO-tX_9UWH1xzeA5GLkf7hVcQ/s1600/cosmic_gc3_new_med.gif" width="200" /></a><br />
<h3>
Matifutbol worlwide
</h3>
<br />
<ul>
<li><a href="http://pinterest.com/matifutbol/matifutbol-worldwide-matifutbol-en-el-mundo-matifu/" target="_blank" title="Matifutbol worldwide">Watch now who's following our website</a></li>
<li><a href="https://www.facebook.com/pages/Matifutbolcom/417774584902761" target="_blank" title="Matifutbol on Facebook">Visit our profile on Facebook</a></li>
<li><a href="http://twitter.com/matifutbolcom" target="_blank" title="Matifutbol on Twitter">Keep in contact with us on Twitter</a></li>
<li><a href="http://pinterest.com/matifutbol/new-issues-nuevas-publicaciones-nous-problemes/" target="_blank" title="Matifutbol on Pinterest">Find Matifutbol on Pinterest</a></li>
<li><a href="https://plus.google.com/115955093014908454406/posts" target="_blank" title="Matifutbol en Google+">Add Matifutbol to your circles</a></li>
<li><a href="http://matematicasyfutbol.blogspot.com.es/" target="_blank">Matifutbol in Spanish</a></li>
<li><a href="http://matesifutbol.blogspot.com.es/" target="_blank">Matifutbol en catalán</a></li>
</ul>
</div>
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<div align="center">
<div style="background-color: #eaffdd; border: 1px double #000000; color: black; padding: 10px; text-align: justify; width: 630px;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-9NhGciTdecVJ-Yut0eo7nMxYx7dMbqL1dmgsv8Mh9kym1cD4kmM-PV_L2mXE6w5WwOGZeUODH7zjk9p324GhAXBfrsY9qLmaHi7re9WQWy8DmEJLyGdXb0bIUAZruE0U0GMgdiQq3aA/s1600/Vicente_del_Bosque_Euro_2012_final.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-9NhGciTdecVJ-Yut0eo7nMxYx7dMbqL1dmgsv8Mh9kym1cD4kmM-PV_L2mXE6w5WwOGZeUODH7zjk9p324GhAXBfrsY9qLmaHi7re9WQWy8DmEJLyGdXb0bIUAZruE0U0GMgdiQq3aA/s1600/Vicente_del_Bosque_Euro_2012_final.jpg" width="132" /></a><br />
<h3>
Coach
</h3>
<br />
<ul>
<li><a href="http://www.linkedin.com/profile/view?id=149486372&trk=hb_tab_pro_top&locale=en_US" target="_blank" title="Coach">My profile in Linkedin</a></li>
<li><a href="http://pinterest.com/profeshermyguad/" target="_blank" title="Coach">My dashboard in Pinterest</a></li>
<li><a href="https://plus.google.com/u/0/115979955961342404369/posts?tab=XX" target="_blank" title="Coach">My profile in Google+</a></li>
<li><a href="http://herminiolopez.blogspot.com.es/p/my-profile.html" target="_blank" title="Coach">My profile in blogspot</a></li>
<li><a href="http://about.me/herminiolopez" target="_blank">Mi perfil in about.me</a></li>
<li><a href="http://herminiolopez.blogspot.com.es/" target="_blank">All my editions</a></li>
</ul>
<br />
<br /></div>
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-91621605031209365202015-12-28T22:02:00.001+00:002015-12-28T22:15:25.689+00:002015 summary<!-- Entry text -->
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<div style="display: none; text-align: justify;">
<b>A summary of the posts issued in 2015. I hope you enjoyed them all. </b></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="text-align: justify;">
First of all, I want to <b>thank you</b> for being there one more year.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
In this 12 months we have practised some new mathematical issues. I hope you have enjoyed them.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Here you have a list of the posts of this year 2015:</div>
<br />
<ul>
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<a href="http://mathsball.blogspot.com.es/2015/01/flavius-josephus-problem.html" style="font-weight: bold; text-align: justify;" target="_blank"><span style="color: #cc0000; text-align: justify;">Flavius Josephus and the puisi-nanoq game</span></a>.<br />
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We started the year in Greenland, among seals and bears, with a puzzle about modular arithmetic in which we discovered the Flavius Josephus problem.
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<a href="http://mathsball.blogspot.com.es/2015/04/power-factor-number.html" style="font-weight: bold; text-align: justify;" target="_blank"><span style="color: #cc0000; text-align: justify;">A galactic prize</span></a>.<br />
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Later we traveled to the far west, to try to build enormous numbers with a few figures. This entry was rewarded with the first prize of the mathematical Spanish contest "Edición 6.2 Número Pi" del Carnaval de Matemáticas.
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<span style="text-align: justify;"><a href="http://mathsball.blogspot.com.es/2015/06/the-awkward-question.html" style="font-weight: bold; text-align: justify;" target="_blank"><span style="color: #cc0000;">The awkward question</span></a></span>. <br />
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We went to Guanzhong, in China, where we learnt some mathematical methods to ask questions about awkward issues.
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<a href="http://mathsball.blogspot.com.es/2015/09/sabotage-shop.html" style="font-weight: bold; text-align: justify;" target="_blank"><span style="color: #cc0000;">Sabotage in the stores</span></a></span>. <br />
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We returned to Madrid, to face a problem of simplification.
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<a href="http://mathsball.blogspot.com/2015/12/spiral-arquimedes.html" style="font-weight: bold; text-align: justify;" target="_blank"><span style="color: #cc0000;">Spirals and roulettes</span></a></span>. <br />
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And to close the season, we take a quick look at the spirals and helices at Monaco.
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<a href="https://www.blogger.com/blogger.g?blogID=8244131673148079224" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=8244131673148079224" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=8244131673148079224" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a>I hope that next year you'll also visit my blog. And that you will enjoy my new issues.
Thank you very much. I wish you a happy 2016!</div>
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<br />WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-5009478394237654532015-12-20T15:10:00.001+00:002015-12-20T15:47:47.407+00:00Spirals and roulettes<!-- Entry image of the story -->
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7rqCxqhUfmK32MRevtagvj8QP2ASLRM7wL90LWl40c6jz6O8aYF9KXN1faXwLRQ_aVv0F8V6LS_b4cU7wlCmN9Mas6_9W0E-RZnQmUxGKcWc8-Nj1ommqaxgrWHmEHcg4lERtTyapz2k/s1600/Portada+eng+del+post.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Joe Vitruvius is going to cellebrate a relaxing and cheap New Year's Eve in Monaco." border="0" height="426" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7rqCxqhUfmK32MRevtagvj8QP2ASLRM7wL90LWl40c6jz6O8aYF9KXN1faXwLRQ_aVv0F8V6LS_b4cU7wlCmN9Mas6_9W0E-RZnQmUxGKcWc8-Nj1ommqaxgrWHmEHcg4lERtTyapz2k/s640/Portada+eng+del+post.jpg" title="Joe Vitruvius is going to cellebrate a relaxing and cheap New Year's Eve in Monaco." width="640" /></a></div>
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<b>The end of the year wouldn't be the same without the confettis, such spirals that become helices when we launch them to the air. There are thousands of examples of mathematical curves all around us that we hardly notice: Archimedes spirals, logarithmic spirals, spirals of Fermat... if we talk about two dimensions. And cylindrical, conical, spherical helices... if we think in three dimensions. Before midnight, several of these curves will get in the path of Joe Vitruvius. </b></div>
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(This post participates on the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">130th edition</span></a> of the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="http://bit-player.org/" target="_blank"><span class="negrita unidas">Bit-player</span></a>.)</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlMjHswTqpP5Jr_JfL8ngduR_beDHMi8HzXr-2UT7JbB5bUCpd7ZxrvRRv1fm0rjUrCcYgnGT7E8esJKJygVySClOHeMwZ6B4ydox2rOZCls4op670qybkj6pOs2s7vpIYKeqm4sOdXio/s1600/Monaco+vs.+Mobsters.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="If balls are not found, they should play a bridge game, for instance." border="0" height="190" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlMjHswTqpP5Jr_JfL8ngduR_beDHMi8HzXr-2UT7JbB5bUCpd7ZxrvRRv1fm0rjUrCcYgnGT7E8esJKJygVySClOHeMwZ6B4ydox2rOZCls4op670qybkj6pOs2s7vpIYKeqm4sOdXio/s320/Monaco+vs.+Mobsters.jpg" title="If balls are not found, they should play a bridge game, for instance." width="320" /></a>
The teams of AS Monaco and Las Vegas Mobsters are going to play a charity match on New Year's Eve, for helping the victims of <b>Monte Carlo method</b>.</div>
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Players jump into the pitch to start the training session before the match, when the delegate of the club realizes that there are no <b>balls</b>.
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The <b>club's kit man</b> has gone off on <b>holidays</b>, so the delegate is about to pick up the keys of the material store from his office, but instead of keys, he finds a Christmas album hanging on the hanger. He tries to telephone him, but there is no way to contact the kit man.</div>
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He's a person who likes <b>mind traps</b>, so the disk may represent not only his New Year's <b>greeting</b> to his peers, but also some kind of clue to find the keys.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIz0NtIUxYqskY7ODHO_SxcxrUMW5Diq2mvACit_AKzuU5BDUbCZM7seAbRqOW1mHTcLY7XYef-At5vFAePyF421KrbxHcev7I32Rw2fK5TsonWU-Ec0kJb4skuj0_13Gr7f4YnF6QEQo/s1600/Disco+colgado.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="It seems to me that this kit man is very strange. Couldn't he have written a Christmas card, like everybody?" border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIz0NtIUxYqskY7ODHO_SxcxrUMW5Diq2mvACit_AKzuU5BDUbCZM7seAbRqOW1mHTcLY7XYef-At5vFAePyF421KrbxHcev7I32Rw2fK5TsonWU-Ec0kJb4skuj0_13Gr7f4YnF6QEQo/s320/Disco+colgado.jpg" title="It seems to me that this kit man is very strange. Couldn't he have written a Christmas card, like everybody?" width="320" /></a>
Joe Vitruvius has come to the stadium to watch the game, so the chairman invites him to go down with him to the locker room to see if they can solve the problem.</div>
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- Wow, an album of U2! New Year's Day! What a beautiful song to celebrate the New Year!</div>
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- Yes, but it would be even more beautiful if it was a <b>key</b>. We can't open the door of the room where we store the balls, with this disk.</div>
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- And what do you want me to do? It would be better to call a locksmith.</div>
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- Yes, but today is <b>New Year's Eve</b>, and there is no open shop. We will not find any locksmith who wants to come to unlock the door before the time of the start of the match is over.</div>
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- I know nothing about how to open locked doors...<br />
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- But you know a lot about mind puzzles. And we are convinced that the disk is one of them. In addition, the kit man loves <b>Maths</b>, so we're sure that the album must represent a mathematical clue.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: "verdana" , sans-serif;"><img alt="I calculate, therefore I am" border="0" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" title="I calculate, therefore I am" width="640" /></span></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmCj70fq21dhbhU_dvDXlGn-FwjisObWsDBJ-3fwiGDqX1MsAxYhpkO8aqFIm3DGYhQ4twTSUd3n5F_WcqLtaCOpTpFB9JrNvcMr_evsEtkUOVLCor3j-RxzjgxUNwCaPNznUHWbkNhOQ/s1600/Tocadiscos+2.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="It was very lucky that they still keep an old gadget." border="0" height="112" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmCj70fq21dhbhU_dvDXlGn-FwjisObWsDBJ-3fwiGDqX1MsAxYhpkO8aqFIm3DGYhQ4twTSUd3n5F_WcqLtaCOpTpFB9JrNvcMr_evsEtkUOVLCor3j-RxzjgxUNwCaPNznUHWbkNhOQ/s200/Tocadiscos+2.JPG" title="It was very lucky that they still keep an old gadget." width="200" /></a>
- Have you tried to play the disk just to see if it includes any <b>recorded message</b> indicating the place where he has left keys?</div>
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- Luckily we still have an old record player in the office. We played the album, and the only we heard is the song. It's very strange that he used a <v>vinyl record</v>, instead of something more modern such as a CD or a digital format.</div>
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- Maybe this is the clue.</div>
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- Oh, yes?</div>
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- Yes, it's possible. Do you know how a vinyl record is recorded?</div>
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- Not entirely.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqCDBEbNAtvIq00W59mqyMXmzSX5lnv8uJlfn7Y0o2KSZIYkrmDNtK4qRyd_SoQMKsmuo7ucgdmTioq_zDcTTDg93Sb6EA4lYigVbOW10l07MuyhmJ7TiYqd0QjiqDLpaGPjbvX9TsMi4/s1600/Vinilo+2.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="I was never able to remove the lints clinged to the disk because of the static electricity." border="0" height="164" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqCDBEbNAtvIq00W59mqyMXmzSX5lnv8uJlfn7Y0o2KSZIYkrmDNtK4qRyd_SoQMKsmuo7ucgdmTioq_zDcTTDg93Sb6EA4lYigVbOW10l07MuyhmJ7TiYqd0QjiqDLpaGPjbvX9TsMi4/s320/Vinilo+2.JPG" title="I was never able to remove the lints clinged to the disk because of the static electricity." width="320" /></a>
- Unlike compact disks, the information in vinyl disks is recorded in <b>a single track</b>. It's a spiral track, that a needle reads moving from the edge of disk to its center.</div>
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- So it's only one single groove, right?</div>
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- In fact, this type of curve has a special name in Mathematics. It's called <b>Archimedean spiral</b>, as this ancient wise Greek man described it in his treatises.
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- Were there music albums in those days?</div>
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- No. Archimedes used this spiral for his studies of <b>squaring the circle<b>.</b></b></div>
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- And has it got any other uses?</div>
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- Of course. It has multiple applications. It's used in the mechanisms of <b>air conditioning</b>, in certain tests for the diagnosis of <b>neurological diseases</b>, to calculate the <b>concentrations of bacteria</b>, or for really important issues such as counting the meters of <b>toilet paper</b> on a roll, for example.</div>
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- I see. I imagine that the formulas used for its description will be very complex, right?</div>
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- It depends. If we define it with <b>polar coordinates</b> (r,<i style="font-size: 110%;">θ</i>), the expression we have is very simple:
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<span style="font-size: 130%;">r = a + b·<i>θ</i></span></div>
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- It's true, it seems simple, but I don't understand anything.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidyRv8Ap6Doha8hFXdh396mDE7rZqPN8favCr1bFI4XnTiUyto91uE2mGmc3Iw3cwXJ_C-inecDuQlNITG_tr019yRKY9KSKwIDyqI8eq-DnpYhFsHJ5Ju4vI9aMBXSpJCWTb4d-CVpjk/s1600/espirala1_normal.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="A spiral thanks to the El Zombie de Shcröndiger." border="0" height="176" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidyRv8Ap6Doha8hFXdh396mDE7rZqPN8favCr1bFI4XnTiUyto91uE2mGmc3Iw3cwXJ_C-inecDuQlNITG_tr019yRKY9KSKwIDyqI8eq-DnpYhFsHJ5Ju4vI9aMBXSpJCWTb4d-CVpjk/s200/espirala1_normal.gif" title="A spiral thanks to the El Zombie de Shcröndiger." width="200" /></a>
</div>
<div style="text-align: justify;">
- The formula says that the distance (r) of each point of the curve from the origin, depends on the turns (<i>θ</i>) and on a coefficient (b). The larger this coefficient is, the greater the distance between the turns of the spiral.
</div>
<br />
<div style="text-align: justify;">
- Well, it was not that complicated. But I think we are giving around the bush like in an Archimedean spiral, without actually any conclusion.</div>
<br />
<div style="text-align: justify;">
- Well, I think we should search for somewhere in the stadium where we can find <b>another spiral of Archimedes</b>.</div>
<br />
<div style="text-align: justify;">
- Hmmm. I don't know.</div>
<br />
<div style="text-align: justify;">
- I think that, at the entrance, I saw a magnificent <b>Christmas tree</b>. I've noticed the lights arrangement, and I saw that they formed a perfect spiral of Archimedes.<br />
<br /></div>
<div style="text-align: justify;">
- Oh, yes?<br />
<br />
- Well, not exactly. It would be rather a <b>conical helix</b>, as it is a continuous curve which develops in three dimensions. But if we look at the tree <b>from above</b>, and project the helix in only two dimensions, we would see that the lights are arranged according to an Archimedean spiral.
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6uT6X1vcVcZPN8fIMMRxViXNx33VuUkxUMT3Ebfxx7w1hPLta8NX0n9-w5-P2Uxnnd2LO6PPodCZ7Fk3IziikvFGnDDFu_nzD7piSyUgVkxnZGKkmO4aJYgWWld1A5SDKVDFispVfauA/s1600/%25C3%2581rbol+fuera+de+estadio.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="An amazing zenital picture of the stadium. But I can't distinguish anything." border="0" height="147" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6uT6X1vcVcZPN8fIMMRxViXNx33VuUkxUMT3Ebfxx7w1hPLta8NX0n9-w5-P2Uxnnd2LO6PPodCZ7Fk3IziikvFGnDDFu_nzD7piSyUgVkxnZGKkmO4aJYgWWld1A5SDKVDFispVfauA/s200/%25C3%2581rbol+fuera+de+estadio.jpg" title="An amazing zenital picture of the stadium. But I can't distinguish anything." width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3YO6gNbMxvObuZa33NAoIxeN5ODOnRXM8QSr7cqQLBD0NjW8NRODxEYqy8pks3VIJyjQaC5QQKVJNUToXkqhrPOAb49pMKFCr10O9ZFb_GvX0xkLdaH4Zo6H3OQMGbCrxwW9pmdoYN3Q/s1600/Vista+cenital+de+%25C3%25A1rbol+de+Navidad+en+calle.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Now we can see perfectly the amazing Arquimedean spiral formed by the lights, and the numbered balls." border="0" height="254" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3YO6gNbMxvObuZa33NAoIxeN5ODOnRXM8QSr7cqQLBD0NjW8NRODxEYqy8pks3VIJyjQaC5QQKVJNUToXkqhrPOAb49pMKFCr10O9ZFb_GvX0xkLdaH4Zo6H3OQMGbCrxwW9pmdoYN3Q/s320/Vista+cenital+de+%25C3%25A1rbol+de+Navidad+en+calle.jpg" title="Now we can see perfectly the amazing Arquimedean spiral formed by the lights, and the numbered balls." width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7LzBHU6GNquV4eQ8mktLceE33dCdBNog9ngdEGX-vHECgiE7xDiss06f_QQy1JBCmg7jGUsU0hc1aj419HLVfWTFtNRuC_GBOHlSXofCwuVzJzGXORFyKk5i3jJnPSt5fNSXmfqx0a6c/s1600/Vista+cenital+de+%25C3%25A1rbol+de+Navidad+en+calle.jpg" imageanchor="1" style="margin-bottom: 1em; margin-right: 1em;"></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_JUvDaI_nx7kysZIbAzxk8-hDTaiie7QJ1nCYsGryMtnjTRZk2mCcxdw2Gt0QSjSrgMv-pqV3vzZP14TjQyZd69sYMon01v6950Sjlt4LZoZD3I6YmctHXWzKE6fCOl0rlm3EM3ibY6E/s1600/%25C3%2581rbol+fuera+de+estadio+3.jpg" imageanchor="1" style="margin-bottom: 1em; margin-left: 1em;"></a></div>
<span style="text-align: justify;">
- Who put up this tree?</span><br />
<br />
<div style="text-align: justify;">
- The kit man.</div>
<br />
<div style="text-align: justify;">
- Well, we're approaching the solution.</div>
<br />
<div style="text-align: justify;">
- But where can he have hidden the keys? Hanging from a <b>branch</b>?
</div>
<br />
<div style="text-align: justify;">
- I don`t think so. They may be within any of the balls that adorn the tree. If you look, at them, all <b>have a number</b>.</div>
<br />
<div style="text-align: justify;">
- Yes, but which ball will we choose? We have no time to break all the balls to see if we find the keys within any of them.
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz6bHLB2NMMIOZ-LHGWBVyjT1RYdyQlNt6jqI79_xaozYMp5oqNzUdH7fyGrCodY92iW-Fj_WI3K-xNV9INoY9MXr2TDS1EqBemUm95ohaoA2J5mgPionsRm1Z_zXF93Vu-skbG-nuwyg/s1600/Ruleta+2.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Place your bets, ladies and gentlemen! No more bets!" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz6bHLB2NMMIOZ-LHGWBVyjT1RYdyQlNt6jqI79_xaozYMp5oqNzUdH7fyGrCodY92iW-Fj_WI3K-xNV9INoY9MXr2TDS1EqBemUm95ohaoA2J5mgPionsRm1Z_zXF93Vu-skbG-nuwyg/s1600/Ruleta+2.JPG" title="Place your bets, ladies and gentlemen! No more bets!" /></a>
- He hasn't left us more clues, except the disk with the Archimedean spiral. So we should focus in it. This spiral is also known by another name: <b>Archimedean roulette</b>...</div>
<br />
<div style="text-align: justify;">
- Ah! Now that you mention it, I remember that before he worked for our club, he was employed as a <b>croupier</b> at the <b>Grand Casino</b>. Perhaps there is some sort of relationship with the roulette you say.</div>
<br />
<div style="text-align: justify;">
- Certainly. The roulette is a game invented by the Mathematician <b>Blaise Pascal</b>, who established its rules on the 17th century. This rules haven't experienced any outstanding change since then. The most important new was introduced to include, among the 36 existing numbers, the red and black ones, an extra green number, zero, exclusive for the casino, which allows it to get some profits.</div>
<br />
<div style="text-align: justify;">
- Given that he worked as a croupier at the casino, perhaps we should then open the ball <b>number 37</b>.<br />
<br />
- What an ugly number!<br />
<br />
- There are no ugly numbers. For example, 37 is a fascinating number, because it's a prime number that is factor of all three-digit repdigit numbers.
<br />
<br />
- <b>Repdigit</b> numbers?<br />
<br />
- Yes, numbers that are composed by repetition of the same figure: 111, 222, 333...</div>
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<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheaqQjnhVgKU-vAOXNQ22OxT0G_4goHJY-4f-FdTd1QLO1ICbb9HslB4mgPFUULVA6sm1kF0AN1HZlqcSnhXsBpLh6jWLxScyr4UHnAsw3u6BZQNUNLZbIDyzlleyT2JOJcwTMJD_86Zc/s1600/Bola+37.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Number 37 was deserving a good tribute." border="0" height="159" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheaqQjnhVgKU-vAOXNQ22OxT0G_4goHJY-4f-FdTd1QLO1ICbb9HslB4mgPFUULVA6sm1kF0AN1HZlqcSnhXsBpLh6jWLxScyr4UHnAsw3u6BZQNUNLZbIDyzlleyT2JOJcwTMJD_86Zc/s200/Bola+37.jpg" title="Number 37 was deserving a good tribute." width="200" /></a>
- Ah! It's okay. Let's try with ball number 37.</div>
<br />
<div style="text-align: justify;">
- Correct: here are the keys! Go quickly to open the door and get the balls so that the match can be played.</div>
<br />
<div style="text-align: justify;">
- And we're going to the VIP box to open a few bottles to celebrate the resolution of the problem, and to wish all our followers a <b>Happy New Year</b>!
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If you want to know more about the content of this post, you can also look at these amazing articles: <a href="http://www.onlineroulette.com.au/history" target="_blank">From physics to online fun: The history of Roulette</a>, <a href="http://www.mathematische-basteleien.de/spiral.htm" target="_blank">Spirals</a>, <a href="http://xahlee.info/SpecialPlaneCurves_dir/ArchimedeanSpiral_dir/archimedeanSpiral.html" target="_blank">Arquimedean spiral</a>, <a href="http://www.disfrutalasmatematicas.com/graficos/coordenadas-polares-cartesianas.html" target="_blank">Polar and Cartesian Coordinates</a>.<br />
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<span style="text-align: justify;">And don't forget to take a walk by the </span><a href="http://aperiodical.com/carnival-of-mathematics/" style="text-align: justify;" target="_blank"><span class="negrita">130th Carnival of Mathematics</span></a><span style="text-align: justify;">. There you'll find lots of excellent math posts that you'll surely like too.</span></div>
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</div>WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-89483226249213578812015-09-28T16:33:00.001+01:002015-10-22T22:19:08.849+01:00Sabotage in the stores<!-- Entry image of the story -->
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFj_jSTYXUbzIpHEDXIz1GbU6JH8tG2p__gYLIC36y0SXJfzmmiu6yfiWrIEdOPWwX80Sf4SHvEiGGNUtmi_y6Nzyfh50XemYf8WL-bp10J-Gnj8Yz6v7I3R4exFAVDVulZdKCFuauikc/s1600/Sabotage+in+the+stores.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Joe Vitruvius and Madrid nightlife" border="0" height="380" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFj_jSTYXUbzIpHEDXIz1GbU6JH8tG2p__gYLIC36y0SXJfzmmiu6yfiWrIEdOPWwX80Sf4SHvEiGGNUtmi_y6Nzyfh50XemYf8WL-bp10J-Gnj8Yz6v7I3R4exFAVDVulZdKCFuauikc/s640/Sabotage+in+the+stores.jpg" title="Joe Vitruvius and Madrid nightlife" width="640" /></a></div>
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<b>The method of simplifying consists on transforming one expression into another one simpler, more useful to work, or easier to memorize. We simplify fractions, polynomials, powers and radicals... It's a great Mathematical tool, but it can become a dangerous method if you don't use it properly.</b></div>
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(This post participates on the <a href="http://blog.ifem.co.uk/carnival-of-mathematics-127/" target="_blank"><span class="negrita">127th edition</span></a> of the <a href="http://blog.ifem.co.uk/carnival-of-mathematics-127/" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="http://blog.ifem.co.uk/" target="_blank"><span class="negrita unidas">Mathematics and Coding</span></a>.)
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<span style="font-family: Verdana, sans-serif;">FIRST HALF</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyVkI3KQxoW_d5yzd2NIeXmEs4Qp1pXyZxG50PKi_MHLKqNfedzumzOwh4gDUmxm_wCyxAiOtwJkofzShwMRpC4mYh-h3I8UhjYGeUhX_L1P6bw0m5PGKOfs1EMN3308SM8iur9Dl91Es/s1600/Iker+Casillas%252C+del+Real+Madrid+al+Oporto.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Iker Casillas signs for Porto" border="0" height="193" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyVkI3KQxoW_d5yzd2NIeXmEs4Qp1pXyZxG50PKi_MHLKqNfedzumzOwh4gDUmxm_wCyxAiOtwJkofzShwMRpC4mYh-h3I8UhjYGeUhX_L1P6bw0m5PGKOfs1EMN3308SM8iur9Dl91Es/s200/Iker+Casillas%252C+del+Real+Madrid+al+Oporto.JPG" title="Iker Casillas signs for Porto" width="200" /></a>Real Madrid president feels anxious. He’s not able to <b>balance the books</b> of the sales in the stores.<br />
<br />
After the signing of Iker Casillas for Porto, Florentino wants to <b>clear the stock</b> of the goalkeeper t-shirts as soon as possible.<br />
<br />
To achieve this, he has instructed the two official stores. In the store closer to the Santiago Bernabeu stadium, they will sell <b>2 t-shirts for <span style="white-space: nowrap;">20 €</span></b> in total, while in the other store, the deal will be <b>3 t-shirts for <span style="white-space: nowrap;">20 €</span></b>.<br />
<br />
In the <b>first week</b>, they have sold a total of 6,000 t-shirts, 3,000 in each store. In total, they have made <span style="white-space: nowrap;"><b>50,000 €</b></span>.<br />
<br />
But there are still many shirts to sell, so in the <b>second week</b> they have decided to change the offer.<br />
<br />
If the previous week they sold 2 units for <span style="white-space: nowrap;">20 euros</span> and another 3 units for <span style="white-space: nowrap;">20 euros</span>, now both stores will sell batches of <b>5 t-shirts for <span style="white-space: nowrap;">40 euros</span></b>, and will get the same income for every 5 t-shirts.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRv5z6cpPKmz2-mKMiCZYpZnQOFY4Ro4TKhY8QZJZX5rmmOHsjfYo83fkOXZboOGT6FXN2YQQ0sdBU1_07Q-HvOuoq_oGCsLaDqk8ykq6Fbgx_1lLhvu2ncfD-1rAcfnRUriudkbkomsQ/s1600/Liquidaci%25C3%25B3n+total.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Iker t-shirts on sale" border="0" height="170" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRv5z6cpPKmz2-mKMiCZYpZnQOFY4Ro4TKhY8QZJZX5rmmOHsjfYo83fkOXZboOGT6FXN2YQQ0sdBU1_07Q-HvOuoq_oGCsLaDqk8ykq6Fbgx_1lLhvu2ncfD-1rAcfnRUriudkbkomsQ/s400/Liquidaci%25C3%25B3n+total.jpg" title="Iker t-shirts on sale" width="400" /></a></div>
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During the second week, they have managed to sell the same number of shirts (6,000), also 3,000 in both stores, but the takings have dropped to <span style="white-space: nowrap;"><b>48,000 euros</b></span>.<br />
<br />
The club doesn’t understand why they have got less money. They think that maybe an employee of one of the stores has stolen some notes.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVG-tP8ANWWYsH1U_ZOei4yWujVcghZA5CuJnFdv9QRgXn8UNx4HyQtBCrbTnsBP3mjOqmHmCC7wdJStOCXWwd8huUbEABUkbmaaa8rzTywPG0ifGrkfdfuob8BxPjCP0JbYA2B-LpGu8/s1600/caja+registradora.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Cash register" border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVG-tP8ANWWYsH1U_ZOei4yWujVcghZA5CuJnFdv9QRgXn8UNx4HyQtBCrbTnsBP3mjOqmHmCC7wdJStOCXWwd8huUbEABUkbmaaa8rzTywPG0ifGrkfdfuob8BxPjCP0JbYA2B-LpGu8/s200/caja+registradora.gif" title="Cash register" width="199" /></a>
The fact is that there are still 6,000 Iker t-shirts to sell. This time it's been decided a different deal. They’ll sell batches of <b>3 units for <span style="white-space: nowrap;">15 euros</span></b> at the shop of the stadium, and batches of <b>2 units <span style="white-space: nowrap;">for 25 euros</span></b> at the store of Gran Vía. Thus, for every 5 shirts they will get again a total of <span style="white-space: nowrap;">40 euros</span>.<br />
<br />
Finally, they sold the remaining 6,000 shirts, 3,000 in each store again. But this time they get <span style="white-space: nowrap;"><b>52,500 euros</b></span>.<br />
<br />
Now the board of directors doesn't understand anything. It’s possible that the employee who stole <span style="white-space: nowrap;">2,000 euros</span> last week has returned them to the cash desk, or maybe the money was miscounted last week. But then, what about the extra <span style="white-space: nowrap;">500 euros</span> of the <b>third week</b>?<br />
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Undoubtedly, this is a mystery that only Joe Vitruvius can solve, so Florentino calls him to come and to solve the enigma.<br />
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<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWulUD6vT7T5bjYygbHnsbVGyvMPf5Qfq3ctJPrwnEJUQp6QPYpK6dBPG4UGC08AO5Lx6I8oC8gNbAdQrAsMqsE5yifPmwPVfgZxeahdANbXnI_snjk2DBnEQToD6faePOxtxtJuGTOx8/s1600/Florentino+y+Pepe+Vitruvio.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="Joe Vitruvius with Florentino" border="0" height="203" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWulUD6vT7T5bjYygbHnsbVGyvMPf5Qfq3ctJPrwnEJUQp6QPYpK6dBPG4UGC08AO5Lx6I8oC8gNbAdQrAsMqsE5yifPmwPVfgZxeahdANbXnI_snjk2DBnEQToD6faePOxtxtJuGTOx8/s320/Florentino+y+Pepe+Vitruvio.jpg" title="Joe Vitruvius with Florentino" width="320" /></a>
- Hi, Florentino, how are you?<br />
<br />
- Well, we’ve got a small problem. We sold <b>18,000 t-shirts</b> over the past few weeks. We have asked <span style="white-space: nowrap;">40 €</span> for every 5 shirts, so we should have entered <span style="white-space: nowrap;"><b>144,000 euros</b></span>. And yet, we got <b>6,<span style="white-space: nowrap;">500 euros more</span></b>.<br />
<br />
- That's fine. With that extra money you can sign a new goalkeeper, right? Or you can also buy a next-generation fax...<br />
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- The thing is that during the last three weeks we have sold the <b>same number of t-shirts</b> and yet, every week we have cashed up a <b>different amount</b> of money.<br />
<br />
- I see. Were all units sold at the same price?<br />
<br />
- No. We have two stores, and each one had a different offer.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTFcp3RRxjhCdZoQWkvRAFes_UXR8MgCOszs_64Jtqfa8eDCcZGSN11VZEN5vIxj9kKRoNSjh5PvtVlnOp7pM7o3tDUtYOPDWg8xp3KWrydfvpimTg6JRqnQdH_Q2o4ypuBFDyGv1k5d0/s1600/Las+dos+tiendas+oficiales.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Shops at Gran Vía and near Santiago Bernabéu stadium" border="0" height="162" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTFcp3RRxjhCdZoQWkvRAFes_UXR8MgCOszs_64Jtqfa8eDCcZGSN11VZEN5vIxj9kKRoNSjh5PvtVlnOp7pM7o3tDUtYOPDWg8xp3KWrydfvpimTg6JRqnQdH_Q2o4ypuBFDyGv1k5d0/s400/Las+dos+tiendas+oficiales.gif" title="Shops at Gran Vía and near Santiago Bernabéu stadium" width="400" /></a></div>
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- Well, it may explain everything.<br />
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- No, because for every 5 t-shirts, the money extra we get in one store equals the money we lose in the other.<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnGX-SbpcvtEWTLpmwUJE8RUuloOgTXrpAYe8x0VFbHv2mzYGEv5lEQPe1GI7LaxtcdsGDjM3iak3Va9L2U9RUQMiqsgUpg96Ch5SLJYd8h4Qm1xXcD9WoiP6xDPEX29TjD_cd9TQyrzE/s1600/Lotes+de+5+camisetas.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Batches of 5 Iker t-shirts" border="0" height="258" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnGX-SbpcvtEWTLpmwUJE8RUuloOgTXrpAYe8x0VFbHv2mzYGEv5lEQPe1GI7LaxtcdsGDjM3iak3Va9L2U9RUQMiqsgUpg96Ch5SLJYd8h4Qm1xXcD9WoiP6xDPEX29TjD_cd9TQyrzE/s320/Lotes+de+5+camisetas.gif" title="Batches of 5 Iker t-shirts" width="320" /></a><br />
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- Then it's clear. You've sold more t-shirts in the store where you have a better deal. That’s the difference.<br />
<br />
- Not either. It turns out that despite the offer, we sold the same number of shirts in both shops. So the differences weren't because we sold more shirts in the cheapest store.<br />
<br />
I think that there is an employee who wants to turn us crazy with this, perhaps a staunch supporter of Iker who didn’t like him to leave Real Madrid, or something like this. I would like to<b> find the saboteur</b>.<br />
<br />
- Well, I think that such a person doesn’t exist. It's just a small <b>mathematical paradox</b>, because of the <b>simplification </b>on the calculus.<br />
<br />
- Oh yeah? What is it about?<br />
<br />
- Let’s see. We’ll start with the <b>central week</b>, in which both stores sold batches of 5 t-shirts for <span style="white-space: nowrap;">40 euros</span>. If 6,000 shirts were sold in total, we can say that <b>1,200 batches of 5 units</b> were sold, or what is the same, you sold <b>t-shirts for <span style="white-space: nowrap;">8 euros</span></b>, right?<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaoTTMW0GibiJQbCVBUr5kxSs5TH3E6yfE-QloOW_P67B04Kpc61bOj3452gGydoEEcygklgl7ERYqHAi1PeAJFY7Pa2mjPA4Z0K1nhZ_WZBm3XOmaxYK3LB3Wr4OucsBiwgzybhLW6rA/s1600/Sales+distribution+of+the+second+week.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Sales distribution between the two stores" border="0" height="434" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaoTTMW0GibiJQbCVBUr5kxSs5TH3E6yfE-QloOW_P67B04Kpc61bOj3452gGydoEEcygklgl7ERYqHAi1PeAJFY7Pa2mjPA4Z0K1nhZ_WZBm3XOmaxYK3LB3Wr4OucsBiwgzybhLW6rA/s640/Sales+distribution+of+the+second+week.jpg" title="Sales distribution between the two stores" width="640" /></a></div>
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- Yes, I think it's pretty clear.<br />
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- Now let's see what happened on the <b>first week</b>.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfRvCZp_dT6J8zOKHbNfRnUpHp_2WV3W7KS3t9yFsybKzndOCe7RzBMqabetqkdfRev0iHs-4usfucExV2hNsCEduHd89G8LVF0cHC8EVYdOOJ4WhssfpUkAF7RzDpnXqDyJ0PywOAxmM/s1600/Camiseta+de+Iker+Casillas+a+8+euros.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="8 euros an Iker t-shirt" border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfRvCZp_dT6J8zOKHbNfRnUpHp_2WV3W7KS3t9yFsybKzndOCe7RzBMqabetqkdfRev0iHs-4usfucExV2hNsCEduHd89G8LVF0cHC8EVYdOOJ4WhssfpUkAF7RzDpnXqDyJ0PywOAxmM/s200/Camiseta+de+Iker+Casillas+a+8+euros.gif" title="8 euros an Iker t-shirt" width="163" /></a>
- Then we also sold batches of 5 t-shirts for <span style="white-space: nowrap;">40 euros</span>. In the store close to the stadium we sold 2 t-shirts for <span style="white-space: nowrap;">20 euros</span>, and in Gran Vía we sold the remaining 3 shirts for <span style="white-space: nowrap;">20 euros</span>.<br />
<br />
- There's the trap. In this case we can’t talk about batches of 5 shirts, since it’s not entirely true. You are wrongly simplifying the distribution of your sales.<br />
<br />
- Why?<br />
<br />
- In each store you sold 3,000 shirts, right?<br />
<br />
- Certainly.<br />
<br />
- In this case we can only speak of <b>1,000 batches of 5 t-shirts</b> (sold at an average of<b> <span style="white-space: nowrap;">8 euros</span></b> each unit). And a surplus of <b>1,000 t-shirts</b>, which were sold in the store close to the Bernabeu stadium at 2 shirts for <span style="white-space: nowrap;">20 euros</span>, ie <span style="white-space: nowrap;"><b>10 euros</b></span> per t-shirt.<br />
<br />
Therefore, part of the t-shirts were sold at a higher price. That’s why you got an additional money during the first week.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdkLZox-Y-tQH_T-UzqEU26ZJdkQeCb8YaNLCM-06WBY68WyB83v256-08GKhri1YdL6cmYON8zj4mH30mu_jnpCgLAYQ2DY3fSZ51PX2eVZZfQdasiFZiyeaTFVGhzBwuXydepihpKXc/s1600/Sales+distribution+of+the+first+week.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Sales distribution between the two stores" border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdkLZox-Y-tQH_T-UzqEU26ZJdkQeCb8YaNLCM-06WBY68WyB83v256-08GKhri1YdL6cmYON8zj4mH30mu_jnpCgLAYQ2DY3fSZ51PX2eVZZfQdasiFZiyeaTFVGhzBwuXydepihpKXc/s640/Sales+distribution+of+the+first+week.jpg" title="Sales distribution between the two stores" width="564" /></a></div>
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<br />
- And if you look at the <b>last week</b>, we can see that a total of <b>1,000 batches of 5 t-shirts</b> were sold for a total of <span style="white-space: nowrap;">40 euros (at an average of <b>8 euros</b> a t-shirt)</span>. And there were still <b>1,000 t-shirts</b> in the store of Gran Vía, which were sold at 2 units for <span style="white-space: nowrap;">25 €</span>, ie <span style="white-space: nowrap;"><b>12.5 euros</b></span> per shirt.<br />
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And that's why this week you got even more money.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPIKa4I0cdpVbSBo7Whd4HWegwXsZ8ChqdBJ1UaN3w9zwlRJKot_vimfK-ivE2vXnbnnjcgAXpZry-8ApncDxVURdglaDTWS64LJm416Ihtp5Vzw_1T_QIUpqshLs1g6TR9Py7NIAB0iU/s1600/Sales+distribution+of+the+third+week.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Sales distribution between the two stores" border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPIKa4I0cdpVbSBo7Whd4HWegwXsZ8ChqdBJ1UaN3w9zwlRJKot_vimfK-ivE2vXnbnnjcgAXpZry-8ApncDxVURdglaDTWS64LJm416Ihtp5Vzw_1T_QIUpqshLs1g6TR9Py7NIAB0iU/s640/Sales+distribution+of+the+third+week.jpg" title="Sales distribution between the two stores" width="564" /></a></div>
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<br />
- Now I understand everything. So there is no saboteur...<br />
<br />
- Nope. The error was that you thought that the t-shirts were sold in batches of 5 units all the time, when it happened only one week.<br />
<br />
- And has this ever happened elsewhere?<br />
<br />
- Yes. The writer <b>Malba Tahan</b>, in his book '<b><i>The man who counted</i></b>', writes a case just like this with some pineapple vendors. And <b>Martin Gardner</b>, on his book '<b><i>Aha! Gotcha. Paradoxes to puzzle and delight</i></b>', shows a similar case which takes places in a music store.<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh23rOTzvH416_-0Sx0zLuyXFFFAgUMKlCRcCSowuNyK7DDPjdDV-HBzWojx6U2N_mlN426sFYvTffFB-1jDKV8mHtD6wagk7zKVBO75xzjvMjfeRxsUKoF_x4foxk3TNQgIXfHGdqoHYM/s1600/Matem%25C3%25A1ticas+divertidas.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Recreational Mathematics" border="0" height="185" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh23rOTzvH416_-0Sx0zLuyXFFFAgUMKlCRcCSowuNyK7DDPjdDV-HBzWojx6U2N_mlN426sFYvTffFB-1jDKV8mHtD6wagk7zKVBO75xzjvMjfeRxsUKoF_x4foxk3TNQgIXfHGdqoHYM/s200/Matem%25C3%25A1ticas+divertidas.gif" title="Recreational Mathematics" width="200" /></a>-OK, now I stay calmer. We'll have to see what we can do with the extra money we have got.<br />
<br />
- Well, you can give the extra money to any philanthropic association. Or you can also buy some books of recreational mathematics, and give them to the children who come to the stadium, at the beginning of the next match. Surely you will increase the attendance to the stadium.<br />
<br />
- We will think about it. Bye, Joe, and thank you very much for your help.<br />
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- Till next time, Florentino.<br />
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<span style="text-align: justify;">And don't forget to take a walk by the </span><a href="http://blog.ifem.co.uk/carnival-of-mathematics-127/" style="text-align: justify;" target="_blank"><span class="negrita">127th Carnival of Mathematics</span></a><span style="text-align: justify;">. There you'll find lots of excellent math posts that you'll surely like too.</span></div>
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WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-14033049826718698112015-06-04T16:21:00.000+01:002015-06-04T16:21:45.322+01:00The awkward question<!-- Entry image of the story -->
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsmCwG9iK5qdO3txptW5rxC6tAUsYu6dOls7PM90W61W5FebSvyDOGMvlRr5FD1Kjm2Xjlc-mLC9ngPn257LbuDdPJbzfbvFSeNI5qSEYSs1fNct9heZs6QbPcEj1WRvFBkff6V4RN2VY/s1600/Joe+Vitruvius+at+Guangzhou.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Joe Vitruvius and his new history in Guangzhou: The awkward question." border="0" height="426" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsmCwG9iK5qdO3txptW5rxC6tAUsYu6dOls7PM90W61W5FebSvyDOGMvlRr5FD1Kjm2Xjlc-mLC9ngPn257LbuDdPJbzfbvFSeNI5qSEYSs1fNct9heZs6QbPcEj1WRvFBkff6V4RN2VY/s1600/Joe+Vitruvius+at+Guangzhou.jpg" title="Joe Vitruvius and his new history in Guangzhou: The awkward question." width="640" /></a></div>
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<b>Sometimes it's quite difficult to get a truthful answer from people in surveys. Fortunately, some mathematical methods have been created to overcome the reluctance of people to answer some awkward questions.</b></div>
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(This post participates on the <a href="http://mathematicalmysterytour.blogspot.co.uk/" target="_blank"><span class="negrita">123th edition</span></a> of the <a href="http://www.whitegroupmaths.com/2015/02/119th-carnival-of-mathematics.html" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="http://mathematicalmysterytour.blogspot.co.uk/" target="_blank"><span class="negrita unidas">Mathematical mystery tour</span></a>.)
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<span style="font-family: Verdana, sans-serif;">FIRST HALF</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhebdY_6P49zvlxZe7Wr-iG9F-KEr1gYjl-jyfjfLzrSqygkmQJfHmjciZAyEXYG2w8agPW0YxVVls7ss6saSacU3UQtFZZf99QpuXuVH_FPWZxEZO4l2N118mWxCsmA67Vajfs-11NpUU/s1600/Asian+Champions+League+2.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Semifinal of the Champions League AFC between Guangzhou Evergrande and Kashiwa Reysol." border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhebdY_6P49zvlxZe7Wr-iG9F-KEr1gYjl-jyfjfLzrSqygkmQJfHmjciZAyEXYG2w8agPW0YxVVls7ss6saSacU3UQtFZZf99QpuXuVH_FPWZxEZO4l2N118mWxCsmA67Vajfs-11NpUU/s400/Asian+Champions+League+2.jpg" title="Semifinal of the Champions League AFC between Guangzhou Evergrande and Kashiwa Reysol." width="400" /></a></div>
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Next week there's an interesting semifinal of the AFC 2015 Champions League between the Chinese <b>Guangzhou Evergrande</b> and the Korean <b>Kashiwa Reysol</b>.</div>
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<b>Fabio Cannavaro</b>, coach of the Chinese club, has noticed that, in the last weeks, the level of play of his team has dropped a lot.</div>
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Some people say that some players of Guangzhou <b>go out partying </b>at night, and that's why they don't perform well in training sessions and matches. But nobody has proved it's true.</div>
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The club has hired some private investigators to follow them, but for now the players have always got rid of the detectives' persecution.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFbKhyphenhyphenwvl7dzUB6kpQJBpPkP8TpjJhl0IpsoMrJ61SpNcJbGYadnQVZtq9rUEZFWQ6QB8KdIgALuBg2W4nlDJGHccfrspVFhGwCMeZlmz1k_B4POojz6IX4F2xEBzKorG6YbNjg2so8ro/s1600/Discoteca+china+3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Badge of the Chinese team Guangzhou Evergrande. A curious Spanglish word." border="0" height="170" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFbKhyphenhyphenwvl7dzUB6kpQJBpPkP8TpjJhl0IpsoMrJ61SpNcJbGYadnQVZtq9rUEZFWQ6QB8KdIgALuBg2W4nlDJGHccfrspVFhGwCMeZlmz1k_B4POojz6IX4F2xEBzKorG6YbNjg2so8ro/s400/Discoteca+china+3.JPG" title="Badge of the Chinese team Guangzhou Evergrande. A curious Spanglish word." width="400" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWfeMGDyGnjt0V7xjkNYdF-f-WNBCVeJmuzN2a3TK7Ar5XnWLIO8uJqnhSmjBNKN9F9eq6v8i2QZ0yNnCGxTw5xjUMiTu5fPlvjV3gXmPSRkcTfJr8fmg4Jyc-ny5bJHbAGN5jv2tbBLE/s1600/Escudo+del+Guangzhou+Evergrande.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWfeMGDyGnjt0V7xjkNYdF-f-WNBCVeJmuzN2a3TK7Ar5XnWLIO8uJqnhSmjBNKN9F9eq6v8i2QZ0yNnCGxTw5xjUMiTu5fPlvjV3gXmPSRkcTfJr8fmg4Jyc-ny5bJHbAGN5jv2tbBLE/s200/Escudo+del+Guangzhou+Evergrande.gif" width="145" /></a></div>
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<div style="text-align: justify;">
Fabio is worried about it, and doesn't know what to do. He has called them one by one to his office, and has directly asked each one if rumours are true. But none of them has accepted that goes out at night. And they didn't want to say how many colleagues use to do it.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5jDxuvUDup1K4OMBgLmKsO4_NLy_eI_YrKKbe8dGo68gVoltUEZ5KI3Lu056zP4gfZ-g1bdiBdSpbNfdAGNxa7zZYjhX47i17rGjIjq_cHmuittS5b2G_ZxV6S6sIBU4IJZGTmyNyk9E/s1600/Jugadores+del+Guanzhou+Evergrande+posando+para+los+fot%C3%B3grafos+2.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="Guangzhou Evergrande players posing for the press. I don't know why the two players on the left are crouching." border="0" height="126" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5jDxuvUDup1K4OMBgLmKsO4_NLy_eI_YrKKbe8dGo68gVoltUEZ5KI3Lu056zP4gfZ-g1bdiBdSpbNfdAGNxa7zZYjhX47i17rGjIjq_cHmuittS5b2G_ZxV6S6sIBU4IJZGTmyNyk9E/s200/Jugadores+del+Guanzhou+Evergrande+posando+para+los+fot%C3%B3grafos+2.JPG" title="Guangzhou Evergrande players posing for the press. I don't know why the two players on the left are crouching." width="200" /></a><br />
<div style="text-align: justify;">
It seems clear that they have lied, and that they form a very close group, and that they don't want to betray each other. But Fabio needs to know whether <b>the problem of the trips at night</b> is widespread or not among the team.</div>
<br />
<br />
<div style="text-align: justify;">
Tonight, while dinning at a restaurant in Hong Kong, he has met <b>Joe Vitruvius</b>, who has come to the city to take part in a Conference of Mathematics. Fabio has commented him on his problem, and Joe thinks that he should call his players again to his office.</div>
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiD8WC1nTIJ-i25enEMzcxV5DwUxbyzHHvvwJmsE0Lcakj9Gps7bCRCtZWjCo8Zr-LHM6NJ_KVOlL36-Lh6C-umUk5rtobJH_FcuXaR_GI-UTW8f1owuIIFjvWl46x0mPVlPUJeYBvOc9Q/s1600/Restaurante+de+Hong+Kong.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Hong Kong restaurant where Fabio Cannavaro meets Joe Vitruvius. What a coincidence!" border="0" height="348" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiD8WC1nTIJ-i25enEMzcxV5DwUxbyzHHvvwJmsE0Lcakj9Gps7bCRCtZWjCo8Zr-LHM6NJ_KVOlL36-Lh6C-umUk5rtobJH_FcuXaR_GI-UTW8f1owuIIFjvWl46x0mPVlPUJeYBvOc9Q/s640/Restaurante+de+Hong+Kong.JPG" title="Hong Kong restaurant where Fabio Cannavaro meets Joe Vitruvius. What a coincidence!" width="640" /></a><br />
<br />
<div style="text-align: justify;">
- But they will answer the same again, Joe.</div>
<br />
<div style="text-align: justify;">
- Not necessarily. You can <b>get </b>some of them to tell the <b>truth</b>.</div>
<br />
<div style="text-align: justify;">
- Well, I don't see how. I don't want to <b>threaten them</b>, nor to offer them any kind of <b>reward </b>to betray their teammates.</div>
<br />
<div style="text-align: justify;">
- Well, there's a way to get them to tell the truth, without threaten nor rewarding them...</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3i2iz3STE42ZWnWmbPzxJZqgPlM_MZtKwpL_kN3lhp9S3ZqExOS5Y_HS4yVBvnEA4SJjOZZ2BR5HkIEv2lAmJJROOamz-8bfHBv5MUa6wwiOo2YNXHQY2XP27oPjew5ya92yyItSlt2M/s1600/Fabio+Cannavaro+1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Fabio Cannavaro, coach of Guangzhou Evergrande, who is closing his eyes like if he already had Chinese eyes." border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3i2iz3STE42ZWnWmbPzxJZqgPlM_MZtKwpL_kN3lhp9S3ZqExOS5Y_HS4yVBvnEA4SJjOZZ2BR5HkIEv2lAmJJROOamz-8bfHBv5MUa6wwiOo2YNXHQY2XP27oPjew5ya92yyItSlt2M/s320/Fabio+Cannavaro+1.JPG" title="Fabio Cannavaro, coach of Guangzhou Evergrande, who is closing his eyes like if he already had Chinese eyes." width="277" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhhqgc3M_OfXIrH-Vyz1QqppFfZy_W4DTDyTopKwH7u7-2V4eLQHBgety2seyAZXpK0739Kwf304CYk2dWtPlzlM_4S92PWXrzk5Y6JAyIogJsydixuzYrEKHDHoENnwC9kC23XblPaRc/s1600/Pepe+Vitruvio+en+China+-+Joe+Vitruvius+in+China+-+Pep+Vitruvi+a+la+Xina.GIF" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Joe Vitruvius, after having lost his luggage at the airport." border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhhqgc3M_OfXIrH-Vyz1QqppFfZy_W4DTDyTopKwH7u7-2V4eLQHBgety2seyAZXpK0739Kwf304CYk2dWtPlzlM_4S92PWXrzk5Y6JAyIogJsydixuzYrEKHDHoENnwC9kC23XblPaRc/s400/Pepe+Vitruvio+en+China+-+Joe+Vitruvius+in+China+-+Pep+Vitruvi+a+la+Xina.GIF" title="Joe Vitruvius, after having lost his luggage at the airport." width="195" /></a></div>
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<div style="text-align: justify;">
Can you imagine how they will know what's happening with the players?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img alt="I calculate, therefore I am" border="0" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" title="I calculate, therefore I am" width="640" /></span></a></div>
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<div class="rojo negrita" style="float: left; margin-left: 2em;">
<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
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<!-- Main article -->
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivj-MNmAwVuuWjpHMjRYfPg-OCuIo39o8vjw8D11D_1sZVpfonyPbQ4oRmAzjLexWZOd-vNP0iLJs-Si21dmWk8IDaxmhqw-HGJBjKdd5p1Dj5xfORuWYaXnR45TrYaZK1cb7dt81OHTg/s1600/Estadio+de+Guanzhou+2.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Guangzhou Evergrande's football stadium is wonderful." border="0" height="149" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivj-MNmAwVuuWjpHMjRYfPg-OCuIo39o8vjw8D11D_1sZVpfonyPbQ4oRmAzjLexWZOd-vNP0iLJs-Si21dmWk8IDaxmhqw-HGJBjKdd5p1Dj5xfORuWYaXnR45TrYaZK1cb7dt81OHTg/s200/Estadio+de+Guanzhou+2.JPG" title="Guangzhou Evergrande's football stadium is wonderful." width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3pG2vEIOPGrTU_LVjoKhUWlNoYPw8b45PbNG3N-KTgJ0NjErzJGz7NIFnQ5TZS7-sSqkfcsAWiFHHYmrXaxGT0tczrPuuqwwtlvDhkG1HvWWdw-nyUn5nDLrDS4Jc8cdUAHx8XsLw9Fg/s1600/Skyline+de+Los+Angeles.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><br />
<div style="text-align: justify;">
- Let's see, Fabio, which were the <b>questions </b>you did?</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- First, I asked: Do you<b> go out</b> for party at night? And everyone told me 'no'.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
And then I asked: Do you know <b>how many playmates like partying all night</b>? And they answered me that none.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- Well, I think you should call them back one at a time and ask them the same questions.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- I don't understand, Joe. If I ask them the same questions, I will get the same answers.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- No, because now we've got a <b>magical coin</b>.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieYe-bSyjNZG8z0wXfb89y50JQTO7ISCiRfLRMhdEjZjEvAjhot5TpIxvhdyk10Ux_Xq5i6nzky9gPOl-HwKwXkG0KScEyb-Zj321EY3YuLlcDbn4_zgCFEwX_uFocJoJncP0Moc9AgJI/s1600/Moneda+china+con+panda+y+serpiente+2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="50 yuan Chinese coin. I think that Joe Vitruvius's coin is counterfeit." border="0" height="151" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieYe-bSyjNZG8z0wXfb89y50JQTO7ISCiRfLRMhdEjZjEvAjhot5TpIxvhdyk10Ux_Xq5i6nzky9gPOl-HwKwXkG0KScEyb-Zj321EY3YuLlcDbn4_zgCFEwX_uFocJoJncP0Moc9AgJI/s400/Moneda+china+con+panda+y+serpiente+2.gif" title="50 yuan Chinese coin. I think that Joe Vitruvius's coin is counterfeit." width="400" /></a></div>
</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- I've already told you that you won't be able to bribe them. And even less with a simple coin, no matter how pretty or magical it is.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- I'm sure this coin will help us to know the truth, Fabio. </div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- I can't see how.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- It's very simple. We'll call them one by one to your office, and we'll propose this: we'll give them this coin, and we'll tell them to <b>toss it </b>secretly. </div>
<div style="text-align: justify;">
<div class="separator" style="clear: both; text-align: center;">
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<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
If they get a <b>panda bear</b>, they must answer <b>truthfully </b>about whether if they go out on a spree or not.<br />
<br />
But if they get the <b>snake</b>, they <b>always</b> have to answer that <b>they do</b> go out at night, regardless of the truth.</div>
<div style="text-align: justify;">
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghpwLy5oOHH8vmiD0ESNRaCZ9UhjdKck6ultmS6dzc0Y71mcC9aAoehf2fdc_Tv6A3pUMlDh2QD6qrFvEmAWigHzAeVgZSfGTX_Rq3uwG3qxlUxi4aRUu7svuaWeCTiZdsycEq5fHQtVQ/s1600/Salir+o+no+salir.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="It's clear: with the tender panda they will tell the whole truth; with the evil serpent they will say in all cases they like to party. Easy, right?" border="0" height="228" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghpwLy5oOHH8vmiD0ESNRaCZ9UhjdKck6ultmS6dzc0Y71mcC9aAoehf2fdc_Tv6A3pUMlDh2QD6qrFvEmAWigHzAeVgZSfGTX_Rq3uwG3qxlUxi4aRUu7svuaWeCTiZdsycEq5fHQtVQ/s640/Salir+o+no+salir.jpg" title="It's clear: with the tender panda they will tell the whole truth; with the evil serpent they will say in all cases they like to party. Easy, right?" width="640" /></a></div>
<div style="text-align: justify;">
<br />
They'll flip the coin again. If they get a <b>panda</b>, they must <b>tell us the number of players</b> they know going out by night, while if they get a <b>snake</b>, they can <b>invent </b>the amount, and tell any figure <b>between 0 and 20</b>.</div>
<div style="text-align: justify;">
<br />
We will never <b>know, anyway</b>, if each individual player has got a panda or a snake, that is, we won't know if he's lying or not, so those who has got a panda will have no fear of telling us the truth. Do you think they will accept this deal?</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- I think so. It seems that the method doesn't compromise them.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- Here we go. Tell the first player to come in...</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi459gpM4KGc2_BuEn6c61D9oqcafnzVZ-yiTOR5iRLT76VDJkLV01fZQksB0iby5yizbvOSfszAj4y3Wf7sVyUCtbi_oPnL3x8YaNZECAlUPVDhPJaE2MeFAbaXh2FmtCWbD-wzN1NagI/s1600/Despacho+de+Fabio+Cannavaro.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Fabio Cannavaro office at Guangzhou Evergrande stadium. I mean, its door." border="0" height="427" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi459gpM4KGc2_BuEn6c61D9oqcafnzVZ-yiTOR5iRLT76VDJkLV01fZQksB0iby5yizbvOSfszAj4y3Wf7sVyUCtbi_oPnL3x8YaNZECAlUPVDhPJaE2MeFAbaXh2FmtCWbD-wzN1NagI/s640/Despacho+de+Fabio+Cannavaro.jpg" title="Fabio Cannavaro office at Guangzhou Evergrande stadium. I mean, its door." width="640" /></a></div>
<br /></div>
<div style="text-align: justify;">
- ...we've<b> finally completed the survey</b>, Joe!<br />
<br />
- Well, now we can get a clear idea about what's happening on your team.</div>
<div style="text-align: justify;">
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<br /></div>
<div style="text-align: justify;">
- Oh, yes? You'll have to explain it to me. We don't know if each player has lied or has answered truthfully.<br />
<br /></div>
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsrgeH3LCry0-IRDeFyeO9FMIAAdkCHRP3WMwRqeFFQZpuXDSqfetjrnKID_GBcTnwdfWdfz_0BR-4e7z-UerVYenpt26Yc27QY5vAMyAuevz0l7mVMIwo74NXz3S_cqj7ETbJViRGxy0/s1600/Results+of+the+survey.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Results of the survey carried out among the players. I hope you haven't been confused with the snake and the panda." border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsrgeH3LCry0-IRDeFyeO9FMIAAdkCHRP3WMwRqeFFQZpuXDSqfetjrnKID_GBcTnwdfWdfz_0BR-4e7z-UerVYenpt26Yc27QY5vAMyAuevz0l7mVMIwo74NXz3S_cqj7ETbJViRGxy0/s1600/Results+of+the+survey.JPG" title="Results of the survey carried out among the players. I hope you haven't been confused with the snake and the panda." width="292" /></a></div>
<div style="text-align: justify;">
- Let's see the results of our survey. Regarding the first question, in which we directly asked if they go out or not, we have a total of <b>12 positive answers and 8 negative ones</b>. </div>
<div style="text-align: justify;">
</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
We have a <b>50-50</b> chance of getting a panda or a snake, so the most likely is that half of the players have told the truth and half the players have lied.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
This means that, about <b>10 players</b> have said that they go partying at night because they have got the <b>snake</b>. And among the other <b>10 players</b> who got the <b>panda</b>, and therefore had to say the truth, 2 have answered that they go out, and 8 have answered that they don't.<br />
<br />
They are 2 from 10, that's it, <b>a 20% </b>probability, so in the total of 20 players of your squad, we could think that there will be only 4 players go on a spree at night.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- Yes, but if it happens to get 20 pandas by chance, then all the answers would be true, and I'd have a problem with 12 players.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqNm8oGJv9XDY-JZFtAZr7SodPpZPMw3PNiFjRbZnf8poLOe0k5vU8g7kfQQgweEqw7gctjw-od2IJmn3Lo27mB5JG9iMrHgCLn93xMJavLZSObuqkSpmYt8MZlLln8QYU8fIdrZCLp7A/s1600/Probability+of+gettien+n+heads+or+tails.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="Probability of getting a certain number of faces and tails when tossing a coin 20 times, even in the case that the coin is Chinese." border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqNm8oGJv9XDY-JZFtAZr7SodPpZPMw3PNiFjRbZnf8poLOe0k5vU8g7kfQQgweEqw7gctjw-od2IJmn3Lo27mB5JG9iMrHgCLn93xMJavLZSObuqkSpmYt8MZlLln8QYU8fIdrZCLp7A/s1600/Probability+of+gettien+n+heads+or+tails.JPG" title="Probability of getting a certain number of faces and tails when tossing a coin 20 times, even in the case that the coin is Chinese." width="345" /></a>
- It's true. But look at this other table. If we calculate the <b>probability of getting a certain number of pandas</b>, when we toss a coin 20 times, we can see that, first, the most likely option is that we get 10 pandas, a 17.62% of the times.<br />
<br /></div>
<div style="text-align: justify;">
</div>
<div style="text-align: justify;">
And, on second place, we notice that the chance of getting 20 pandas is about a 0.0001%, that is, once every million times we do the test. In fact, there is a probability of almost 98% that we get 14 pandas at maximum.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
By the way, it seems a little risky to draw conclussions from <b>a sample of only 10 players</b>. So we should analyze the answers to the second question, in order to check if our assumptions are correct.</div>
<div style="text-align: justify;">
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<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9h5TB-XBWq5W2Ik8MCPKfqNLZm3gScRZpIA4u3YlGiNBdeZGdLd5eKKZUU8NyKpkDRryo5eSyddYZy_g_qVz-fDthbxo1QOrbWj0hq0rzoOg9TJiWjSMFTzdSAsSD0OHoypsCi8plODA/s1600/Survey+with+sorted+data.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Granny, why we have ordered the survey data? Just to study them better..." border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9h5TB-XBWq5W2Ik8MCPKfqNLZm3gScRZpIA4u3YlGiNBdeZGdLd5eKKZUU8NyKpkDRryo5eSyddYZy_g_qVz-fDthbxo1QOrbWj0hq0rzoOg9TJiWjSMFTzdSAsSD0OHoypsCi8plODA/s640/Survey+with+sorted+data.JPG" title="Granny, why we have ordered the survey data? Just to study them better..." width="275" /></a>
- But here, Joe, we find the same problem. We don't know which data are true and which data are invented. In fact, we have a set of answers, quite diverse. Some have answered that no player goes out, and other have said that there are 20 revellers.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- We can work on it, Fabio. If we calculate <b>the average</b> of the answers we have, we get that 5.25 people like partying. But the <b>coefficient of variation</b>, which measures the data dispersion, is enormous. We should work hard with sample data to obtain more acceptable statistical values.<br />
<br /></div>
<div style="text-align: justify;">
We know that about half of the responses are invented, and therefore only half of the data are reliable. So we should try to <b>eliminate some data</b>, to get a more realistic average.<br />
<br />
- But, <b>how can we separate right data from invented data</b>? If the second answers were related to the first responses, we would know that the data from those who said that they don't go out are correct, but they tossed the coin again before answering the second questions, so we can't get any help from this point.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- That's true. I've made them flip the coin twice just in order to avoid they lie when answering the second question for those who have answered truthfully the first one and have answered that they don't party late at night, because therefore we know that they got the panda and are obliged to tell the truth.<br />
<br />
Fortunately, in Statistics <b>there are some methods</b><b> to eliminate invalid data</b> that can distort the average.</div>
<div style="text-align: justify;">
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1lb4BsGav6RyYVGdfEjnp8w__bj8yiTDcaE_ejCXJ8j54QUeZ17dC60x_66jTnyQofiqyJVjDqCFf_8dN0I7r8B3s8wY72Pipg0jYGNl5mEyzWoW2FAwimrB1POOU2ffLGfqY1EenyC4/s1600/Sorted+data%252C+eliminating+quartiles+or+furthest+data.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="There are data that are not real. How can we remove them?" border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1lb4BsGav6RyYVGdfEjnp8w__bj8yiTDcaE_ejCXJ8j54QUeZ17dC60x_66jTnyQofiqyJVjDqCFf_8dN0I7r8B3s8wY72Pipg0jYGNl5mEyzWoW2FAwimrB1POOU2ffLGfqY1EenyC4/s640/Sorted+data%252C+eliminating+quartiles+or+furthest+data.JPG" title="There are data that are not real. How can we remove them?" width="531" /></a><br />
<br /></div>
<div style="text-align: justify;">
Some statisticians remove those <b>values furthest from the average </b><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyWfLRwjsUee2CbKW_Qxj7ve3CS0apCNGmP0KBfne1mpuHBrf9ZvdNhUJ8IDST5roqlwc7ds2Tu9l9yCPN_pRAozGNbytouGR4b_1-vC_YPJ9k4GmUHlCFvxzF4nmqwBU_7tuE77mgPv4/s1000/media+aritm%25C3%25A9tica.JPG" imageanchor="1" style="margin-left: 0em; margin-right: 0em; text-align: center;"><img alt="Average" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyWfLRwjsUee2CbKW_Qxj7ve3CS0apCNGmP0KBfne1mpuHBrf9ZvdNhUJ8IDST5roqlwc7ds2Tu9l9yCPN_pRAozGNbytouGR4b_1-vC_YPJ9k4GmUHlCFvxzF4nmqwBU_7tuE77mgPv4/s1000/media+aritm%25C3%25A9tica.JPG" title="Average" /></a>, distanced from it by a certain multiple of the standard deviation <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhU5lKUf3SH9OQ88ekSjmsizl2oHLZz2iGH_wzMw6EFvHwnOR_Bs835v8YRqBWLb8bfwg-nJ9j3k0KSiY_cFVcwtj4X_sZvNWRWlkb2doOQz-Wvrsyq1swmIv1uDUchJSSwMPcxmx992k/s1000/desviaci%25C3%25B3n+t%25C3%25ADpica.JPG" imageanchor="1" style="margin-left: 0em; margin-right: 0em; text-align: center;"><img alt="Standard deviation" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhU5lKUf3SH9OQ88ekSjmsizl2oHLZz2iGH_wzMw6EFvHwnOR_Bs835v8YRqBWLb8bfwg-nJ9j3k0KSiY_cFVcwtj4X_sZvNWRWlkb2doOQz-Wvrsyq1swmIv1uDUchJSSwMPcxmx992k/s1000/desviaci%25C3%25B3n+t%25C3%25ADpica.JPG" title="Standard deviation" /></a>, and keep only the data in the interval <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6OBSZ-MWn7YRLEONjqRrLQGCf78VRwy9rnYxJ-oEjbFZp2Cb_OWW5W0klHTbUiJm2a1QC1BB0mdptqe74_b9RYqImfLZ7j0A8akhlMMNsiGRBH7DAau5iPr7-2wIWi_xd9kRqSO-yk0Q/s800/intervalo.JPG" imageanchor="1" style="margin-left: 0em; margin-right: 0em; text-align: center;"><img alt="Data interval around the average" border="0" height="24" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6OBSZ-MWn7YRLEONjqRrLQGCf78VRwy9rnYxJ-oEjbFZp2Cb_OWW5W0klHTbUiJm2a1QC1BB0mdptqe74_b9RYqImfLZ7j0A8akhlMMNsiGRBH7DAau5iPr7-2wIWi_xd9kRqSO-yk0Q/s800/intervalo.JPG" title="Data interval around the average" width="150" /></a>. </div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
In other cases, we can sort the data from the lowest to the highest, and <b>remove the first and the fourth quartile</b>, leaving only the data of the two central quartiles, closer to the <b>median</b>.<br />
<br />
With either of these two options, we can see how the standard deviation drops down considerably to a somewhat more <b>acceptable </b>value.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Although in our case, as we only have 20 data, we must be cautious when removing some of them. By the way, we have 3 values that we can reject because they're <b>impossible</b>.<br />
<br />
- And, which are they?</div>
<div style="text-align: justify;">
<div class="separator" style="clear: both; text-align: center;">
</div>
<br /></div>
<div style="text-align: justify;">
- If there are 8 people who say that they don't go out, it's impossible that there are 17 or 20 lively players. And there's no way that 0 players go partying because, in that case, all the players who have got the panda should have answered that they don't know anyone who goes out, and there should be several zeros, not only one, except in the very unlikely event that they got 20 snakes.<br />
<br />
- Yes, once in a million times that we did the survey, right?</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6mAR-tU98M0FSLNUgietESEYwYeuF_EJlyuvTqO6I63q9GrUJCXeT6LrWqn7OF9IPhLjO0lsuoYy5PRLOFTpCckQ-ZphqXYctGO-Ex6EgO0Z2S-5g5G5gykYJXfEXB3zX4wmid0JoSM4/s1600/Sorted+data+without+incorrect+values.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="More conclusions on the survey. I promise this is the last table of the story." border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6mAR-tU98M0FSLNUgietESEYwYeuF_EJlyuvTqO6I63q9GrUJCXeT6LrWqn7OF9IPhLjO0lsuoYy5PRLOFTpCckQ-ZphqXYctGO-Ex6EgO0Z2S-5g5G5gykYJXfEXB3zX4wmid0JoSM4/s640/Sorted+data+without+incorrect+values.JPG" title="More conclusions on the survey. I promise this is the last table of the story." width="396" /></a>
- OK. We could go on the process by <b>eliminating those less likely results</b>, the furthest from the median or the average. Even in that case we could get confused, because not all the players know what they playmates do at night.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
You know that <b>not all of them are friends</b> of each other, and that they don't have fun all together. That way, if we eliminate, for example, the datum of 2, maybe this number has been provided by a person who was telling the truth, but it's also incorrect, because he only knows 2 people who do it, but there are much more.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
As the sample we have is very small, perhaps the <b>bias </b>that we generate when performing a <b>screening </b>of the remaining data would be more prejudicial than beneficial, so we will simply make the average of the data we have so far.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- Then, at the end, which of the methods of data cleaning will be use?<br />
<br />
- If we eliminate some data, we can see that the method with a lowest <b>coefficient of variation</b> is that of the central quartiles. Anyway, the average of all the methods is close to the value of 4. And this figure coincides with the result of the first question, so we can conclude that around <b> 20%</b> of your players goes out at night.<br />
<br /></div>
<div style="text-align: justify;">
Therefore, you can be happy, Fabio. It's very likely that only 4 players like to go for partying!</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- That's right, and this means that the poor performance can be combated with more physical and tactical sessions. Now, the last work is to convince the 4 merry players to control themselves until we have won the championship...<br />
<br /></div>
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj98Qutg5q5aLlur-wdF2mteCzXhtptT_m220FkwHbMFs1cjh9B-rWbjw-0Rxs3CcoVibHS-4Rxpu8kU36s9btH4-bXnuU0V4x2XcRm7cq90TyuFDNR1n30Cze_VPrkU351iVYxJEnMST4/s1600/Jugadores+del+Guanzhou+Evergrande+levantando+el+t%C3%ADtulo.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="Chinese team players celebrating the championship. Tonight nobody will tell them not to go partying." border="0" height="166" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj98Qutg5q5aLlur-wdF2mteCzXhtptT_m220FkwHbMFs1cjh9B-rWbjw-0Rxs3CcoVibHS-4Rxpu8kU36s9btH4-bXnuU0V4x2XcRm7cq90TyuFDNR1n30Cze_VPrkU351iVYxJEnMST4/s320/Jugadores+del+Guanzhou+Evergrande+levantando+el+t%C3%ADtulo.JPG" title="Chinese team players celebrating the championship. Tonight nobody will tell them not to go partying." width="320" /></a></div>
<div style="text-align: justify;">
And, are you sure that this <b>method </b>we have employed is <b>reliable</b>?</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- Of course. This ingenious method is attributed to <b><a href="http://people.math.umass.edu/~cattani/" target="_blank">Eduardo Cattani</a></b>, an Argentinian professor of Mathematics and Statistics at the University of Massachusetts, as Adrián Paenza refers in his book “Matemática... ¿estás ahí?". Morevoer, <b>Stanley L. Warner</b>, an American mathematician, published in March of 1965 an article about <a href="http://www.tandfonline.com/doi/abs/10.1080/01621459.1965.10480775" target="_blank">randomized responses and survey techniques for eliminating evasive answer bias</a> in the Journal of the American Statistical Association.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Clearly, there are certain <b>questions on sensible iussues</b> like <b>drugs</b> consumption, <b>sexual</b> behaviour, illegal or forbidden topics, violence, bullying, socially frowned <b>conducts</b>, etc., in which respondents use to reply with incorrect answers.</div>
<div style="text-align: justify;">
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg88HOv0XPiWCVywByJYcLUbAV-6TTpKfJb-hiIbevQAfofy43hx2q2NyxlqUh4bAlKySt3ngkdseoH6jlAzB3tUokzMURyHhvsM3zRyFkWkBeZ1LVbVO1mTuuq36O9iTzgKBkq3N5Nhtc/s1600/Pepe+Vitruvio+en+el+centro+de+Cant%25C3%25B3n.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Joe Vitruvius in front of the Shangxiajiu pedestrian street, wondering if he'd rather dinner a hamburger or a spring roll." border="0" height="264" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg88HOv0XPiWCVywByJYcLUbAV-6TTpKfJb-hiIbevQAfofy43hx2q2NyxlqUh4bAlKySt3ngkdseoH6jlAzB3tUokzMURyHhvsM3zRyFkWkBeZ1LVbVO1mTuuq36O9iTzgKBkq3N5Nhtc/s320/Pepe+Vitruvio+en+el+centro+de+Cant%25C3%25B3n.jpg" title="Joe Vitruvius in front of the Shangxiajiu pedestrian street, wondering if he'd rather dinner a hamburger or a spring roll." width="320" /></a>
The one way to ensure the <b>anonymity </b>and the <b>privacy</b>, and therefore to win the <b>confidence </b>of the respondents, is using these <b>techniques of randomized answers</b>, although they don't always work, sometimes because respondents just don't understand the mechanism, sometimes because they don't trust in all the procedure, and sometimes because, despite everything, they don't respond truthfully.</div>
<div style="text-align: justify;">
<br />
In our case, as we have done two questions on the same matter, we can be more sure of the result of our enquiry.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- Great, Joe. Many thanks for all. Have a good time in China!</div>
<div style="text-align: justify;">
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
</div>
<div style="text-align: justify;">
- Sure, Now, I go for a walk through the center of Guangzhou (Canton) to see if I meet any of my Chinese
<a href="http://blog.sina.com.cn/s/blog_1473418f10102vh91.html" target="_blank">friends</a>
or perhaps one of your players.<br />
<br />
I hope you have good luck on your next matches, Fabio. Bye!</div>
<div style="text-align: justify;">
<br /></div>
<br />
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If you're interested in <b>learning more </b>about this topic, you can visit any of this wonderful articles: <a href="http://isites.harvard.edu/fs/docs/icb.topic1463715.files/Randomized%20response%20approach%20to%20sensitive%20issues%20in%20surveys.pdf" target="_blank">Randomized response approach to sensitive issues in surveys</a>, <a href="http://users.iems.northwestern.edu/~ajit/papers/84%29%20Randomized%20Response.pdf" target="_blank">Randomized Response Techniques for Multiple Sensitive Attributes</a>, <a href="http://imai.princeton.edu/research/files/randresp.pdf" target="_blank">Design and Analysis of the Randomized Response Technique</a>, <a href="http://www.soz.unibe.ch/content/ueber_uns/jann/presentations/e20783/e20792/files20794/rrt_online07_asa_talk.pdf" target="_blank">Sensitive Questions in Online Surveys</a>.</div>
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And don't forget to take a walk by the <a href="http://mathematicalmysterytour.blogspot.co.uk/" target="_blank"><span class="negrita">123th Carnival of Mathematics</span></a>. There you'll find lots of excellent math posts that you'll surely like too.
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WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-21692028500638215982015-04-07T21:42:00.000+01:002015-04-13T22:08:58.328+01:00A galactic prize<!-- Link to Carnival of Mathematics -->
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcopTe4Zte_xzuPxcAlnGFBsthZWS_cY89-j1NCuwbvJQxTQowhqHg68NTYjlQ7ZOctF6nooZnhOSDlVsvw_XQhsIyYidS_9CT8l4lQ1GqNkAKQJdjmFxbwFN9E08Em6V56ckUiU1AYqg/s1600/Joe+Vitruvius+in+A+galactic+prize.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="The new story of Joe Vitruvius: a galactic prize" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcopTe4Zte_xzuPxcAlnGFBsthZWS_cY89-j1NCuwbvJQxTQowhqHg68NTYjlQ7ZOctF6nooZnhOSDlVsvw_XQhsIyYidS_9CT8l4lQ1GqNkAKQJdjmFxbwFN9E08Em6V56ckUiU1AYqg/s1600/Joe+Vitruvius+in+A+galactic+prize.jpg" height="264" title="The new story of Joe Vitruvius: a galactic prize" width="640" /></a></div>
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</div>
<br />
<b>The mathematical language is so amazing that, with a few figures, and a pair of operations, we can create formidable numbers.</b><br />
<br />
(This post participates on the <a href="https://lifethroughamathematicianseyes.wordpress.com/2015/04/12/carnival-of-mathematics-121/" target="_blank"><span class="negrita">121th edition</span></a> of the <a href="http://www.whitegroupmaths.com/2015/02/119th-carnival-of-mathematics.html" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="https://lifethroughamathematicianseyes.wordpress.com/" target="_blank"><span class="negrita unidas">Life through a mathematician's eyes</span></a>.)
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<span style="font-family: Verdana, sans-serif;">FIRST HALF</span></div>
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<!-- Main article -->
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyNBhR-3WH52JooShWdhhtxWdierfwwJzHsB5w3FJHfLHgrUQ_KU12CRL9l9EOJ2hW6evk1UNOcRVp9t1BNDFt4jxTtIWPzUKOIFcZNsUb2SBT-_grehAQzBC1P0md2Y_QOB8-jNrsPmI/s1600/El+equipo+de+Los+Angeles+Galaxy+gana+la+Major+League+Soccer.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Los Angeles Galaxy team wins the Major League Soccer" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyNBhR-3WH52JooShWdhhtxWdierfwwJzHsB5w3FJHfLHgrUQ_KU12CRL9l9EOJ2hW6evk1UNOcRVp9t1BNDFt4jxTtIWPzUKOIFcZNsUb2SBT-_grehAQzBC1P0md2Y_QOB8-jNrsPmI/s1600/El+equipo+de+Los+Angeles+Galaxy+gana+la+Major+League+Soccer.jpg" height="234" title="Los Angeles Galaxy team wins the Major League Soccer" width="320" /></a><br />
<div style="text-align: justify;">
The players of Los Angeles Galaxy team are very happy. They've just <b>won the Major League Soccer Cup</b>. </div>
<br />
<div style="text-align: justify;">
Once the celebrations for the victory have finished, president Chris Klein has promised them an <b>economical reward</b> for their victory:</div>
<br />
<div style="text-align: justify;">
- I've come to the changing room to tell you our decision of giving you a gift for your success.</div>
<br />
<div style="text-align: justify;">
We're going to give you as many cents of dollar as the <b>greatest number you may be able to form</b> with three of your shirts (you can't use those with double figures, nor use the number zero or the infinity symbol, right?).</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-pE5g511XDfiX3J7yJOnFiS5mOLAZdaUoZAYz3IbAwdl7PSU-gE4e8tg1LmwjyYaqQZ0HrFaUJ0GZ6ocS5WMG1rMl9B_tFDbDdlvcj5VolQV7A5FtPV4sBPS0T2h-z_lKZnS8Lr2kk_Y/s1600/Chris+Klein+pensando+en+una+divisi%C3%B3n+entre+cero.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Chris Klein considering a division by zero" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-pE5g511XDfiX3J7yJOnFiS5mOLAZdaUoZAYz3IbAwdl7PSU-gE4e8tg1LmwjyYaqQZ0HrFaUJ0GZ6ocS5WMG1rMl9B_tFDbDdlvcj5VolQV7A5FtPV4sBPS0T2h-z_lKZnS8Lr2kk_Y/s1600/Chris+Klein+pensando+en+una+divisi%C3%B3n+entre+cero.jpg" height="249" title="Chris Klein considering a division by zero" width="320" /></a></div>
<div style="text-align: justify;">
- For example, if you put togheter the shirts with numbers 5, 2 and 7, we'll give you 527 cents of dollar.</div>
<br />
- For everyone?<br />
<br />
<div style="text-align: justify;">
- Oh, no! Our budget is not so big. It will be 5.27 dollars for all the team. You can share the amount the way you decide.</div>
<br />
- Can we <b>repeat </b>numbers?<br />
<br />
<div style="text-align: justify;">
- No, you can't use three shirts of the same player.</div>
<br />
- And can use <b>mathematical symbols</b>?<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQi290dq9gqAW8tW_mBwGMFzseKucS1eA4QHgxv-JxsZLhZVj0zrN_AL38slskYvdAqURJ-LvVdc5SGzfZrX7V9bxPqxnpWcDiD0YOp-wuKHx08iiD1dB6kNMD-naTJqov3l81HWnxruE/s1600/Chris+Klein+pensando+en+una+multiplicaci%C3%B3n.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Chris Klein considering a multiplication" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQi290dq9gqAW8tW_mBwGMFzseKucS1eA4QHgxv-JxsZLhZVj0zrN_AL38slskYvdAqURJ-LvVdc5SGzfZrX7V9bxPqxnpWcDiD0YOp-wuKHx08iiD1dB6kNMD-naTJqov3l81HWnxruE/s1600/Chris+Klein+pensando+en+una+multiplicaci%C3%B3n.jpg" height="232" title="Chris Klein considering a multiplication" width="320" /></a></div>
<div style="text-align: justify;">
- Ok, you can, as long as you don't repeat them. And if you use one symbol, you must remove one number.<br />
<br /></div>
<div style="text-align: justify;">
- That is, if you use, for example, the plus sign, then you must remove one figure, so you can use only two digits: 5 + 9.
</div>
<div style="text-align: justify;">
<br />
- So for every sign we use, we must use one digit less, right?</div>
<br />
- Right.<br />
<br />
- And when should we tell you the answer?<br />
<br />
<div style="text-align: justify;">
- Tomorrow morning. The wishes of the club are that you can enjoy your well deserved reward as soon as possible...</div>
<br />
- All right. We see you tomorrow.<br />
<br />
<div style="text-align: justify;">
The players will meet in the evening, after the training session, to see what amount they will ask the club.<br />
<br /></div>
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKcZuwNFgH0qi92DDAR98fmb0fV6wnxDjOG6Mxir147QBBg-L7MhEgaS8enLTrQR2RkfAmqd5DaTi0I6W6Zq86FJGmBKlHX80wYpOOetoiBSmJ9JdmDGe2BE1d5ZskqSkNFC0BPiUqvA0/s1600/Pepe+Vitruvio+en+Hollywood.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Joe Vitruvius in Hollywood" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKcZuwNFgH0qi92DDAR98fmb0fV6wnxDjOG6Mxir147QBBg-L7MhEgaS8enLTrQR2RkfAmqd5DaTi0I6W6Zq86FJGmBKlHX80wYpOOetoiBSmJ9JdmDGe2BE1d5ZskqSkNFC0BPiUqvA0/s1600/Pepe+Vitruvio+en+Hollywood.jpg" height="356" title="Joe Vitruvius in Hollywood" width="640" /></a><br />
<br /></div>
<div style="text-align: justify;">
In addition, they've called <b>Joe Vitruvius</b>, who is on holiday by the West coast, to help them with this problem.</div>
<div style="text-align: justify;">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img alt="I calculate, therefore I am" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" height="252" title="I calculate, therefore I am" width="640" /></span></a></div>
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<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
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<!-- Main article -->
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3pG2vEIOPGrTU_LVjoKhUWlNoYPw8b45PbNG3N-KTgJ0NjErzJGz7NIFnQ5TZS7-sSqkfcsAWiFHHYmrXaxGT0tczrPuuqwwtlvDhkG1HvWWdw-nyUn5nDLrDS4Jc8cdUAHx8XsLw9Fg/s1600/Skyline+de+Los+Angeles.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Los Angeles skyline" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3pG2vEIOPGrTU_LVjoKhUWlNoYPw8b45PbNG3N-KTgJ0NjErzJGz7NIFnQ5TZS7-sSqkfcsAWiFHHYmrXaxGT0tczrPuuqwwtlvDhkG1HvWWdw-nyUn5nDLrDS4Jc8cdUAHx8XsLw9Fg/s1600/Skyline+de+Los+Angeles.JPG" height="125" title="Los Angeles skyline" width="320" /></a><br />
<div style="text-align: justify;">
The sun starts to fall down, and the football players met on the top floor of a skycraper in the city of Los Angeles.</div>
<br />
- Well, let's see what we can do.<br />
<br />
<div style="text-align: justify;">
- I see that,<b> if we're going to use only numbers</b>, it's clear that the highest number is 987. So we'll get <b>987 </b>cents, that is, 9.87 dollars. A high fortune, to be shared between all of us!</div>
<br />
<div style="text-align: justify;">
- Yes. And if we use some mathematical symbols, it seems that we'll only lose money.</div>
<br />
<div style="text-align: justify;">
- We can leave apart the <b>subtraction sign</b>. If we use it, we'll also lose money. The best option, without using the zero, would be 9 - 1 = 8 cents. </div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnUnqDUAYLMoy5VAXeoCuMWkoLZ4oNRCBaU__FRJbya4dhK8C2F0cqJ5kYl6l-l18r5hDmw9jgUTdgdhOv0qNVi6BiJCbNSTo9eQIBfRGJiLAlyp319NEu5PKilje2CjjIK2hqB09L48A/s1600/Signos+matem%C3%A1ticos.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Basic mathematical symbols" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnUnqDUAYLMoy5VAXeoCuMWkoLZ4oNRCBaU__FRJbya4dhK8C2F0cqJ5kYl6l-l18r5hDmw9jgUTdgdhOv0qNVi6BiJCbNSTo9eQIBfRGJiLAlyp319NEu5PKilje2CjjIK2hqB09L48A/s1600/Signos+matem%C3%A1ticos.gif" height="86" title="Basic mathematical symbols" width="320" /></a></div>
<span style="text-align: justify;"><br /></span>
<br />
<div style="text-align: justify;">
- The <b>sign of addition</b> doesn't seem to be very helpful. From our chosen number, if we replace any of the digits with this sign, we certainly lose a lot of money again. At best, we'll get 9 + 8 = 17 cents, far from the initial 987.<br />
<br />
- And with the multiplication symbol (here we'll use a dot) we can't either improve our prize. In this case, among all the possible products, the most beneficial one would be 9·8 = 72 cents. Still far from the 987 cents.</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuAZoT2In07IlBQiIxKeU2fQzIwsYjag6V4yfW09C__zJuVabiWJZ0zeRT9tpn9qLR6Z0ClcvDqMMIzy2wsUerm6x_rZi-GssWPJ7dOUu-XI5zDjJrx0h04DxEG9Wer84z6aeVwcymGXc/s1600/Camisetas+987.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Shirts with numbers 9, 8 and 7" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuAZoT2In07IlBQiIxKeU2fQzIwsYjag6V4yfW09C__zJuVabiWJZ0zeRT9tpn9qLR6Z0ClcvDqMMIzy2wsUerm6x_rZi-GssWPJ7dOUu-XI5zDjJrx0h04DxEG9Wer84z6aeVwcymGXc/s1600/Camisetas+987.gif" height="133" title="Shirts with numbers 9, 8 and 7" width="320" /></a></div>
<br />
<div style="text-align: justify;">
- Obviously, if we introduce the <b>division symbol</b>, we won't achieve anything positive, not to mention the <b>square root</b>...</div>
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMgettqel-debr-10AHS4cbDqVtPaeVf9RnNVupOrfyerKMx1xM6BIFPp853toF9vMSqucg5286pekkWVtDRSTumbwXhTrP46OTavETXqeLlFpp0jFuyoIVlFXYcK6XqsO0yH2NeqUwpU/s1600/Pepe+Vitruvio+con+sombrero+vaquero.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Joe Vitruvius with a cowboy hat" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMgettqel-debr-10AHS4cbDqVtPaeVf9RnNVupOrfyerKMx1xM6BIFPp853toF9vMSqucg5286pekkWVtDRSTumbwXhTrP46OTavETXqeLlFpp0jFuyoIVlFXYcK6XqsO0yH2NeqUwpU/s1600/Pepe+Vitruvio+con+sombrero+vaquero.gif" height="200" title="Joe Vitruvius with a cowboy hat" width="162" /></a>- Don't think more about it, We'll ask 987 cents, and problem solved!<br />
<div>
<br /></div>
<div style="text-align: justify;">
- Do you agree with us, Joe? You've been very quiet, during our discussions...</div>
<br />
- Oh, I haven't said anything because all your arguments were right. All, except the final decision you've taken.<br />
<br />
- Why?<br />
<br />
- I think you've forgotten a very useful mathematical operator: <span style="text-align: justify;"> the</span><b style="text-align: justify;"> exponentiation</b><span style="text-align: justify;">. It's a particular case of product in which one of the numbers (the</span><span style="text-align: justify;"> </span><b style="text-align: justify;">exponent</b><span style="text-align: justify;">) shows how many times we have to multiply the other number (the</span><span style="text-align: justify;"> </span><b style="text-align: justify;">base</b><span style="text-align: justify;">) by itself.</span><br />
<br />
Thus, we have for example 9<sup>3</sup>, which means 9·9·9 = 729.<br />
<br />
All we have to do is to place some numbers at a higher level from the others, and that's all.<br />
<br />
<div style="text-align: justify;">
Althought it has not always been like this. Babylonians, Greeks (especially Diophantus), Arabs, or medieval mathematicians used different symbols and words to describe the squares, the cubes, and the rest powers of a number.</div>
<br />
<div style="text-align: justify;">
<b>Michael Stifel</b>, in his <i>‘Integra Arithmetica’</i> of 1544 introduced the concept of exponent. <b>James Hume</b>, in 1636, decided to set the exponent on a higher level in relation to the base, although he used Roman numbers to designate them. And finally <b>Descartes</b>, in his work <i>‘Géométrie’</i> replaced the uncomfortable Roman numerals for the Hindu-Arabic numbers.</div>
<br />
<div style="text-align: justify;">
So, thanks to Descartes, this operation will be free for us, because <b>we don't have to use any extra character</b> so we don't have to eliminate one of the figures. Now, our problem is how to place the digits in the most favourable order.</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJhOKMPhPYHJ65mac5ynBNdLUhO_W_fcoHDLK7P3ucAGWWVHEF889r5LafAonQ43IOBFhEdJduJDOTEr_3YNXh-cQt6IaJsiTY1RZP8XzcwO381TbcWHzlhelINuFqa4Ts20pReA7jqOA/s1600/Distintas+combinaciones+de+potencias.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible powers with numbers seven, eight and night" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJhOKMPhPYHJ65mac5ynBNdLUhO_W_fcoHDLK7P3ucAGWWVHEF889r5LafAonQ43IOBFhEdJduJDOTEr_3YNXh-cQt6IaJsiTY1RZP8XzcwO381TbcWHzlhelINuFqa4Ts20pReA7jqOA/s1600/Distintas+combinaciones+de+potencias.jpg" height="325" title="Possible powers with numbers seven, eight and night" width="400" /></a></div>
<br />
As you can see, I've eliminated some combinations. For example, we know that 89<sup>7</sup> has to be smaller than 98<sup>7</sup>, so we don't calculate it.<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSN7VKsVmZ2EL1mAsrZCMvRR_QIBK9BWIGVHjYyP0mKok1H-KDJG23HphgE5GFD2EW4Fi6g7EMEi22Q-Q8qfgQpYNLHSUHbtqKnrZF4iX9tddzc5YxGUBPpPnbBVphr0bKgfo-bUr35Bs/s1600/Calculadora+cient%C3%ADfica.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Scientific calculator" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSN7VKsVmZ2EL1mAsrZCMvRR_QIBK9BWIGVHjYyP0mKok1H-KDJG23HphgE5GFD2EW4Fi6g7EMEi22Q-Q8qfgQpYNLHSUHbtqKnrZF4iX9tddzc5YxGUBPpPnbBVphr0bKgfo-bUr35Bs/s1600/Calculadora+cient%C3%ADfica.gif" height="200" title="Scientific calculator" width="90" /></a><br />
- Joe, I've got a scientific calculator in my backpack.<br />
<br />
<div style="text-align: justify;">
- Yes, and I downloaded a calculator app for my mobile. We could use it.</div>
<br />
<div style="text-align: justify;">
- I'm afraid that none of them will be useful for our purpose. Both apps will get blocked with the operations.</div>
<br />
<div style="text-align: justify;">
In fact, there are lots of internet websites that offer services of<b> calculation for huge numbers</b>, and they wouldn't be able to make these calculations. But there's a powerful one. This is the <a href="http://mrob.com/pub/comp/hypercalc/hypercalc-javascript.html" target="_blank">HyperCalc JavaScript</a> website. We'll use it for getting the results of the various options.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp-gk2Da65wNfYjgeaRT6enIjYqqLAPSDUGkH13Cmg47pmIGIahD5UNUyYEijjujWV28vmVRMJSGuVJvHrL1D9EWhLWzUz7nhleE4zGzxpFVF41l7dDVg0kwb83AOt1rUYBNCNaRAJxb8/s1600/Formulas+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Some huge numbers obtained thanks to power function" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp-gk2Da65wNfYjgeaRT6enIjYqqLAPSDUGkH13Cmg47pmIGIahD5UNUyYEijjujWV28vmVRMJSGuVJvHrL1D9EWhLWzUz7nhleE4zGzxpFVF41l7dDVg0kwb83AOt1rUYBNCNaRAJxb8/s1600/Formulas+3.jpg" height="106" title="Some huge numbers obtained thanks to power function" width="320" /></a></div>
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<br /></div>
<div style="text-align: justify;">
Now we're going for the last row of my table of possible combinations: <span style="text-align: center;">9</span><sup style="text-align: center;">8<sup>7</sup></sup>, <span style="text-align: center;">9</span><sup style="text-align: center;">7<sup>8</sup></sup>, <span style="text-align: center;">8</span><sup style="text-align: center;">9<sup>7</sup></sup>, <span style="text-align: center;">8</span><sup style="text-align: center;">7<sup>9</sup></sup>, <span style="text-align: center;">7</span><sup style="text-align: center;">9<sup>8</sup></sup>and <span style="text-align: center;">7</span><sup style="text-align: center;">8<sup>9</sup></sup>. In this case, each combination of numbers can be calculated by 2 different ways. If we take number <span style="text-align: center;">9</span><sup style="text-align: center;">8<sup>7</sup></sup>, we can calculate first <span style="text-align: center;">9</span><sup style="text-align: center;">8 </sup>and raise the result to 7, or we can calculate first <span style="text-align: center;">8</span><sup style="text-align: center;">7 </sup>and then take number 9 and raise it to the result we got.</div>
<br />
Let's see the results:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCTVe3H2z-0vrdCzoZ17G7fx3Rg1liJESAriSBRRrWoKgFax1lxejmz8HacjUj7hZsE8YoyaUDQuGVm8A9lHIgXssPD4hte_wk4l8FbiU6YNvHiTQcmXX2dXFL-e3WYLmOBJETawJZTlY/s1600/Formulas+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Some huge numbers obtained thanks to power function" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCTVe3H2z-0vrdCzoZ17G7fx3Rg1liJESAriSBRRrWoKgFax1lxejmz8HacjUj7hZsE8YoyaUDQuGVm8A9lHIgXssPD4hte_wk4l8FbiU6YNvHiTQcmXX2dXFL-e3WYLmOBJETawJZTlY/s1600/Formulas+2.jpg" height="223" title="Some huge numbers obtained thanks to power function" width="400" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibhk-iWYlIdAjSiBILIjMOczcP1xe6Jn0YNaei-1LIhgOIpUbfFfSrNapsN76AlDAB2avREzF5l2YsLjVbCgCkjG96Ehu86smGTNZa6uszEXaywRyFOVb7iTaXDIC231NEAbOeMIAcGuY/s1600/Pepe+Vitruvio+vestido+de+sheriff.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: justify;"><img alt="Joe Vitruvius dressed as a sheriff" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibhk-iWYlIdAjSiBILIjMOczcP1xe6Jn0YNaei-1LIhgOIpUbfFfSrNapsN76AlDAB2avREzF5l2YsLjVbCgCkjG96Ehu86smGTNZa6uszEXaywRyFOVb7iTaXDIC231NEAbOeMIAcGuY/s1600/Pepe+Vitruvio+vestido+de+sheriff.gif" height="320" title="Joe Vitruvius dressed as a sheriff" width="136" /></a>- According to this, we get the highest number with the combination <span style="text-align: center;">7</span><sup style="text-align: center;">(8<sup>9</sup>)</sup>. So we'll ask the club<b><span style="font-size: x-large;"> <span style="text-align: center;">7</span><sup style="text-align: center;">8<sup>9</sup></sup></span></b> dollars.<br />
<br />
- Great! We can now go home!<br />
<br />
<div style="text-align: justify;">
- Don't hurry, strangers! There's still something to be done. </div>
<br />
- What do you mean?<br />
<br />
- Well, I think that you're fogetting another mathematical symbol.<br />
<br />
- Ah, yes? I can't think of anything.<br />
<br />
<div style="text-align: justify;">
- Well. There's a symbol that will cause <b>admiration </b>to some of you... It's the sign: </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-FJozjnQ_t3Ag1oUFNbuRF6fEyRN-yYnI5_ckXMrcgSE-y4hTA68EVXNRqlcOskOnU1M1QVyHudRf96Y4ML6-nbt7-nNuarDUstU2KyiBC7f58lAVYpAnO8NE3q1MqhM7gktkj0L8OEI/s1600/Signo+de+exclamaci%C3%B3n.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Exclamation mark" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-FJozjnQ_t3Ag1oUFNbuRF6fEyRN-yYnI5_ckXMrcgSE-y4hTA68EVXNRqlcOskOnU1M1QVyHudRf96Y4ML6-nbt7-nNuarDUstU2KyiBC7f58lAVYpAnO8NE3q1MqhM7gktkj0L8OEI/s1600/Signo+de+exclamaci%C3%B3n.gif" height="320" title="Exclamation mark" width="320" /></a></div>
<br />
<br />
- Is it a mathematical sign, too? And what is it for?<br />
<br />
<div style="text-align: justify;">
- This symbol represents the <b>factorial function </b>of a number. The factorials consist of the product of all positive integers from 1 to the related number.</div>
<br />
<div style="text-align: justify;">
That way, the factorial of 5, written<i> 5!</i>, will be the product of the first five integer numbers: </div>
<br />
<div style="text-align: center;">
5! = 1·2·3·4·5 = 120</div>
<br />
And the factorial of 9 will be:<br />
<br />
<div style="text-align: center;">
9! = 1·2·3·4·5·6·7·8·9 = 362,880</div>
<br />
<div style="text-align: justify;">
- We know when to use the signs of addition or multiplication. But, when can we use these factorials?</div>
<br />
<div style="text-align: justify;">
- They've been used for a long time, especially in <b>Combinatorics </b>and in <b>Mathematical Analysis</b>. Hindu Mathematicians used them yet in the twelfth century. And they're used, for instance, to calculate the number of different ways of <b>arranging</b> some distinct objects.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTGmT6AwZO4ukdHa5AmfazLV8CVPROcxTDRfqqSSP9BTpysJ2iwuwOjs-ZDg3N2IpLCKYbl85bbv5i2qRIreJgfKxX7TAQLINqH4HKilWUH61bz1vKluqZkp-Fw3c5IokE09heE9tQcOY/s1600/salida+de+jugadores+al+terreno+de+juego.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Players entering the pitch one by one" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTGmT6AwZO4ukdHa5AmfazLV8CVPROcxTDRfqqSSP9BTpysJ2iwuwOjs-ZDg3N2IpLCKYbl85bbv5i2qRIreJgfKxX7TAQLINqH4HKilWUH61bz1vKluqZkp-Fw3c5IokE09heE9tQcOY/s1600/salida+de+jugadores+al+terreno+de+juego.JPG" height="240" title="Players entering the pitch one by one" width="320" /></a></div>
<div style="text-align: justify;">
For example, if we want to know <b>how many differents forms</b> you can enter the pitch (the 11 players going one by one, not by pairs, and assuming that the starting lineup is always the same), we get a total of:</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
11! = 1·2·3·4·5·6·7·8·9·10·11 = 39,916,800 different ways of entering the pitch.</div>
<br />
<div style="text-align: justify;">
I mean that, if you play a match every 2 hours, 24 hours a day without resting, you could be over 4,556 years playing football without repeating the entry order to the field.</div>
<br />
- Blow it!<br />
<br />
<div style="text-align: justify;">
- And why are factorials represented by the <b>exclamation mark</b>?</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi50UmjJW9E6Bqvg0uur3Tn6ISxcgUHzhW4x9QKFUstuPo_7BtH6cKoNYKRCSeTD2LedDMAnjbMzJbfAaVjJ4gcb_ABY5V2Zn2BwuXk-02kTmkFK-AtuWS7eQbfOZsV7ai_qgF9yfxk1M8/s1600/Christian+Kramp+y+el+s%C3%ADmbolo+factorial.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Christian Kramp and factorial symbols" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi50UmjJW9E6Bqvg0uur3Tn6ISxcgUHzhW4x9QKFUstuPo_7BtH6cKoNYKRCSeTD2LedDMAnjbMzJbfAaVjJ4gcb_ABY5V2Zn2BwuXk-02kTmkFK-AtuWS7eQbfOZsV7ai_qgF9yfxk1M8/s1600/Christian+Kramp+y+el+s%C3%ADmbolo+factorial.jpg" title="Christian Kramp and factorial symbols" /></a></div>
<div style="text-align: justify;">
- Well, the first person who used this symbol was <b>Christian Kramp</b> in <b>1808</b>. He was a Math teacher at Strasburg, who made a in-depth study on factorials. Initially he used a symbol in the form of a right angle, placed on the lower right corner of the number.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
But he had problems with the printer's, so for the edition of his book <i>Elements d'arithmétique universelle</i>, he decided to use the exclamation mark just after the number.</div>
<br />
<div style="text-align: justify;">
What I ignore is why he chose this sign, used to express happiness. The symbol comes from the Latin word <b><i>iocundum</i></b>, which means happy, pleasant, nice. First it was shortened to the expression (<b>io</b>), that became an expression of happiness, and then the capital '<i>I</i> 'was placed over the small '<i>o</i>', and that's how this symbol was born, as a sign of exclamation.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigwY4mEHXUh-ey2yvWsRPlm3WhrgyEbswWe6pyvjsVJl8SgXI23ito2y0s7dEkX40S9T2ab_hcwTBhN-we2fN7cNt0NZ9wgaWjNx70v1gf1HgTmfXVR33pqB9J0Qqhtg8qKF1SBHGJNk0/s1600/La+palabra+latina+iocundum+y+el+signo+de+admiraci%C3%B3n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="The Latin word iocundum and the exclamation mark" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigwY4mEHXUh-ey2yvWsRPlm3WhrgyEbswWe6pyvjsVJl8SgXI23ito2y0s7dEkX40S9T2ab_hcwTBhN-we2fN7cNt0NZ9wgaWjNx70v1gf1HgTmfXVR33pqB9J0Qqhtg8qKF1SBHGJNk0/s1600/La+palabra+latina+iocundum+y+el+signo+de+admiraci%C3%B3n.jpg" height="243" title="The Latin word iocundum and the exclamation mark" width="400" /></a></div>
<br />
<div style="text-align: justify;">
Maybe the relationship between the symbol and the <b>factorial function </b>is the wonder caused in people when calculating the result of the operation. Just look at what happens when we change one of our digits, i.e. number 7, by an exclamation mark:</div>
<br />
<div style="text-align: center;">
98! = 9.4268 · 10<sup>153</sup></div>
<br />
That is, a 154-digits number!<br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8JJeMiEOdTvCldcTTyvORJbe32tR2zNPhAvH6PZKFLTFR-wgNXHiLw4pEnnEFuI-9WFR9okCeQ4QzSTl08hfspcf3Nqxk2PtDSATOmIQnwzmM1BUtBHrFZWnhZ8IVQW7BP8C8J06ewS8/s1600/Factorial+de+98.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Factorial of 98" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8JJeMiEOdTvCldcTTyvORJbe32tR2zNPhAvH6PZKFLTFR-wgNXHiLw4pEnnEFuI-9WFR9okCeQ4QzSTl08hfspcf3Nqxk2PtDSATOmIQnwzmM1BUtBHrFZWnhZ8IVQW7BP8C8J06ewS8/s1600/Factorial+de+98.gif" height="330" title="Factorial of 98" width="640" /></a></div>
<br />
<div style="text-align: justify;">
- Yes, it's a huge number. But I remember that with potences we got a higher amount:</div>
<br />
<div style="text-align: center;">
<span style="text-align: center;">7</span><sup style="text-align: center;">8<sup>9</sup></sup><span style="text-align: center;"> = 7</span><span style="text-align: center;">,3785</span><span style="text-align: center;"> · 10</span><sup style="text-align: center;">113,427,138</sup> > <span style="text-align: center;">98! = 9,4268 · 10</span><sup style="text-align: center;">153</sup></div>
<br />
- Then it seems quite clear: we get <span style="text-align: center;">7</span><sup style="text-align: center;">8<sup>9</sup></sup><span style="text-align: center;"> cents. </span>Or do you have one more surprise for us, Joe?<br />
<br />
<div style="text-align: justify;">
- Yes, we can still <b>mix both operations</b> to try to get the highest number.</div>
<br />
<div style="text-align: justify;">
If we want to observe the imposed conditions, it seems that there are six possible combinations. But it's very difficult to choose one of them.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjO08VP1-AOWepu830dFUjtmdj1sXKdsn9mkVUrHiOZbUcjuGiO5iwgbsDut6mlhpu2KPRg2hqdUYrYQxx2ftgEH6U8lFUAlzd0UVaucj6Mso7K12wQuCBbYHMq8GAdyC1wyIDRuuY5Plc/s1600/Combinaciones+posibles+de+factoriales+y+potencias.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible combinations of factorials and powers" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjO08VP1-AOWepu830dFUjtmdj1sXKdsn9mkVUrHiOZbUcjuGiO5iwgbsDut6mlhpu2KPRg2hqdUYrYQxx2ftgEH6U8lFUAlzd0UVaucj6Mso7K12wQuCBbYHMq8GAdyC1wyIDRuuY5Plc/s1600/Combinaciones+posibles+de+factoriales+y+potencias.jpg" height="233" title="Possible combinations of factorials and powers" width="320" /></a></div>
<br />
<div style="text-align: justify;">
I don't know what's better: to put the highest digit as the base or as the coeficient:</div>
<br />
<div style="text-align: center;">
2<sup>3</sup> (8) <b><</b> 3<sup>2</sup> (9) but 3<sup>4</sup> (81) <b>></b> 4<sup>3</sup> (64)</div>
<br />
<div style="text-align: justify;">
And I don't know where to place the factorial symbol, after the base or after the exponent:</div>
<br />
<div style="text-align: center;">
2!<sup>3</sup> (8) <b><</b> 2<sup>3!</sup> (64) but 3!<sup>2</sup> (36) <b>></b> 3<sup>2!</sup> (9)</div>
<br />
We'll have to use again the <a href="http://mrob.com/pub/comp/hypercalc/hypercalc-javascript.html" style="text-align: justify;" target="_blank">HyperCalc JavaScript</a> website and get the different results:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiekTaH3RmnaAUNmU2uio0cEQ9j3tbr_StKZ0PSet1XP5e-BzMOYf_uFwqbk8Gz-3eEWta3GRddPeCVDncgjfRkf3FsvzBWs40SfRKQIa_H1cihmPWeZKClD28untcpRKU0rVtyCvfp-h0/s1600/Formulas+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="With factorials and powers we get some huge numbers" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiekTaH3RmnaAUNmU2uio0cEQ9j3tbr_StKZ0PSet1XP5e-BzMOYf_uFwqbk8Gz-3eEWta3GRddPeCVDncgjfRkf3FsvzBWs40SfRKQIa_H1cihmPWeZKClD28untcpRKU0rVtyCvfp-h0/s1600/Formulas+1.jpg" height="62" title="With factorials and powers we get some huge numbers" width="640" /></a></div>
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<div style="text-align: center;">
<br /></div>
<div style="text-align: justify;">
- I think that we finally will choose the option of <b>8<sup>9</sup>! </b>better than <span style="text-align: start;"><span style="text-align: center;">7</span><sup style="text-align: center;">8<sup>9</sup></sup></span><span style="text-align: start;"> </span><br />
<b><br /></b>
<br />
<div style="text-align: center;">
<span style="text-align: center;">8</span><sup style="text-align: center;">9</sup><span style="text-align: center;">! = 1,7828 · 10</span><sup style="text-align: center;">1.032.606.169</sup><span class="Apple-tab-span" style="text-align: center; white-space: pre;"> > </span><span style="text-align: center;">7</span><sup style="text-align: center;">8<sup>9</sup></sup><span style="text-align: center;"> = 7</span><span style="text-align: center;">,3785</span><span style="text-align: center;"> · 10</span><sup style="text-align: center;">113.427.138</sup></div>
<br />
- It seems a great amount of cents, but I'm sure it won't be a problem for Los Angeles Galaxy, right?</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9NmVGgqKeuKtkedo4gcICje2MzTgtPGzMVY6_tN6qIebqmtzFLWMyEp8BYXDo5TK4R4J-T3HcQk0cei67lanmuKKrL0BpPBCUntxFRFXKCf6fB8_7IaT0LpJPm-oTRBS4Q5wdDstKPbg/s1600/Camisetas+89!.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Shirts with numbers eight and night, and an exclamation mark" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9NmVGgqKeuKtkedo4gcICje2MzTgtPGzMVY6_tN6qIebqmtzFLWMyEp8BYXDo5TK4R4J-T3HcQk0cei67lanmuKKrL0BpPBCUntxFRFXKCf6fB8_7IaT0LpJPm-oTRBS4Q5wdDstKPbg/s1600/Camisetas+89!.gif" height="166" title="Shirts with numbers eight and night, and an exclamation mark" width="400" /></a></div>
<br />
<br />
<div style="text-align: justify;">
- I'm afraid this is going to be a problem. Are you aware of what this number means?</div>
<br />
- Hmmm…<br />
<br />
<div style="text-align: justify;">
- It's a number of over <b>1,032 billion digits</b>.</div>
<br />
- It's not very big, isn't it?<br />
<br />
<div style="text-align: justify;">
- Do you know how many <b>atoms </b>are there <b>in the universe</b>? About 10<sup>87</sup>, that is, a number 1 followed by 88 zeros.</div>
<br />
<div style="text-align: justify;">
- Well, I can't see much difference between both numbers. Just a few more digits in the exponent.</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitCfa3P6xELiqbjWetLKm1cMajrUcyHV84wJ-YmRb27n69oQJWvAJZDBq0vstPhN3ts7OY88TyVwZLn2m5IRXBXmOvOsj93-oA7gUg3a4bVzGCkIP0Hy2MnOECaOka4T49NfYxYByoomc/s1600/Comparaci%C3%B3n+entre+cifras+elevadas.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Comparison between high figures" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitCfa3P6xELiqbjWetLKm1cMajrUcyHV84wJ-YmRb27n69oQJWvAJZDBq0vstPhN3ts7OY88TyVwZLn2m5IRXBXmOvOsj93-oA7gUg3a4bVzGCkIP0Hy2MnOECaOka4T49NfYxYByoomc/s1600/Comparaci%C3%B3n+entre+cifras+elevadas.gif" height="205" title="Comparison between high figures" width="320" /></a></div>
<br />
<div style="text-align: justify;">
- With scientific notations it's quite difficult to be conscious of the magnitude and the proportion of numbers. To get started on this subject, there's a great website that explains it very well: <a href="http://apod.nasa.gov/apod/ap120312.html" target="_blank">The scale of the universe</a>.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Look: the <b>volume of an electron</b> is estimated to be 9.36·10<sup>−44</sup> m<sup>3</sup>, and the <b>volume of the universe</b> is about 7·10<sup>81</sup> m<sup>3</sup>. Well, if we fill the whole universe with electrons, the amount of these would be of <b>7.4786 · 10<sup>124</sup></b>. This will be as long as we leave no space between them, and we know this is not possible because of the laws of <a href="http://www.matifutbol.com/es/balones.html" target="_blank">sphere packing</a>. </div>
<div style="text-align: justify;">
<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHEr_sy7HJ9JbQVVRqcZh9vqWVUoskIy7OQ23-Msgy5NqfxnYkXxsvlW84pIcaZxwSFkfW23YK-2L9z7s7SVPk_76qMB34QCAIM7ayt8icmoR8KqK9NMK-NrldPqXbeLZQQx_z3zXhRTs/s1600/Universo+lleno+de+electrones.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Universe filled with electrons" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHEr_sy7HJ9JbQVVRqcZh9vqWVUoskIy7OQ23-Msgy5NqfxnYkXxsvlW84pIcaZxwSFkfW23YK-2L9z7s7SVPk_76qMB34QCAIM7ayt8icmoR8KqK9NMK-NrldPqXbeLZQQx_z3zXhRTs/s1600/Universo+lleno+de+electrones.jpg" height="408" title="Universe filled with electrons" width="640" /></a></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Now let's imagine that all the universe, full of electrons, was compressed to the size of an electron, and we fill up again another universe like ours with those little universes full of electrons.</div>
<br />
<div style="text-align: justify;">
And imagine we repeat this process 8 million times. The final universe, full of small universes, full of small universes, full of small universes….(8 million times)… full of small universes full of electrons would contain as many electrons as the cents the club should give us.</div>
<div style="text-align: justify;">
<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLntyqsBUSgGoHoOet1ScP89B6MusF3jg5B-jeSXalP08mdQ7ZmlMPd1NdEWa7gzaVMAMldfawHS2QnZOkWn32KvTPgkwTj6ArIf3_XE5FgCiau829rO_K842Y-aWsqNFvt_lWANB0YBg/s1600/8+million+universes.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="A universe filled with universes filled with universes... filled with electrons" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLntyqsBUSgGoHoOet1ScP89B6MusF3jg5B-jeSXalP08mdQ7ZmlMPd1NdEWa7gzaVMAMldfawHS2QnZOkWn32KvTPgkwTj6ArIf3_XE5FgCiau829rO_K842Y-aWsqNFvt_lWANB0YBg/s1600/8+million+universes.jpg" height="324" title="A universe filled with universes filled with universes... filled with electrons" width="640" /></a></div>
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<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- That way, it does seem a very high amount. Perhaps we'd better choose another combination of numbers to get an amount that the club can afford…</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- All the money in circulation worldwide is estimated to be about <b>80 trillion dollars</b> (80·10<sup>12</sup> $). So there you have the upper limit of your request.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8QzbrD5tLpUrng3HDvzvrrFDqGq7VZU-4oUskKREDEedkzLthrUqTK_6LNQyqsxh4oaqcBHnqaMXo4gN8HwKZxT5O56kQ8O-g9sGR4VhKhFh7Q3G_WpFREXAg3K_9VYDFyG2HQxtCyX8/s1600/La+sonrisa+de+los+j%C3%B3venes+aficionados.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="The smile of the young supporters" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8QzbrD5tLpUrng3HDvzvrrFDqGq7VZU-4oUskKREDEedkzLthrUqTK_6LNQyqsxh4oaqcBHnqaMXo4gN8HwKZxT5O56kQ8O-g9sGR4VhKhFh7Q3G_WpFREXAg3K_9VYDFyG2HQxtCyX8/s1600/La+sonrisa+de+los+j%C3%B3venes+aficionados.jpg" height="281" title="The smile of the young supporters" width="320" /></a></div>
<div style="text-align: justify;">
And you also know which is the lower limit: when I was entering in the skyscraper, I saw lots of <b>young fans</b> at the main entrance because they knew you've met here, and have begged me to ask if you can sign some <b>autographs</b> for them.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- Yes, perhaps we'd better forget those big numbers, because we already have got enough <b>reward </b>with the smile of these supporters. I think that our satisfaction and the children's can be also calculated with a potence, don't you think so, Joe?<br />
<br />
- Let's go down immediately!</div>
<br />
<br />
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If you're interested in <b>learning more </b>about this topic, you can visit any of this wonderful articles: <a href="http://mathserver.neu.edu/~bridger/U201/History_of_Math_Mathematical_Symbols.htm" target="_blank">The History of Mathematical Symbols</a>, <a href="http://math.wikia.com/wiki/History_of_mathematical_notation" target="_blank">History of Mathematical Notation</a>, <a href="http://math.unipa.it/~grim/mathnotation.pdf" target="_blank">The origins and development of mathematical notation</a>, <a href="http://www.ehow.com/about_5134780_history-exponents.html" target="_blank">The History of Exponents</a>.</div>
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WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-30263239184603681382015-01-28T17:27:00.001+00:002015-02-10T21:24:17.180+00:00Flavius Josephus and the puisi-nanoq game<!-- Page resume -->
<span style="display: none;">A historical fact of 20 centuries ago led to a mathematical problem that still gives enough to talk: the Josephus problem. It combines recursive algorithms, permutations, modular arithmetics and the process of removing the elements of a set.
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsUG3lmmgidHQQLyANGHkLBybpTSnPHpSkpO5nUqhyphenhyphenfC0fT_xbG7x3AJpyJcgXQHrjlkVdjXtd_lLyJuw_DRelybc8wrSH-CXrKc8JjLWgDp-Y90gpWt6W8VqC7JykyJCgVIjnms2zg3k/s1600/cuadro+resumen+de+la+historia+groenlandesa.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Joe Vitruvius faces in Greenland his new challenge: the puisi-nanoq game" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsUG3lmmgidHQQLyANGHkLBybpTSnPHpSkpO5nUqhyphenhyphenfC0fT_xbG7x3AJpyJcgXQHrjlkVdjXtd_lLyJuw_DRelybc8wrSH-CXrKc8JjLWgDp-Y90gpWt6W8VqC7JykyJCgVIjnms2zg3k/s1600/cuadro+resumen+de+la+historia+groenlandesa.jpg" height="266" title="Joe Vitruvius faces in Greenland his new challenge: the puisi-nanoq game" width="640" /></a></div>
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(This post participates on the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">119th edition</span></a> of the <a href="http://www.whitegroupmaths.com/2015/02/119th-carnival-of-mathematics.html" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="http://www.whitegroupmaths.com/" target="_blank"><span class="negrita unidas">White Group Mathematics</span></a>.)
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_zLUng9W1yMysgH8cb5JbkwByszprbdX6x5AmHepnXO1pwLx45wHeJtJFCcPmC5N3WYJ8MBlpsa5tcrv20vuAx7H13gxUxIlyfz2X5bnUOZDERbsUrZXdDjCft2huoU0xfjALN_7_Lo8/s1600/Bandera+y+escudo.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="The Greenland Football Federation badge, and Greenland's flag" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_zLUng9W1yMysgH8cb5JbkwByszprbdX6x5AmHepnXO1pwLx45wHeJtJFCcPmC5N3WYJ8MBlpsa5tcrv20vuAx7H13gxUxIlyfz2X5bnUOZDERbsUrZXdDjCft2huoU0xfjALN_7_Lo8/s1600/Bandera+y+escudo.jpg" height="192" title="The Greenland Football Federation badge, and Greenland's flag" width="320" /></a><br />
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The coach of the <b>Greenland</b>'s national team is very worried. Next weekend they play a decisive match to qualify for the <b>Olympic Games</b> in Rio de Janeiro, and she has a serious problem with the team's captain.</div>
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There's a tradition on the team of <b>drawing lots </b>to choose who should <b>shoot the penalties</b>, among all the players who want to do it.</div>
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The problem is that, for some time, <b>all penalties have been kicked by the captain</b>, and she has missed the last 10.</div>
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<b>Pipaluk</b>, the coach, has decided to call her friend <b>Joe Vitruvius</b>, to see if he helps her solve this mystery, and thus they can face the next match with the highest guarantees.</div>
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Joe answers the call of Pipaluk, and she goes to the Nuuk airport to welcome him.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhD0n0jsamC0lCWU2A_Je3poxWkIlf9TblYOB4n6nRV9j2yb5OrwcNycpQ6Ff77Po9Lwsc_PHk_OYKM6ex_-FVfyGPx7PxDxImqow35YlG29lNt38V80AsZD-d0xvV0TCk8aGZk5InFzpk/s1600/Pepe+Vitruvio+en+el+aeropuerto+de+Nuuk.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Joe Vitruvius upon arrival at the airport in Nuuk" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhD0n0jsamC0lCWU2A_Je3poxWkIlf9TblYOB4n6nRV9j2yb5OrwcNycpQ6Ff77Po9Lwsc_PHk_OYKM6ex_-FVfyGPx7PxDxImqow35YlG29lNt38V80AsZD-d0xvV0TCk8aGZk5InFzpk/s1600/Pepe+Vitruvio+en+el+aeropuerto+de+Nuuk.jpg" height="428" title="Pepe Vitruvius upon arrival at the airport in Nuuk" width="640" /></a><br />
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- Hi Joe, I'm very happy you've come to help me.</div>
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- It's a pleasure. Tell me what's the problem.</div>
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- You better see it for yourself on the pitch. Let's come tomorrow to our training session.</div>
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- Ok, see you there.</div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilqe9-XC2Q-WvxfGKP4j7EfUGh1rY8TzjFd_UdX7-_ibHeP87huYGYj9SH3JTziCpmG37c-fp_RDqTkhoM85bSMAI9wGBus_ZYef1ybrnhTdSveVKr4ug1M0OG4GiMzxebiyaN78SX4Hk/s1600/jugadoras+en+c%C3%ADrculo.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Soccer players in a circle" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilqe9-XC2Q-WvxfGKP4j7EfUGh1rY8TzjFd_UdX7-_ibHeP87huYGYj9SH3JTziCpmG37c-fp_RDqTkhoM85bSMAI9wGBus_ZYef1ybrnhTdSveVKr4ug1M0OG4GiMzxebiyaN78SX4Hk/s1600/jugadoras+en+c%C3%ADrculo.jpg" title="Soccer players in a circle" /></a><br />
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On the following day, Joe goes to the stadium, and watches the curious <b>way of choosing the player</b> who will shoot the penalty.</div>
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- What's that what they do when they have to take a penalty? Why do they form a <b>circle</b>?</div>
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- That's the way they choose the penalty-taker: they do <b>the puisi-nanoq game</b> (the seal-bear game in Greenlandic).</div>
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- And what's that?</div>
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- All the players willing to take the penalty place themselves in a circle, and<b> they're eliminated one by one </b>as follows. They start counting from one of them, saying: puisi-nanoq-puisi-nanoq... (seal-bear-seal-bear...) And all the players who get a <b>bear</b>, are <b>removed</b>. The last ‘seal’ will be who kicks the penalty.</div>
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- How come?<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWeAhG68nx6kJUYGIVQhefnsCINehAEZbs10O3JFmDytl26mmScPk-8jYdiDxAZgKNK2OQnjFEHY46237B4yVgnMoH8e-2gJ81chgVIXNOjmJodyazQYwVkz5B3iKoK0gD8ki_ldBnTMw/s1600/Foca+polar.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="puisi - seal in Groenlandic" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWeAhG68nx6kJUYGIVQhefnsCINehAEZbs10O3JFmDytl26mmScPk-8jYdiDxAZgKNK2OQnjFEHY46237B4yVgnMoH8e-2gJ81chgVIXNOjmJodyazQYwVkz5B3iKoK0gD8ki_ldBnTMw/s1600/Foca+polar.gif" height="200" title="puisi - seal in Groenlandic" width="191" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMdIb65VC2KcKgkbAlVj5qSPUAqe_uh7-y0DR0QZHgRmNHYtBmvW-LHDd9rnhQRQ35ETQKYuyhVpCfzEMBtzqeji0aMeh5o3cw-l2I0Aml0DeP-mD4GC8U9vPjYWdyBylrn8Bky8gMVOI/s1600/Osa+polar.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="nanoq - bear in Greenlandic" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMdIb65VC2KcKgkbAlVj5qSPUAqe_uh7-y0DR0QZHgRmNHYtBmvW-LHDd9rnhQRQ35ETQKYuyhVpCfzEMBtzqeji0aMeh5o3cw-l2I0Aml0DeP-mD4GC8U9vPjYWdyBylrn8Bky8gMVOI/s1600/Osa+polar.gif" height="195" title="nanoq - bear in Greenlandic" width="200" /></a></div>
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- Yes, it's very simple, we play it since kindergarten. We start counting at one of them (we assign number 1 to her, so that this first player, which corresponds to a <b>seal</b>, <b>stays in the circle</b>, and the second one (bear) goes out. The third player (seal) also stays in the circle, and the fourth one (bear) is taken out. And we continue this way, eliminating one player of every two, and therfore <b>the circle becomes smaller</b>, till <b>only one remains</b>, who will be who will take the penalty.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgW01QbjXBwIdRXdh38GEKHk_0-NxwXbZBbp9hGPFu0brRGktjVYWoPdwQRHOimf7R08Wx-i_f8gbXYcDv-e5Yk3VtTsedM7HlTgdTlKHGJ2oeLJnDZIN8XSXDedZiovNoMYZoEa0d689k/s1600/Juego+1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="The players linked to a bear are eliminated" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgW01QbjXBwIdRXdh38GEKHk_0-NxwXbZBbp9hGPFu0brRGktjVYWoPdwQRHOimf7R08Wx-i_f8gbXYcDv-e5Yk3VtTsedM7HlTgdTlKHGJ2oeLJnDZIN8XSXDedZiovNoMYZoEa0d689k/s1600/Juego+1.gif" height="312" title="The players linked to a bear are eliminated" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTEyVpgbcZpeEUzeu4qWVKQ9PSAmucr8ETpDGWkj_PUF_FJkeaaga6mD97_YGsVIdQPr-N5erl4hVOPeg4MaPXGX5HToLKnRbdSeimFqVzO6EwQ3fgrOZ9LfziuhjBxsX2ehenaD-m5YA/s1600/flecha+amarilla.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTEyVpgbcZpeEUzeu4qWVKQ9PSAmucr8ETpDGWkj_PUF_FJkeaaga6mD97_YGsVIdQPr-N5erl4hVOPeg4MaPXGX5HToLKnRbdSeimFqVzO6EwQ3fgrOZ9LfziuhjBxsX2ehenaD-m5YA/s1600/flecha+amarilla.png" height="108" width="200" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjETSGN2wpwiUYi0rNDCtcRBsOr1rk2O3J8aLrBp1SMN7rOjT6OLIAE4zxddYlfvmiXpVfhmXzJa1aLsxLEYmoANdUztUTveof7Y92EGLU28F_wt469UqsOoHm1zsEGV7LrzfMpztZ-Aak/s1600/Juego+2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="We follow with this procedure until there's only one player" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjETSGN2wpwiUYi0rNDCtcRBsOr1rk2O3J8aLrBp1SMN7rOjT6OLIAE4zxddYlfvmiXpVfhmXzJa1aLsxLEYmoANdUztUTveof7Y92EGLU28F_wt469UqsOoHm1zsEGV7LrzfMpztZ-Aak/s1600/Juego+2.gif" height="312" title="We follow with this procedure until there's only one player" width="320" /></a></div>
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- I can't see where's the problem. The 3 penalties have been kicked by different players. According to what I've seen, <b>the number of players</b> who want to shoot the penalty <b>varies every time</b> they have to kick one. Also, <b>each time they stand in a different order</b> when forming the circle, and <b>they start counting from a different player</b>.<br />
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- Yes, that's the way they do it on <b>training sessions</b>. But when playing <b>official matches</b>, the captain has the right of being the one who does the counting, and who decides from which player she will start from. And surprisingly, in the last 10 times she has always been the player selected to kick all the penalties.</div>
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- This wouldn't be a problem if she were a good penalty-taker. But she's the worst in this facet of the game, and we've already lost some matches for this cause. Next saturday we play to qualify for the Olympic Games, and I don't want take any risk. She says that she doesn't cheat, that she's <b>very lucky</b>. Is it really so, Joe?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" height="252" width="640" /></span></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiD3Un9b_zujEHmBPfG0HcYsxWgKZ0-GMszPhbkgnGwFe-T_WRXVtq36k6HVtzwqcAs7avxX1FB7Lbbc2KoKh-6rrqyDnewSweMC-OBJrhlH4BPDAkxNt_vdmx4a3t8dzGrwgd9jYbp1sE/s1600/Flavio+Josefo.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Bust of Flavius Josephus" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiD3Un9b_zujEHmBPfG0HcYsxWgKZ0-GMszPhbkgnGwFe-T_WRXVtq36k6HVtzwqcAs7avxX1FB7Lbbc2KoKh-6rrqyDnewSweMC-OBJrhlH4BPDAkxNt_vdmx4a3t8dzGrwgd9jYbp1sE/s1600/Flavio+Josefo.jpg" height="200" title="Bust of Flavius Josephus" width="133" /></a><br />
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- Just one question, Pipaluk: is the captain a Mathematician?</div>
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- No. She's got a degree on <b>History </b>from the University of Nuuk.</div>
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- Well, this would also explain her ability. I'm sure she knows<b> the history of Flavius Josephus</b>.</div>
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- Who's that?</div>
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<div style="text-align: justify;">
- Well, it's an old history, from the <b>I century A.D.</b>, when <b>Rome </b>ruled all the Mediterranean Sea. The <b>Jews had rebelled</b> against the Empire, but the Roman army was very powerful.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwz2XGhrdAbBpFRQgZXQ3dIkcRPbpUt0jsWEW9nshDD8QwL4rN8hT09uAuk354IKYJOpBLj-qgxZrtDOBRdpSSatk-Zfz345QptHulc4QaxI1D6OjiimkEdTcHqAMhXm4TiOCs_UkNHg4/s1600/mapa+del+imperio+romano.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Extension of the Roman Empire in the I century A.D." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwz2XGhrdAbBpFRQgZXQ3dIkcRPbpUt0jsWEW9nshDD8QwL4rN8hT09uAuk354IKYJOpBLj-qgxZrtDOBRdpSSatk-Zfz345QptHulc4QaxI1D6OjiimkEdTcHqAMhXm4TiOCs_UkNHg4/s1600/mapa+del+imperio+romano.jpg" height="435" title="Extension of the Roman Empire in the I century A.D." width="640" /></a></div>
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- <b>At the command of the Galilean troops</b> was<b> Flavius Josephus</b>, who descended from a family of priests bound to the Israel kingdom. They were defending from the attack of the Roman legions at the impregnable <b>fortress </b> of <b>Yodfat</b> for some months, but they could no longer resist the Roman siege.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFnUZknIJqitGngnVUbfB-mDX-IfCOkhzIULtuhKOTf-jpk-UBlGphPaMMnfwZ2wUQOO5rf7mDtMp95pzxpuR1Mc6v4uVig-b8wX2fXYvmqWM0xc0_dc0sTAWF4WFh0jpcxyyWULR6ZEE/s1600/fortaleza+de+jotapata.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Illustration of the Yodfat siege" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFnUZknIJqitGngnVUbfB-mDX-IfCOkhzIULtuhKOTf-jpk-UBlGphPaMMnfwZ2wUQOO5rf7mDtMp95pzxpuR1Mc6v4uVig-b8wX2fXYvmqWM0xc0_dc0sTAWF4WFh0jpcxyyWULR6ZEE/s1600/fortaleza+de+jotapata.jpg" height="452" title="Illustration of the Yodfat siege" width="640" /></a><br />
<br />
<div style="text-align: justify;">
- When they realized that they would be captured, they decided to commit suicide rather than surrender dishonorably to the enemy. But Jewish law forbade them to suicide, so they adopted <b>a curious process</b> to carry out their purpose, which is known in Mathematics and Computing Science as <b><span style="color: blue; font-size: 130%;">the Josephus problem</span></b>.</div>
<br />
<div style="text-align: justify;">
- The 41 Jewish soldiers still alive <b>formed a circle</b>, such as your players do. In their case, they counted in threes, so that the every third soldier was killed by his partner on the right. And they continued the procedure until Flavius Josephus was the last person alive, so he had to commit suicide, as they've agreed. But he didn't do so, but <b>he gave himself to the Roman troops</b>.</div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSVytO6nQEePhfG047TIwtixNOFpAUlNOTqAYjClh7G4s1fk1O1YRncFO7StOfuXIOBNSb4Cy75pRshj3hY9H4FrndUpWM52zW6f93AmVHPtcSWLdBm5WlxXIHAdiwUWv0lxr4M14dUQw/s1600/Flavio+Josefo+y+Vespasiano.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="Busts of Flavius Josephus and Vespasian in front of the Roman Coliseum" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSVytO6nQEePhfG047TIwtixNOFpAUlNOTqAYjClh7G4s1fk1O1YRncFO7StOfuXIOBNSb4Cy75pRshj3hY9H4FrndUpWM52zW6f93AmVHPtcSWLdBm5WlxXIHAdiwUWv0lxr4M14dUQw/s1600/Flavio+Josefo+y+Vespasiano.jpg" height="191" title="Busts of Flavius Josephus and Vespasian in front of the Roman Coliseum" width="320" /></a><br />
<div style="text-align: justify;">
- He was sent to Rome as a <b>slave </b>of the imperial family Flavia, from which he took his name. Thanks to his academic education and his diplomatic skills, was finally <b>released by the emperor Vespasian</b>, and developed a broad work in Rome as historian and writer. Among his works, we can find the chronicle of this event that marked him for life.</div>
<br />
<div style="text-align: justify;">
- Some people think he saved his life by chance, or thanks to the Divine Providence, like he used to proclaim, but it seems clear that he knew where he had to stand to get rid of the fatal outcome.<br />
<br /></div>
<div style="text-align: justify;">
- And that's exactly what happens to your <b>captain</b>. She's the one that <b>points</b> who's saved and who leaves the circle. So she firmly knows well how to be the player who kicks the penalty.<br />
<br />
- Yes, but she's the first player to place herself in the circle. She doesn't know how the other players are going to stand in it, nor how many will be.</div>
<br />
<div style="text-align: justify;">
- That's true, but she's who <b>decides which player she starts from</b>. And there's the trick. She knows the correct position to stand in relation to the first player, so to be the last to go.</div>
<br />
<div style="text-align: justify;">
- Ah! And how?</div>
<br />
<div style="text-align: justify;">
- I'll explain it to you with an example of a group of 6 players. We're going to number them, <b>beginning by the top, and following clockwise</b>.</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsBVGd0B936X1KuRJy6sqXGsYPskgkWkszoGPXtIrfmIO3kjHk6sdREwHpayuWQwF6eljMv8w-ndT4mfF9Xp_51E47u_REqbiyAIBugWhljUFAcqhHF72arf5iBT3bzofQSkprTMX8wwQ/s1600/juego+con+6+jugadoras.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="We start from the player above, and we eliminate players who get a bear" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsBVGd0B936X1KuRJy6sqXGsYPskgkWkszoGPXtIrfmIO3kjHk6sdREwHpayuWQwF6eljMv8w-ndT4mfF9Xp_51E47u_REqbiyAIBugWhljUFAcqhHF72arf5iBT3bzofQSkprTMX8wwQ/s1600/juego+con+6+jugadoras.gif" height="400" title="We start from the player above, and we eliminate players who get a bear" width="370" /></a></div>
<br />
<br />
<div style="text-align: justify;">
- And now we go ahead and play your seal-bear game. We start from player number <b>1</b>, who's assigned a <b>seal</b>. We tag player number <b>2</b> as a <b>bear </b>and we remove her. Number <b>3</b> gets a <b>seal</b>, and <b>4</b> is named a <b>bear </b>and is counted out too.</div>
<br />
<div style="text-align: justify;">
- Now there are only 4 players left in the circle. We follow on with number <b>5</b> (<b>seal</b>), who saves, and we take out number<b> 6 </b>(<b>bear</b>). We keep on going around the circle, so that number <b>1</b> gets again a <b>seal</b>. Number <b>3</b> is a <b>bear</b>, so we eliminate her. Number <b>5</b> is a <b>seal</b> and stays, and now number <b>1</b> is a <b>bear </b>and goes out. It will be number <b>5</b> who will kick the penalty.</div>
<br />
<div style="text-align: justify;">
- We know that, with 6 players, the choice will be number 5. If we follow this procedure with different amounts of players, we get the following results:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYmSvbRwPAX23L5_jBn6FuY7Pr1pbv3asJXoGB25LGI2EpJw417w6fdCSxR7QKUPiJge2c9ZKVaE4nlDym4RMwT0WonOhSqXyu4DZfHIUslhfFg6LNlaZO1pUZwdy8aog157Rj0r7_DlI/s1600/table+of+winning+players+with+steps+of+2+by+2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Table of winning players with steps of 2 by 2" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYmSvbRwPAX23L5_jBn6FuY7Pr1pbv3asJXoGB25LGI2EpJw417w6fdCSxR7QKUPiJge2c9ZKVaE4nlDym4RMwT0WonOhSqXyu4DZfHIUslhfFg6LNlaZO1pUZwdy8aog157Rj0r7_DlI/s1600/table+of+winning+players+with+steps+of+2+by+2.JPG" title="Table of winning players with steps of 2 by 2" /></a></div>
</div>
<div style="text-align: justify;">
- Now I understand, the captain has <b>learnt all this <a href="https://oeis.org/A006257" target="_blank">sequence</a> of numbers</b>, to know where she has to place herself depending on how many of players form the circle, right?</div>
<br />
<div style="text-align: justify;">
- Maybe, but it's not necessary to learn them. We can <b>find a formula</b>.</div>
<br />
<div style="text-align: justify;">
- And how would it be?</div>
<br />
<div style="text-align: justify;">
- If we look at the table, we see that when the number of players is a <b>power of 2</b> (1, 2, 4, 8, 16, 32), the <b>player number 1</b>, from which we start to count, is the <b>winner</b> player.<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjO_s1uAoghyphenhyphenfdLbo2AYTVeK1dUBhcYxWQUytzxkoQx69LOtxKG7XLWLkK89ttHLhF0aMByhFgtbCy4-FWLUijVhf_wZZA_pPTHNWFeE7Zj9aY8EhJI1nfZv7CSswt02GsVvDj65Rcrtyg/s1600/table+of+winning+players+with+steps+of+2+by+2+and+powers+of+2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Table of winning players with steps of 2 by 2" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjO_s1uAoghyphenhyphenfdLbo2AYTVeK1dUBhcYxWQUytzxkoQx69LOtxKG7XLWLkK89ttHLhF0aMByhFgtbCy4-FWLUijVhf_wZZA_pPTHNWFeE7Zj9aY8EhJI1nfZv7CSswt02GsVvDj65Rcrtyg/s1600/table+of+winning+players+with+steps+of+2+by+2+and+powers+of+2.JPG" title="Table of winning players with steps of 2 by 2" /></a></div>
<br />
<div style="text-align: justify;">
- Yes, it's true.</div>
<br />
<div style="text-align: justify;">
- It's easy to understand if we study it a little. Imagine we've got 16 players (16 is a power of 2: 2<sup>4 </sup>= 16).</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigvA-l434CgOPJujhtI7Sa4n8p587HyBdI6Ji-xvtTOMlKAZygBuW2p8P_YQTEBd1nn_Uyh0P8mahTGn3ZTAGnx1Ld0W428oOeQVtTUhVmgGsXHMmuWR-wBN3RNc8r_qFiMdjRpGvNyLY/s1600/hexadecagono+2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Circle with 16 players" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigvA-l434CgOPJujhtI7Sa4n8p587HyBdI6Ji-xvtTOMlKAZygBuW2p8P_YQTEBd1nn_Uyh0P8mahTGn3ZTAGnx1Ld0W428oOeQVtTUhVmgGsXHMmuWR-wBN3RNc8r_qFiMdjRpGvNyLY/s1600/hexadecagono+2.gif" height="283" title="Circle with 16 players" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<div style="text-align: justify;">
- In the <b>first round</b> to the circle, all the players occupying <b>even positions</b> will be <b>eliminated</b>, and we'll have half of the initial players.<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhC-bV_NPXSQ32HlUpsY0kvIFEZKgnmTYl4-wLTI0XR5NDTBcmXIZqp5ren6UfDf1WaAQw1THwGTzY96taO-K292Rfg6IvQoIZQoq-JnOynaP88P__hqQ-Hz_TBXhQMuR24OeY4b6cP0zQ/s1600/first+round.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="First round of eliminations" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhC-bV_NPXSQ32HlUpsY0kvIFEZKgnmTYl4-wLTI0XR5NDTBcmXIZqp5ren6UfDf1WaAQw1THwGTzY96taO-K292Rfg6IvQoIZQoq-JnOynaP88P__hqQ-Hz_TBXhQMuR24OeY4b6cP0zQ/s1600/first+round.JPG" height="64" title="First round of eliminations" width="640" /></a></div>
<br />
<div style="text-align: justify;">
- We go on with the game, and all the players occupying even positions get removed again in the <b>second lap</b>.<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2D53K_WoChIRZaRene6gwO9iL9P4YcyPzPbgRtbH8HKVUtDaJ5-GQsu7UmTzfUzoQeh5E8yC8O7x2OUjKSaDokLrVDrO_pKJoS_USOjfxYVSTL73EA7MB7QUcKKF22o4ppuxs-cBa9eA/s1600/second+round.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Second round of eliminations" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2D53K_WoChIRZaRene6gwO9iL9P4YcyPzPbgRtbH8HKVUtDaJ5-GQsu7UmTzfUzoQeh5E8yC8O7x2OUjKSaDokLrVDrO_pKJoS_USOjfxYVSTL73EA7MB7QUcKKF22o4ppuxs-cBa9eA/s1600/second+round.JPG" height="84" title="Second round of eliminations" width="640" /></a></div>
<br />
<div style="text-align: justify;">
- We do the <b>third round</b> to the circle, and number 1 avoids the elimination once more. And finally, when there are only 2 players left (number 1 and number 9), the second one is also removed. So<b> number 1 is saved</b> in all the laps. And this will <b>always happen when the amount of players is a power of 2</b>, because after each pass there's always an even number of players.<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqr0DNITmDmJOlAAy0rSGLmg0F1Y9eWedrcQ_6qruW6HVMe7Voa22LmDjuKyiXYP-QZ3SCRvYYN1Q28-BpSq4NhxOpBhpOncLBC9gsMqXEYDOaTsw8DOFcRHPd0F70EX4Y50Kz1aBzCKY/s1600/third+and+last+round.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Third and last round of eliminations" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqr0DNITmDmJOlAAy0rSGLmg0F1Y9eWedrcQ_6qruW6HVMe7Voa22LmDjuKyiXYP-QZ3SCRvYYN1Q28-BpSq4NhxOpBhpOncLBC9gsMqXEYDOaTsw8DOFcRHPd0F70EX4Y50Kz1aBzCKY/s1600/third+and+last+round.JPG" height="158" title="Third and last round of eliminations" width="640" /></a></div>
<br />
<div style="text-align: justify;">
- Yes, but what can we do when the number of participants is not a power of 2?</div>
<br />
<div style="text-align: justify;">
- Then we must be skillful enough to calculate how many players have to <b>go out till we have a number of players that is a power of 2</b>, so that in this moment we'll be in the position that would be number 1 of the new circle formed after eliminating the last of the players who exceed the power of 2.</div>
<br />
- What did you say?<br />
<br />
<div style="text-align: justify;">
- Suppose that we have 11 players instead of 8. We must achieve to be in position number 1 when we're only 8, so we can apply the method we've seen before, since 8 is power of 2 (2<sup>3</sup> = 8).</div>
<br />
<div style="text-align: justify;">
- So we have to <b>count 3 'bears' backwards</b> from our position. And so we get to place ourselves in position number 7, so that we'll be number 1 of the new circle of 8 players formed after we take out the first 3 players in the group of 11.</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQxiHC-VMUuUUaGy4QwAut1ysYUDCTgWNLvlvtCRDkFVcpjIhIMEMB4GgQPB3qDbS1aXwzpKxuIMrSOJnRKns59evvVyifIGoP8RzZuBHbNualI8a8tTgfFJeYgsbB8k0SxAyKEUndrKU/s1600/endecagono.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="We have three bears back from position number 1" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQxiHC-VMUuUUaGy4QwAut1ysYUDCTgWNLvlvtCRDkFVcpjIhIMEMB4GgQPB3qDbS1aXwzpKxuIMrSOJnRKns59evvVyifIGoP8RzZuBHbNualI8a8tTgfFJeYgsbB8k0SxAyKEUndrKU/s1600/endecagono.gif" height="640" title="We have three bears back from position number 1" width="585" /></a></div>
<br />
<div style="text-align: justify;">
- If we want to generalize the procedure, we have to think on the <b>total number of players</b>, which we'll call <b>n</b> <span style="color: blue;">(in our case 11)</span>. Then we'll calculate <b>how much (k) exceeds</b> this number <b>the previous power of 2</b>:</div>
<br />
<div style="text-align: center;">
<span style="font-size: 130%;"><b>k = n - 2<sup>m</sup></b></span>, where <b>m</b> is a positive natural number such that <span style="font-size: 130%;"><b>2<sup>m</sup> < n < 2<sup>m+1</sup></b></span></div>
<br />
<div style="text-align: center;">
<span style="color: blue;">(in our example k = 11 - 2<sup>3</sup> = 3, with 2<sup>3</sup> < 11 < 2<sup>4</sup>)</span></div>
<br />
<div style="text-align: justify;">
- And our <b>optimal position</b> (<b>p</b>) in the circle is given by the formula: </div>
<br />
<div style="text-align: center;">
<span style="font-size: 130%;"><b>p = 2 · k + 1 </b></span>
<br />
<span style="font-size: 130%;"><b><br /></b></span>
<span style="color: blue;">(in our case p = 2 · 3 + 1 = 7)</span></div>
<br />
<div style="text-align: justify;">
- And that's what your captain does. She places herself on the seventh position of the circle, that is, she starts counting at the seventh player on her right.</div>
<br />
<div style="text-align: justify;">
- And don't you know an <b>easier way to calculate</b> it?<br />
<br />
- Yes, if you know well the <b>binary system</b>. You just have to convert the number of people within the circle into binary. Then you transfer the first 'one' to the last place ot the binary number, and you re-convert the resulting number to the decimal system. And so we get the place where you have to stand for being the winner.<br />
<br />
- For example, for 21 people, the optimal place is the eleventh, as you can see:</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigtHEXap8es6TXoiJOo-rTcbmDTHqzstR2n4QTGtxyctqLDPaSwcxkuol-0vElMMn6M8T7Ryr9TCOBN2lX_wfI9iB5YHg_lCEgSxn7Ikprs8j23fsIH0drT0DB9ooFf7IWw2IrI9yWoa4/s1600/conversi%C3%B3n+binaria.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Binary solution for the Josephus problem" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigtHEXap8es6TXoiJOo-rTcbmDTHqzstR2n4QTGtxyctqLDPaSwcxkuol-0vElMMn6M8T7Ryr9TCOBN2lX_wfI9iB5YHg_lCEgSxn7Ikprs8j23fsIH0drT0DB9ooFf7IWw2IrI9yWoa4/s1600/conversi%C3%B3n+binaria.gif" height="245" title="Binary solution for the Josephus problem" width="320" /></a></div>
<br />
<div style="text-align: justify;">
- I understand. And, how could we <b>avoid</b> this? What if, instead of eliminating every second player, they make it with a <a href="https://oeis.org/A054995" target="_blank">step of three</a> (seal-seal-bear-seal-seal-bear...), or with any other number?</div>
<div style="text-align: justify;">
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCCm6HvNORjGRfpT8PVRNyQDG0y0raqFYDm95BPxsZWZoFK9TVkFNsSTDJeXMANxoKVe4C41CEgNVq0lhZ9Ivc3YsskT8CmLB_Yl79_W7JbQKFF2fO1MStFtFnq2Ehqi43oX29YY5scFI/s1600/table+of+winning+players+with+steps+of+3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Table of winning players with steps of three" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCCm6HvNORjGRfpT8PVRNyQDG0y0raqFYDm95BPxsZWZoFK9TVkFNsSTDJeXMANxoKVe4C41CEgNVq0lhZ9Ivc3YsskT8CmLB_Yl79_W7JbQKFF2fO1MStFtFnq2Ehqi43oX29YY5scFI/s1600/table+of+winning+players+with+steps+of+3.JPG" height="170" title="Table of winning players with steps of three" width="640" /></a></div>
<br />
<div style="text-align: justify;">
<br />
- It'll be the same, although with higher numbers its calculation becomes more difficult. With a computer, we can find the solution in several websites like <a href="http://webspace.ship.edu/deensley/mathdl/Joseph.html" target="_blank">this</a> or <a href="http://webspace.ship.edu/deensley/mathdl/Joseph.html" target="_blank">this other</a>.</div>
<br />
<div style="text-align: justify;">
- And it's useless to entrust this task to another player, because sooner or later she will <b>learn the algorithm</b>.</div>
<br />
<div style="text-align: justify;">
- So one solution is to <b>change the system of choice</b>. You should choose the player who has the <b>best statistics on penalty shots</b>, although this may generate some problems, as happened in the story <a href="http://www.matifutbol.com/es/penalti.html" target="_blank">And now, who kicks the penalty?</a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://www.matifutbol.com/en/penalty.html" target="_blank"><img alt="And now, who should kick the penalty?" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigDLMINqCMIzRCtKXxlyyOBiVRB5v3IPYg9MMYWBh54FI0E61HV3fHl6zvFjt9dKssq3b4yZjlLBme_xUBCXPQjwXZ94rSfh9V853DDcUzQEi2PpBn_pLzI6Mf-QsyhOcV5PWh1B-GhGA/s1600/And+now+who+should+kick+the+penalty.jpg" height="425" title="And now, who should kick the penalty?" width="640" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: justify;">
- No, never. The seal-bear game is an old tradition in our land.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh09BYc9xumRZg_zW5RGvIQl6h2Tvc0uZCXkj5W0pjeWupE9QKxdXU7QOlhYdQjHKMdbJ9DDc-F-kBFz0B240he5622Il2O3k6pplOoh1ZANbf0UVOdnVz23po_fsDFgklRDipTDxIZARE/s1600/bandera+de+groenlandia,+osa+y+foca.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Greenlandic seal and bear" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh09BYc9xumRZg_zW5RGvIQl6h2Tvc0uZCXkj5W0pjeWupE9QKxdXU7QOlhYdQjHKMdbJ9DDc-F-kBFz0B240he5622Il2O3k6pplOoh1ZANbf0UVOdnVz23po_fsDFgklRDipTDxIZARE/s1600/bandera+de+groenlandia,+osa+y+foca.jpg" height="192" title="Greenlandic seal and bear" width="400" /></a></div>
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<div style="text-align: justify;">
- I can think on other solution. Listen:</div>
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<div style="text-align: justify;">
- After doing the first couple seal-bear, so that the captain can eliminate the worst penalty-taker, then she'll look at <b>the number</b> on the <b>player's back</b>, and use this number to reach the next ‘bear’ to be removed. Thus, if it's a 5, she'll sing 'seal-seal-seal-seal-<b>bear</b>'.</div>
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<div style="text-align: justify;">
- If the next excluded player wears a 7, she'll count 6 seals and 1 bear, and so on. That way it will be almost impossible to figure out who will be the last survivor, although the game is determined from the first choice.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE_GCHqEfHZ7O9DlBEC4X17zb9qw8GQ7cCCIkJ99JdwyoACukzpMF9j6meapmp8Nug2mPw58V5MIOH4war2G3gcAuprCizXjTkgPUY5-W9u7BikxUKkIe9o4WcvDY93KWZIN6fBw_Rfpo/s1600/table+of+elimination+for+7+players.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Searching the winner with steps depending on the shirt number" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE_GCHqEfHZ7O9DlBEC4X17zb9qw8GQ7cCCIkJ99JdwyoACukzpMF9j6meapmp8Nug2mPw58V5MIOH4war2G3gcAuprCizXjTkgPUY5-W9u7BikxUKkIe9o4WcvDY93KWZIN6fBw_Rfpo/s1600/table+of+elimination+for+7+players.JPG" height="294" title="Searching the winner with steps depending on the shirt number" width="640" /></a></div>
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<div style="text-align: justify;">
<br />
- But, what if she's able to calculate mentally and quickly who'll be the last one, because she <b>knows the squad numbers</b> of all her playmates?</div>
<br />
<div style="text-align: justify;">
- Even though, it will be <b>impossible </b>in many cases <b>to find a winning combination </b>that allows us to be always the last one.</div>
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<div style="text-align: justify;">
- Suppose, for example, that we put on the circle the players with shirt numbers 5, 6, 7, 8 and 9, in this order, and that the captain has number 5, let's see what would happen to our game with our new rules we've set.<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj31xjgNVHwVY0QjPWkRrzC4n7DVnur0g3TC1kkYwGC5NFR7wUHxEHrX-3nsf6mmwe-dlDzkg7xnWYxoMQk4z7I7jrgIhDmqRtgqfJl1REIR8iqeTKlxkuhCIzvPEpHIdUVX-H0OCqfPj4/s1600/table+of+winning+players+with+steps+depending+on+1st+eliminated+and+squad+numbers.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Table of winning players with steps depending on 1st eliminated and squad numbers" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj31xjgNVHwVY0QjPWkRrzC4n7DVnur0g3TC1kkYwGC5NFR7wUHxEHrX-3nsf6mmwe-dlDzkg7xnWYxoMQk4z7I7jrgIhDmqRtgqfJl1REIR8iqeTKlxkuhCIzvPEpHIdUVX-H0OCqfPj4/s1600/table+of+winning+players+with+steps+depending+on+1st+eliminated+and+squad+numbers.JPG" height="266" title="Table of winning players with steps depending on 1st eliminated and squad numbers" width="640" /></a></div>
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<div style="text-align: justify;">
<br />
- We can check that the last winner depends on who's removed first, but in neither case players 5 and 7 will be the last 'seals', and on the contrary, players 6 and 9 will be more likely to be the winners.</div>
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<div style="text-align: justify;">
- Well I like this variant you've invented. I'm sure that we'll finally prevent the captain kicks all the penalties, and we will <b>qualify for the Olympic Games</b>. Thank you very much for you help, Joe.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmL5nXF75iXQTTTEvj1m_xrI-wVsIOhIGzsspAJq0nTXoqyF55NUe82iKtsOyzHgLhuPWoPkE4BTEkiMHlhRdQY6C_EY92MHcKpouhbj4ccJKXOfPUPR6pY7ka7TDQUYXFIknrceoSQj4/s1600/groenlandia+en+los+juegos+ol%C3%ADmpicos.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Opening ceremony of the Olympic Games in Rio de Janeiro" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmL5nXF75iXQTTTEvj1m_xrI-wVsIOhIGzsspAJq0nTXoqyF55NUe82iKtsOyzHgLhuPWoPkE4BTEkiMHlhRdQY6C_EY92MHcKpouhbj4ccJKXOfPUPR6pY7ka7TDQUYXFIknrceoSQj4/s1600/groenlandia+en+los+juegos+ol%C3%ADmpicos.jpg" height="252" title="Opening ceremony of the Olympic Games in Rio de Janeiro" width="400" /></a></div>
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<span style="text-align: justify;">- I'm sure you will succeed, Pipaluk. Good luck for the match!</span><br />
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<a href="http://feedburner.google.com/fb/a/mailverify?uri=MatifutbolEsp-Matematicasyfutbol&loc=es_ES" style="margin-left: 1em; margin-right: 1em; text-align: center;" target="_blank"><img class="nopin" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQexSwE_cGJfSY0yLcf1h2I68kQZb4k-ZMNDWXQWF3YzTji1DiBnKbKed6bMlXPA44s1ZhDoxhnZLT22KyAGn5Gw0P-RtcKQBRZ11ffblNyMH16wjKLcQ_vh6HMmzMFNmphvpNI307EuA/s1600/banner+de+suscripcion+3.eng.gif" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px !important; margin-top: 20px;" width="640" /></a><br />
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If you're interested in <b>learning more </b>about this topic, you can visit any of this wonderful articles: <a href="http://www.cs.man.ac.uk/~shamsbaa/Josephus.pdf" target="_blank">Formulating the extended Josephus Problem</a>, <a href="https://www.bibnum.education.fr/sites/default/files/flavius-joseph-analysis-en.pdf" target="_blank">On the Josephus problem</a>, <a href="http://mathworld.wolfram.com/JosephusProblem.html" target="_blank">Josephus problem</a>, <a href="http://www.cut-the-knot.org/recurrence/r_solution.shtml" target="_blank">Josephus Flavius Problem: Recursive Solution</a>, <a href="http://www.exploringbinary.com/powers-of-two-in-the-josephus-problem/" target="_blank">Powers of two in the Josephus problem</a> <a href="http://user.math.uzh.ch/halbeisen/publications/pdf/jos.pdf" target="_blank">The Josephus problem</a>.</div>
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Don't forget to take a walk by the <a href="http://www.whitegroupmaths.com/2015/02/119th-carnival-of-mathematics.html" target="_blank"><span class="negrita">Carnival of Mathematics</span></a>. There you'll find lots of excellent math posts that you'll surely like too.
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Also, if you think of any other way to play the puisi-nanoq game in which the captain can't choose in advance who will win, you can share your idea by writing a comment below.<br />
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WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-85512993886735845762014-12-06T15:16:00.000+00:002014-12-16T22:51:53.196+00:00A very special greetings card<!-- Page resume -->
<span style="display: none;">A greetings card for the new year takes us into the history of the binary numbering system, the steganographic encryption methods, the numerology in Renaissance works and the origins of the sign of subtraction.
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8YX9idqkqYn-bSV5hdwXA12lvxaRPlzoDzNrt6gzs2SmQANNfdAbZFIyi3MKxxJUfr64oPG78jo4Wi2xyttdWzTMzhR-j2EL854w3b3UfVPo8EmiAHa5OvP17JFQfiOmGVkonhAk46pE/s1600/2+tuit+feliz+a%C3%B1o+nuevo+2015+Luis+Su%C3%A1rez.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8YX9idqkqYn-bSV5hdwXA12lvxaRPlzoDzNrt6gzs2SmQANNfdAbZFIyi3MKxxJUfr64oPG78jo4Wi2xyttdWzTMzhR-j2EL854w3b3UfVPo8EmiAHa5OvP17JFQfiOmGVkonhAk46pE/s1600/2+tuit+feliz+a%C3%B1o+nuevo+2015+Luis+Su%C3%A1rez.jpg" /></a>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRIJ6QR5OXgSyFmnM-BhV0hQULjYBnmaavRu9LRq0VV65qnHFz4cPOFKrets4knXD5IVNmHRXUaPqCfv8vtpe4OOVN5SFK8UYcKRbp0miVGs55HYAhyMl-jpQyfLIooB9GHwxxCmfF_h8/s1600/arbolito+2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRIJ6QR5OXgSyFmnM-BhV0hQULjYBnmaavRu9LRq0VV65qnHFz4cPOFKrets4knXD5IVNmHRXUaPqCfv8vtpe4OOVN5SFK8UYcKRbp0miVGs55HYAhyMl-jpQyfLIooB9GHwxxCmfF_h8/s1600/arbolito+2.jpg" /></a>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0sl2xgBsx7-2rlj08p89PjN0ZSLPeInmWtOSbXDFx3v84X71DBSe2R_YGxNmp4drxA3H0UapCukWFgVL_TpeDj415x374PXfEsoa-6b7ZsYB2srBCjd_cBH5YGXz_LsX0Gy9xLEwk4mg/s1600/19+Francis+Bacon.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0sl2xgBsx7-2rlj08p89PjN0ZSLPeInmWtOSbXDFx3v84X71DBSe2R_YGxNmp4drxA3H0UapCukWFgVL_TpeDj415x374PXfEsoa-6b7ZsYB2srBCjd_cBH5YGXz_LsX0Gy9xLEwk4mg/s1600/19+Francis+Bacon.jpg" /></a>
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<a href="http://matesifutbol.blogspot.com.es/2014/09/traspas-impossible-paradoxa-will-rogers.html" style="float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYxdbwexXkGFKRd5fqYMvnJMNLgIRhh0uM66729vdeZ-Abndi7cdXiv5TEkLcy_Twtq1EPBrZls7dwg-RMfCUTkMCeWhXgOfLVvWxIziVzGJUsKuv8wK0fBWurrnUfMl4HCLhWoAhNAlo/s1600/versiocatalana.gif" height="60" width="60" /></a>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhR0r-t9OmFsa0_Xtl-p8zmy8UChhdyr1qblR0VzxNyoSUELn1EGMHkKcB4NMbqxGcspxLUBvsxRTBAr8-bnbLNrqO0Oh-PYU-vHIqoIMQVDCvTkp_N6Uo566hri-V2i5NU4a82KJcDe5M/s1600/1+Galileo+Galilei+Mathematics+alphabet+God+writes+universe.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Galileo Galilei. Mathematics is the alphabet in which God has written the universe." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhR0r-t9OmFsa0_Xtl-p8zmy8UChhdyr1qblR0VzxNyoSUELn1EGMHkKcB4NMbqxGcspxLUBvsxRTBAr8-bnbLNrqO0Oh-PYU-vHIqoIMQVDCvTkp_N6Uo566hri-V2i5NU4a82KJcDe5M/s1600/1+Galileo+Galilei+Mathematics+alphabet+God+writes+universe.jpg" height="192" title="Galileo Galilei. Mathematics is the alphabet in which God has written the universe." width="640" /></a></div>
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(This post participates on the <a href="http://plus.maths.org/content/carnival-mathematics-117" target="_blank"><span class="negrita">117th edition</span></a> of the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="http://plus.maths.org/content/News" target="_blank"><span class="negrita unidas">Plus Magazine</span></a>)</div>
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<span style="font-family: Verdana, sans-serif;">FIRST HALF</span></div>
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<b>Luis Suarez</b> is back! The player confirms his return to the <b>Liverpool F.C.</b> at the end of the season.</div>
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It's been a frantic evening at the editorial office of the Liverpool Post, since Luis Suarez launched this <b>publication</b> on twitter: </div>
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<b>FeLiZ Año NueVO – haPPy NEW YeAr – BON aNY NOU <span style="font-size: x-small;">!</span></b></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8YX9idqkqYn-bSV5hdwXA12lvxaRPlzoDzNrt6gzs2SmQANNfdAbZFIyi3MKxxJUfr64oPG78jo4Wi2xyttdWzTMzhR-j2EL854w3b3UfVPo8EmiAHa5OvP17JFQfiOmGVkonhAk46pE/s1600/2+tuit+feliz+a%C3%B1o+nuevo+2015+Luis+Su%C3%A1rez.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Luis Suarez wishes a Happy New Year 2015." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8YX9idqkqYn-bSV5hdwXA12lvxaRPlzoDzNrt6gzs2SmQANNfdAbZFIyi3MKxxJUfr64oPG78jo4Wi2xyttdWzTMzhR-j2EL854w3b3UfVPo8EmiAHa5OvP17JFQfiOmGVkonhAk46pE/s1600/2+tuit+feliz+a%C3%B1o+nuevo+2015+Luis+Su%C3%A1rez.jpg" height="568" title="Luis Suarez wishes a Happy New Year 2015." width="640" /></a></div>
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10 minutes after he has posted the tweet for <b>wishing a happy new year</b>, the newspaper staff met urgently.</div>
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<div style="text-align: justify;">
- As you can see, on the <b>card</b> there are only 10 players, because Luis has crossed out his photo. This can only mean that <b>he wants to leave</b> his current team.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_w-Q4s3eBwl7qOpmkm1itMdgGigbz8YCefehYYvw0bjJfxmRjdmO80NYKyEIRQk6EFI25wPW75bkFj5MkFLAnRxjZpQONYDwSQNIRcc-Pxvu3Hf0iUiJrTtamTtKXJeAREhvQ4GhSkLA/s1600/3+Detalle+c%C3%ADrculos+jugadores+Barcelona+-+Detail+players+FCBarcelona+-+Detall+jugadors+Bar%C3%A7a.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Detail of FC Barcelona players." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_w-Q4s3eBwl7qOpmkm1itMdgGigbz8YCefehYYvw0bjJfxmRjdmO80NYKyEIRQk6EFI25wPW75bkFj5MkFLAnRxjZpQONYDwSQNIRcc-Pxvu3Hf0iUiJrTtamTtKXJeAREhvQ4GhSkLA/s1600/3+Detalle+c%C3%ADrculos+jugadores+Barcelona+-+Detail+players+FCBarcelona+-+Detall+jugadors+Bar%C3%A7a.jpg" height="100" title="Detail of FC Barcelona players." width="640" /></a></div>
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- Also, in the image we can see lots of <b>red </b>devils, and the name given to the Liverpool FC team is: '<b>the reds</b>'. You must remember that he played for this club during the last 2 seasons, and he felt very happy there.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBSXVB6ecB9IgErYTYJvA5PlPxuoVb42Kx2P0kyiT8toXRo8CTMfMjjbnYvAOefhtv-L1IRIQuoNO5k9kGt2-lLZG-8udli34iu92Fr5KpTS2pTLdyMq-Ej4g_lz2bNEApNsOtHmukNUs/s1600/4+1+-+515+Un+jugador+menos+el+15+de+mayo+-+One+player+less+on+May+15+-+Un+jugador+menys+el+15+de+maig.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="One player less on May 15." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBSXVB6ecB9IgErYTYJvA5PlPxuoVb42Kx2P0kyiT8toXRo8CTMfMjjbnYvAOefhtv-L1IRIQuoNO5k9kGt2-lLZG-8udli34iu92Fr5KpTS2pTLdyMq-Ej4g_lz2bNEApNsOtHmukNUs/s1600/4+1+-+515+Un+jugador+menos+el+15+de+mayo+-+One+player+less+on+May+15+-+Un+jugador+menys+el+15+de+maig.jpg" height="193" title="One player less on May 15." width="640" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8Z2QrSM-CDILdNkuOXS04DqGFtoAUx_qTZbyfYTHQiH1OSn7lU3giRZuhyB87DPllkU68xzVQnfrd5kpc682v1lF5tP8aUYuAJxKuYgmVB8eV-B3IlApIfbceehFEz2-RubtuNIWoVkA/s1600/5+N%C3%BAmero+uno+feliz+-+A+happy+number+one+-+Nombre+un+feli%C3%A7.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Number one with a happy face on it." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8Z2QrSM-CDILdNkuOXS04DqGFtoAUx_qTZbyfYTHQiH1OSn7lU3giRZuhyB87DPllkU68xzVQnfrd5kpc682v1lF5tP8aUYuAJxKuYgmVB8eV-B3IlApIfbceehFEz2-RubtuNIWoVkA/s1600/5+N%C3%BAmero+uno+feliz+-+A+happy+number+one+-+Nombre+un+feli%C3%A7.jpg" height="200" title="Number one with a happy face on it." width="147" /></a><br />
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- On the other hand, we've got the encrypted notation he has written: <b class="unidas">1 – 515</b>. If the <b>Spanish League</b> <b>ends</b> this year on 15/5, the message is clear: the <span class="unidas">F.C. Barcelona</span> will have <b>1</b> player <b>less</b> on <b>May</b> <b>15th</b>.</div>
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<div style="text-align: justify;">
- That's true! And he has titled his image: 'Happy New Year', that is, he's very <b>happy</b> because this coming year he will return back to his former club. We can also see a happy face in the number 1, which represents the player less at the end of the season, none other than himself.</div>
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<div style="text-align: justify;">
- In addition, he has chosen a representation of the <b>hell</b> for the background image. I guess that this year, with the disciplinary punishment and the adaptation to the new team, it has been something like a hell for him, and that's why he wants to return to Liverpool, where he was so well.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4cQltzFC2Ymcvl6LcySJnnuh57JPqCAMPd3q4owKzodrx51YOq9mVkP2buFGfS5W-sHbrBi4jY5h_OUboeXTfsOnHtFGgyKDYylgdM2f0zOOBnRLUwm0fvjrfwKyNQS9euLfIpQJnYa0/s1600/7+Probability+and+sunrise.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Probability and sunrise." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4cQltzFC2Ymcvl6LcySJnnuh57JPqCAMPd3q4owKzodrx51YOq9mVkP2buFGfS5W-sHbrBi4jY5h_OUboeXTfsOnHtFGgyKDYylgdM2f0zOOBnRLUwm0fvjrfwKyNQS9euLfIpQJnYa0/s1600/7+Probability+and+sunrise.jpg" height="240" title="Probability and sunrise." width="320" /></a></div>
- However, I think we should talk to him before <b> publishing the news,</b> in order to <b>confirm</b> everything we're saying.</div>
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<div style="text-align: justify;">
- We've tried it, but he's gone on holiday to a <a href="http://mathsball.blogspot.com.es/2014/03/probability-two-envelopes-paradox-exchange.html" target="_blank">remote Pacific island</a> with his friend Laudrup, and we can't get in touch with him, the same way as in the story <a href="http://mathsball.blogspot.com.es/2014/03/probability-two-envelopes-paradox-exchange.html" target="_blank">Probability and sunrise</a>.</div>
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<div style="text-align: justify;">
- Anyway, and considering that the message is made in a <b>mathematical language</b>, we should ask <b>Joe Vitruvius</b> if he thinks similarly as we do.<br />
<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVHV7qcd8myempj8jWq7_jBLZcTcQ__WUlqdroSy4AreUMcARCOlIvY5j8uT3h5TgMzrBWCONFbB3-UYiZ2Z7Q72lXEumjwQAlRgV9MVGhudgwoMlUxXuXHatqKSd2l-FJa55fjk4d7sI/s1600/8+Pepe+Vitruvio+-+Joe+Vitruvius+-+Pep+Vitruvi.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="The wonderful Joe Vitruvius" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVHV7qcd8myempj8jWq7_jBLZcTcQ__WUlqdroSy4AreUMcARCOlIvY5j8uT3h5TgMzrBWCONFbB3-UYiZ2Z7Q72lXEumjwQAlRgV9MVGhudgwoMlUxXuXHatqKSd2l-FJa55fjk4d7sI/s1600/8+Pepe+Vitruvio+-+Joe+Vitruvius+-+Pep+Vitruvi.jpg" height="200" title="The wonderful Joe Vitruvius" width="97" /></a></div>
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<div style="text-align: justify;">
- O.K. Have you got his telephone number?</div>
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- Yes, now I'm going to ask him for coming here with us.</div>
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<div style="text-align: justify;">
While Joe arrives to the editorial office, do you think that the interpretation of the message that the journalists of the Liverpool Post have thought is correct? </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" height="252" width="640" /></span></a></div>
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<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
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Finally <b>Jose Vitruvius</b> arrives to the central office of the newspaper.</div>
<div>
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<div>
- Hello everybody. What happens?</div>
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<div style="text-align: justify;">
- We've got a doubt with <b>Luis Suarez</b>. Let's have a look at the picture he has published on <b>twitter</b>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8YX9idqkqYn-bSV5hdwXA12lvxaRPlzoDzNrt6gzs2SmQANNfdAbZFIyi3MKxxJUfr64oPG78jo4Wi2xyttdWzTMzhR-j2EL854w3b3UfVPo8EmiAHa5OvP17JFQfiOmGVkonhAk46pE/s1600/2+tuit+feliz+a%C3%B1o+nuevo+2015+Luis+Su%C3%A1rez.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Luis Suarez's greetings card for the new year 2015.." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8YX9idqkqYn-bSV5hdwXA12lvxaRPlzoDzNrt6gzs2SmQANNfdAbZFIyi3MKxxJUfr64oPG78jo4Wi2xyttdWzTMzhR-j2EL854w3b3UfVPo8EmiAHa5OvP17JFQfiOmGVkonhAk46pE/s1600/2+tuit+feliz+a%C3%B1o+nuevo+2015+Luis+Su%C3%A1rez.jpg" height="568" title="Luis Suarez's greetings card for the new year 2015." width="640" /></a></div>
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<div style="text-align: justify;">
- He tells that on 15/5 the <b>FC Barcelona</b> team will have one less player, who will go to the <span class="unidas">Liverpool F.C.</span> to play with 'the reds', and he's very happy for this reason, because this season has been a <b>hell</b> for him.</div>
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<div style="text-align: justify;">
- Hmm... Have you spoken with the player?</div>
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- We can't contact with him. That's why we want to ask you if you can find out something else in these <b>numbers</b>, or if you confirm our analysis, before we publish the news
</div>
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<div style="text-align: justify;">
- Let's see, the message just says <b>Happy New Year</b> in three languages: Spanish, English and Catalan. I can't see what's the problem.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjARRMssHQQjttXB7FbIOisfHC09-PN87wIJ2tUyPIwH4XZD69PO2VyeUtol_jMYvz-a6mBruHxxCC9pRb69G0ChW0xqv5HNU3as-dVzRWIm4jvjfzgbZQJdby-mdd0L4X_yvlxvt0HsU0/s1600/9+Detalle+feliz+a%C3%B1o+nuevo+2015+-+happy+new+year+2015+-+bon+any+nou+2015.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Wishing a Happy New Year." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjARRMssHQQjttXB7FbIOisfHC09-PN87wIJ2tUyPIwH4XZD69PO2VyeUtol_jMYvz-a6mBruHxxCC9pRb69G0ChW0xqv5HNU3as-dVzRWIm4jvjfzgbZQJdby-mdd0L4X_yvlxvt0HsU0/s1600/9+Detalle+feliz+a%C3%B1o+nuevo+2015+-+happy+new+year+2015+-+bon+any+nou+2015.jpg" height="112" title="Wishing a Happy New Year." width="640" /></a><br />
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<div style="text-align: justify;">
- Yes, but look at the picture: <b>1-515</b>. One less on 5/15, the day that the Spanish League ends.</div>
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<div style="text-align: justify;">
- Well, it may mean something different.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0-aoa2xgUGkQEA66QN8kd32-qtLoTcKcBP75XEiQfLHyobvrSmctj0k-vn6kSbQwE6eZ4QLuPQhZhRMx5DtvGLcUxwYQTaY52yw2OlFGZYaUqXwSS6u2QUZjNepjb6VI1iz2tgKRP_Dc/s1600/6+Diablos+rojos+-+Red+devils+-+Diables+vermells.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="Red devils." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0-aoa2xgUGkQEA66QN8kd32-qtLoTcKcBP75XEiQfLHyobvrSmctj0k-vn6kSbQwE6eZ4QLuPQhZhRMx5DtvGLcUxwYQTaY52yw2OlFGZYaUqXwSS6u2QUZjNepjb6VI1iz2tgKRP_Dc/s1600/6+Diablos+rojos+-+Red+devils+-+Diables+vermells.gif" height="173" title="Red devils." width="200" /></a></div>
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<div style="text-align: justify;">
- And what can you tell me about the <b>red </b>demons? It's obvious that this is a clear reference to his <b>former club</b>, the Liverpool.</div>
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<div style="text-align: justify;">
- Yes, they're all <b>red</b>...</div>
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<div style="text-align: justify;">
- And the photos of his teammates, among which he has <b>crossed out</b> his face, this has no other possible explanation...</div>
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<div style="text-align: justify;">
- I don't know. I think there's something hidden on this message. Let me think for a moment... As you told me, the background of the image is the hell. That's why devils are all around. But it's not just a simple <b>hell</b>. It's a representation of the hell that <b>Dante </b>describes on his book <b>The Divine Comedy</b>. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6b-Cf9nHUuMbUjtYNVsKCqVi0JPCUTs87Z4izMLdmYfOfSFPEFA_t9tYmk_GQ-pj8IOk6IJVKBBVvnFQVq7zelRIzWu6Gv7QiF3vqrR1siuGgDV3N_z5XObXFnNxAuvL8-nhKQY4pP_4/s1600/16+Infierno+de+Botticelli+-+Botticelli's%2Binferno%2B-%2BInfern%2Bde%2BBotticelli.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Botticelli's hell." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6b-Cf9nHUuMbUjtYNVsKCqVi0JPCUTs87Z4izMLdmYfOfSFPEFA_t9tYmk_GQ-pj8IOk6IJVKBBVvnFQVq7zelRIzWu6Gv7QiF3vqrR1siuGgDV3N_z5XObXFnNxAuvL8-nhKQY4pP_4/s1600/16+Infierno+de+Botticelli+-+Botticelli's%2Binferno%2B-%2BInfern%2Bde%2BBotticelli.jpg" height="446" title="Botticelli's hell." width="640" /></a></div>
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<div style="text-align: justify;">
- Dante was an Italian poet (1265-1321) whose masterpiece is the Divine Comedy, considered as one of the fundamental works of the <b>universal literature</b>, in which we can study the transition from the Medieval to the Renaissance <b>world</b>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRG0CWFzIyzs5_x_32nV61nFs0YwY1qc1-gHh5pWnX9afUUhh5E4bED0VI06wPS4Sn6fpyo8eYG4sCE4zeBeXXrU6tHf57ki2a7CH8UBhp3vZ4J9g7G_V9tst3aGz5TcRE27i-XwMrC8A/s1600/10+Dante+Alighieri.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="Dante Alighieri." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRG0CWFzIyzs5_x_32nV61nFs0YwY1qc1-gHh5pWnX9afUUhh5E4bED0VI06wPS4Sn6fpyo8eYG4sCE4zeBeXXrU6tHf57ki2a7CH8UBhp3vZ4J9g7G_V9tst3aGz5TcRE27i-XwMrC8A/s1600/10+Dante+Alighieri.jpg" height="250" title="Dante Alighieri." width="320" /></a><br />
<div style="text-align: justify;">
- As for <b>number one</b>, I can get no clue from it for the moment, or perhaps I should say I get too many clues: it represents the unity, is a <a href="http://oeis.org/A007770" target="_blank">happy</a>, <a href="http://oeis.org/A000217" target="_blank">triangular </a>and <a href="http://oeis.org/A000326" target="_blank">pentagonal </a>number, appears 2 times in the <a href="http://oeis.org/A000045" target="_blank">Fibonacci </a>sequence, is a <a href="http://oeis.org/A000110" target="_blank">Bell </a>number, a <a href="http://oeis.org/A000108" target="_blank">Catalan </a>number, a <a href="http://oeis.org/A000142" target="_blank">factorial </a>and <a href="http://oeis.org/A000928" target="_blank">regular </a>number, It's also the neutral or identity element of the product, and is the most common digit on a set of data according to the Benford's Law... <b>Too much data</b>. There's also a face inside it, but I don't know how to interpret it for now. </div>
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<div style="text-align: justify;">
- And as regards the sign that separates number 1 from 515, I doubt if it's a <b>minus sign</b> or if it's a simple <b>dash </b>between both numbers.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKeZoKWoDL1F5mq6KAGWd_v7taNitQ0i6pRH_XSHTxdXoaMhxzCZv2bTiAFFFReNroYqJb87XS5LN3WT6Yqj-pY9w_XSdtBXv5SJCelHRRrBdYj09DB5F7HHgo_qznfDPMIxZxKWCC-IE/s1600/10+Signo+medieval+para+la+resta+-+Medieval+sign+for+subtraction+-+Signe+medieval+per+a+la+resta.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Medieval sign for representing the subtraction." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKeZoKWoDL1F5mq6KAGWd_v7taNitQ0i6pRH_XSHTxdXoaMhxzCZv2bTiAFFFReNroYqJb87XS5LN3WT6Yqj-pY9w_XSdtBXv5SJCelHRRrBdYj09DB5F7HHgo_qznfDPMIxZxKWCC-IE/s1600/10+Signo+medieval+para+la+resta+-+Medieval+sign+for+subtraction+-+Signe+medieval+per+a+la+resta.jpg" height="200" title="Medieval sign for representing the subtraction." width="200" /></a></div>
<div style="text-align: justify;">
- In fact, at the time when Dante wrote the Divine Comedy, nobody used that <b>symbol </b>for representing the <b>subtraction</b>, but the <b>letter M</b> (for the Latin word <i>minus</i>), to which later was added a line written over it to avoid confusions. It was in 1518 when Henricus Grammateus used it for the first time in solitary, eliminating the m, as the symbol of <b>subtraction</b>.</div>
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<div style="text-align: justify;">
- On the other hand, we've got <b>number 515</b>. You say that it refers to the expression <b>5/15</b>, that is, the 15th day of the 5th month, the day when the Spanish League ends. But I think there should be a separator between the 5 and the 15. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBSXVB6ecB9IgErYTYJvA5PlPxuoVb42Kx2P0kyiT8toXRo8CTMfMjjbnYvAOefhtv-L1IRIQuoNO5k9kGt2-lLZG-8udli34iu92Fr5KpTS2pTLdyMq-Ej4g_lz2bNEApNsOtHmukNUs/s1600/4+1+-+515+Un+jugador+menos+el+15+de+mayo+-+One+player+less+on+May+15+-+Un+jugador+menys+el+15+de+maig.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="On player less on May 15." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBSXVB6ecB9IgErYTYJvA5PlPxuoVb42Kx2P0kyiT8toXRo8CTMfMjjbnYvAOefhtv-L1IRIQuoNO5k9kGt2-lLZG-8udli34iu92Fr5KpTS2pTLdyMq-Ej4g_lz2bNEApNsOtHmukNUs/s1600/4+1+-+515+Un+jugador+menos+el+15+de+mayo+-+One+player+less+on+May+15+-+Un+jugador+menys+el+15+de+maig.jpg" height="193" title="On player less on May 15." width="640" /></a><br />
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<div>
<div style="text-align: justify;">
- I think we should focus on this number, the <b>515</b>, because it can give us more <b>clues </b>about what he wants to tell with his <b>message</b>. </div>
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<div>
<div style="text-align: justify;">
- As we said, the picture that is on the background of the tweet is a <b>Botticelli</b>'s painting, inspired by ‘The Divine Comedy’ of Dante. As a good man from Renaissance, Dante, was a writer, but also a good fan of <b>mathematics</b>. Thus, in his work grants a great <b>symbolic value</b> to certain pairs of numbers, such as 3 and 9, 7 and 22, or 515 and 666. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitm70pcfQIigEaiKG2BzRxHwhCa2X3TMwZP3A3HWtBf2NoNpn1JFUMIsOuqU3YaD-lNZcj9PgLDpbz-V39DBhLjAoRtYOH30-JMpf_46h51ThxBe_8OJieNZGpSZv0dY8rwaZbmcWABXE/s1600/12+3+versus+9.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="3 versus 9" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitm70pcfQIigEaiKG2BzRxHwhCa2X3TMwZP3A3HWtBf2NoNpn1JFUMIsOuqU3YaD-lNZcj9PgLDpbz-V39DBhLjAoRtYOH30-JMpf_46h51ThxBe_8OJieNZGpSZv0dY8rwaZbmcWABXE/s1600/12+3+versus+9.jpg" height="127" title="3 versus 9" width="400" /></a></div>
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<div style="text-align: justify;">
- We find <b>number 3</b> in many aspects of the book. It's a number that refers to the Holy Trinity, and that means balance and stability. This way, the poem is composed of tercets, articulated in <b>3 chapters: Hell, Purgatory and Paradise</b>.</div>
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<div>
<div style="text-align: justify;">
- Meanwhile, <b>number 9</b> is the number Dante assigns to <b>Beatrice</b>. 9 is the square of 3, and that way he enhances the beauty and importance of his beloved. In the work, this figure appears as the number of <b>divisions </b>or circles of the <b>Hell, </b>or the amount of different <b>heavens </b>of <b>Paradise</b>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIq8f57q8Woo5Dsoe-0M0VJHE-FU6Gwo6L1m160fLHjOVVk4Uc4WQ8w3Q18hmQxA0eiWBbrhhNRfY9UpdMnwS69N93UoU0oL56I5QpZmQEws75p3ZnxMuZPCxv1O8fga4tFzP0uTHZIos/s1600/13+7+versus+22.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="7 versus 22" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIq8f57q8Woo5Dsoe-0M0VJHE-FU6Gwo6L1m160fLHjOVVk4Uc4WQ8w3Q18hmQxA0eiWBbrhhNRfY9UpdMnwS69N93UoU0oL56I5QpZmQEws75p3ZnxMuZPCxv1O8fga4tFzP0uTHZIos/s1600/13+7+versus+22.jpg" height="127" title="7 versus 22" width="400" /></a></div>
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<div>
<div style="text-align: justify;">
- The <b>number 7</b> is a <b>sacred </b> number (seven liberal arts, seven colors of the rainbow, seven days of the week), which he uses to name the <b>divisions </b>of the <b>Purgatory</b>, while the <b>number 22</b> is a number closely related to the 7, since the result of the ratio <b>22/7</b> is very close to <b>number pi</b>. </div>
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<div>
<div style="text-align: justify;">
- So with these 2 numbers that Dante refers to the <b>circle</b>, the <b>most perfect figure</b> for the Pythagoreans. That way, he defines the heaven and the hell as 2 worlds formed by concentric circles.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPtji7-R5n2I6loB9x1-DLOB0Pia0TOFjri0d5mp4vNoupBzyyQbcOi2MAn90ZHFlQfYCXycqI42c0mM2-F-zVgUEqQpK6MQPMjPPyDtwLow64FknLxXZQ-43SzBL7DG5J2-oGicTYNBI/s1600/14+515+versus+666.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="515 versus 666" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPtji7-R5n2I6loB9x1-DLOB0Pia0TOFjri0d5mp4vNoupBzyyQbcOi2MAn90ZHFlQfYCXycqI42c0mM2-F-zVgUEqQpK6MQPMjPPyDtwLow64FknLxXZQ-43SzBL7DG5J2-oGicTYNBI/s1600/14+515+versus+666.jpg" height="127" title="515 versus 666" width="400" /></a></div>
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<div style="text-align: justify;">
- And finally we arrive to the most interesting numbers. Everybody knows <b>number 666</b> is considered the <b>number of the Beast </b>or Devil in the <b>Apocalypse</b>. His antagonist or enemy must be the 'Veltro', or the one <b>sent by God</b>, who is marked by <b>number 515</b> according to the prediction Beatrice makes in the book: "Un cinquecento diece e cinque, messo di Dio..." </div>
</div>
<div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMVaf0tmqifR82aLwrQgx5ikOn_fViylMK1VgG7YP5P9jCdICOKb_Bd9n4SjV3KOIs-XFdyvIJS_yi5dfR3X4NuPAJg0Zg-aieWCgjbTf2QaxpBuKeWt-VY8osNQ90OOXf4ZC4TbxiLew/s1600/15+El+Duque+luchando+contra+la+Bestia+-+The+Duke+fighting+the+Beast+-+El+Duc+lluitat+contra+la+B%C3%A8stia.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="The Duke fighting the Beast." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMVaf0tmqifR82aLwrQgx5ikOn_fViylMK1VgG7YP5P9jCdICOKb_Bd9n4SjV3KOIs-XFdyvIJS_yi5dfR3X4NuPAJg0Zg-aieWCgjbTf2QaxpBuKeWt-VY8osNQ90OOXf4ZC4TbxiLew/s1600/15+El+Duque+luchando+contra+la+Bestia+-+The+Duke+fighting+the+Beast+-+El+Duc+lluitat+contra+la+B%C3%A8stia.jpg" height="213" title="The Duke fighting the Beast." width="320" /></a><br />
<div style="text-align: justify;">
- If we write 515 en <b>Roman numerals</b> we've got DXV, and if we change the order of the letters we get <b>DVX</b>, the Duke in command of the Holy Roman Empire, who will be chosen to fight the evil represented by the 666. </div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- Therefore, number 515 must be taken as if it was a positive force, opposed to the hell of the picture's background. This may mean that Luis is <b>overcoming </b>the <b>bad times</b> of the past.</div>
</div>
<div>
<br />
- So <b>everything </b>seems to be <b>connected</b>, but we still haven't found what Luis Suarez wants to tell with his message.</div>
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<br /></div>
<div>
<div style="text-align: justify;">
- The one link I see between this number and the new year 2015, is that we can get this number by adding 15 hundreds to 515, but there's no longer relationship between them beyond the concentration of ones and fives in the operation.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFnR9fxieSwhVGvq0msUk4BKKFSej8wCakC-5tVDR4e2e9999ZIX3FUeRR378IDT5UiCWfhSIdGJSQG4urppNchTDQi4HIHCcuSq8rfnzqYpKiv3HU14FYhScKGVS7bMsKircZEttOdjE/s1600/16+1500+++515+=+2015.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="1500 + 515 = 2015" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFnR9fxieSwhVGvq0msUk4BKKFSej8wCakC-5tVDR4e2e9999ZIX3FUeRR378IDT5UiCWfhSIdGJSQG4urppNchTDQi4HIHCcuSq8rfnzqYpKiv3HU14FYhScKGVS7bMsKircZEttOdjE/s1600/16+1500+++515+=+2015.gif" height="77" title="1500 + 515 = 2015" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="text-align: justify;">
- And if we study the <b>properties </b>of the <b>number 515</b>, we get that it's a composite number, with 4 divisors (1, 5, 103 and 515), so it's not a prime number. It's neither a <a href="http://oeis.org/A000045" target="_blank">Fibonacci </a>number, nor a <a href="http://oeis.org/A000110" target="_blank">Bell </a>, <a href="http://oeis.org/A000108" target="_blank">Catalan </a>or <a href="http://oeis.org/A000032" target="_blank">Lucas </a>number. It's not <a href="http://oeis.org/A000396" target="_blank">perfect</a>, nor <a href="http://oeis.org/A005101" target="_blank">abundant</a>, nor <a href="http://oeis.org/A000124" target="_blank">polygonal</a>. But it's <a href="http://oeis.org/A005100" target="_blank">deficient</a>, <a href="http://oeis.org/A046075" target="_blank">undulated</a>, <a href="https://oeis.org/A000069" target="_blank">odious</a>, <a href="http://oeis.org/A002113" target="_blank">palindromic</a>, <a href="http://oeis.org/A005117" target="_blank">square-free</a>... and <a href="http://oeis.org/A031177" target="_blank">unhappy</a>. </div>
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<div>
<div style="text-align: justify;">
- As a curiosity, we've got that 515 is the <b>addition of 9 consecutive prime numbers</b> (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73), that the <b>Moon</b>'s orbital plane<b> </b>is <b>inclined </b>at 5.15 degrees to the plane of Earth's orbit round the Sun,<span style="background-color: white; color: #545454; font-family: arial, sans-serif; font-size: x-small; line-height: 18.2000007629395px; text-align: left;">.</span>and that if we multiply <b>number pi</b> by 0.515 we get approximately <b>the golden number 'phi' </b>(1,618), that you can see on my t-shirt, and that is so related to <b>Fibonacci sequence</b>.</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtbvttYpEiY34wkLI5n_DjNle5nnyfPF9K0uJoTH2qQ-TDSGACQb16snN0fpaXo7eIlttWDU9tlx7g-gqhyphenhyphenHO_vxgUMi7YJUXf3yX7xiwmzgqutguBi7gwnLBG99iTr8kkkW1bWN0Edxg/s1600/17+pi+por+0.515+igual+a+phi+=+pi+per+0.515+equals+phi+-+pi+per+0.515+igual+a+phi.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="pi multiplied by 0.515 approximately equals phi" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtbvttYpEiY34wkLI5n_DjNle5nnyfPF9K0uJoTH2qQ-TDSGACQb16snN0fpaXo7eIlttWDU9tlx7g-gqhyphenhyphenHO_vxgUMi7YJUXf3yX7xiwmzgqutguBi7gwnLBG99iTr8kkkW1bWN0Edxg/s1600/17+pi+por+0.515+igual+a+phi+=+pi+per+0.515+equals+phi+-+pi+per+0.515+igual+a+phi.gif" height="87" title="pi multiplied by 0.515 approximately equals phi" width="400" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVHV7qcd8myempj8jWq7_jBLZcTcQ__WUlqdroSy4AreUMcARCOlIvY5j8uT3h5TgMzrBWCONFbB3-UYiZ2Z7Q72lXEumjwQAlRgV9MVGhudgwoMlUxXuXHatqKSd2l-FJa55fjk4d7sI/s1600/8+Pepe+Vitruvio+-+Joe+Vitruvius+-+Pep+Vitruvi.jpg" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: 1em;"><img alt="The magnificent Joe Vitruvius." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVHV7qcd8myempj8jWq7_jBLZcTcQ__WUlqdroSy4AreUMcARCOlIvY5j8uT3h5TgMzrBWCONFbB3-UYiZ2Z7Q72lXEumjwQAlRgV9MVGhudgwoMlUxXuXHatqKSd2l-FJa55fjk4d7sI/s1600/8+Pepe+Vitruvio+-+Joe+Vitruvius+-+Pep+Vitruvi.jpg" height="200" title="The magnificent Joe Vitruvius." width="97" /></a></div>
<div style="text-align: justify;">
- And if we analyze the <b>number 2015</b>, the new year, we've got that it's a composite number, with 8 divisors: 1, 5, 13, 31, 65, 155, 403 and 2015. It's a <a href="http://oeis.org/A005100" target="_blank">deficient</a>, <a href="https://oeis.org/A001969" target="_blank">evil</a>, <a href="http://oeis.org/A031177" target="_blank">unhappy </a>and <a href="http://oeis.org/A005117" target="_blank">square-free</a> number, but it's not a <a href="http://oeis.org/A000045" target="_blank">Fibonacci </a>number, nor <a href="http://oeis.org/A000110" target="_blank">Bell</a>, nor <a href="http://oeis.org/A000108" target="_blank">Catalan</a>, nor <a href="http://oeis.org/A000928" target="_blank">regular</a>, nor <a href="http://oeis.org/A000396" target="_blank">perfect</a>, nor <a href="http://oeis.org/A000124" target="_blank">polygonal </a>number...</div>
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<div style="text-align: justify;">
<br /></div>
<div>
<div style="text-align: justify;">
- It seems that we won't get any clue from these two numbers, either.</div>
</div>
<div>
<br /></div>
<div>
<div style="text-align: justify;">
- On the other hand, if we focus on the face of one of the devils, we can see that it's <b>Francis Bacon</b>, an amazing English <b>philosopher</b>, politician, lawyer and scientist (1561-1626). </div>
</div>
<div style="text-align: justify;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtsQoSyd6hmz7EkNMFfM-VUc7eRZTaOUz2UOb0GehI_vRyqNTHWNjtKpwRIB_V3zjgGxPBicalM2C4yBW0F6hpICUEKAXspj-kw9hKecFNTRflW9J7UX8z3GB_J_yh0Wvo8r1iK42pfG4/s1600/18+Francis+Bacon.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Red devil with Francis Bacon`s face." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtsQoSyd6hmz7EkNMFfM-VUc7eRZTaOUz2UOb0GehI_vRyqNTHWNjtKpwRIB_V3zjgGxPBicalM2C4yBW0F6hpICUEKAXspj-kw9hKecFNTRflW9J7UX8z3GB_J_yh0Wvo8r1iK42pfG4/s1600/18+Francis+Bacon.jpg" title="Red devil with Francis Bacon`s face." /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0sl2xgBsx7-2rlj08p89PjN0ZSLPeInmWtOSbXDFx3v84X71DBSe2R_YGxNmp4drxA3H0UapCukWFgVL_TpeDj415x374PXfEsoa-6b7ZsYB2srBCjd_cBH5YGXz_LsX0Gy9xLEwk4mg/s1600/19+Francis+Bacon.jpg" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: 1em;"><img alt="The philosopher Francis Bacon." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0sl2xgBsx7-2rlj08p89PjN0ZSLPeInmWtOSbXDFx3v84X71DBSe2R_YGxNmp4drxA3H0UapCukWFgVL_TpeDj415x374PXfEsoa-6b7ZsYB2srBCjd_cBH5YGXz_LsX0Gy9xLEwk4mg/s1600/19+Francis+Bacon.jpg" height="258" title="The philosopher Francis Bacon." width="320" /></a></div>
<div style="text-align: justify;">
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<div>
<div style="text-align: justify;">
- His work <b>Novum Organum</b> was a starting point for modern science, with the development of the <b>empirical method</b> based on the observation and experimentation.</div>
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<div>
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<div>
<div style="text-align: justify;">
- Yes, we've already seen it. We find no relationship between the philosopher and Luis Suarez, but there's some connection with other <b>Francis Bacon</b> (1909-1992), the <b>modern painter</b> of the 20th century, whose paintings are exhibited at the museum of Liverpool, and is so admired by Luis.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhadVXxXfZF0-FIU6a_hV0Pq-0UUw63O7nnSISItiirwh1kaeymSwfR7NZ1yiGH4s649seCEeGyxRVN8ZheOKTIr0poBqbpW_UHCC_CPYcNsMeKU3BybSo8BdEV42WMAN4Nc6tuTpNdRyQ/s1600/20+Cuadros+de+Francis+Bacon+-+Pictures+of+Francis+Bacon+-+Quadres+de+Francis+Bacon.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Francis Bacon's paintings." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhadVXxXfZF0-FIU6a_hV0Pq-0UUw63O7nnSISItiirwh1kaeymSwfR7NZ1yiGH4s649seCEeGyxRVN8ZheOKTIr0poBqbpW_UHCC_CPYcNsMeKU3BybSo8BdEV42WMAN4Nc6tuTpNdRyQ/s1600/20+Cuadros+de+Francis+Bacon+-+Pictures+of+Francis+Bacon+-+Quadres+de+Francis+Bacon.jpg" height="142" title="Francis Bacon's paintings." width="400" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFtk32rAbG_YyoOFtzymjtpGBl5glvt9hsIYmEkzkGxfW4VpR8zGiWHIQm9N8iasm7lqc99k1BGCJ26PlHLyj2h-KGZQGsBFYbYHTLn0MC-619c0-cIrrKnhZjkxaADAk6QlNxlvDK8k0/s1600/21+Love+is+the+devil.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFtk32rAbG_YyoOFtzymjtpGBl5glvt9hsIYmEkzkGxfW4VpR8zGiWHIQm9N8iasm7lqc99k1BGCJ26PlHLyj2h-KGZQGsBFYbYHTLn0MC-619c0-cIrrKnhZjkxaADAk6QlNxlvDK8k0/s1600/21+Love+is+the+devil.jpg" height="200" width="141" /></a></div>
<br />
<div style="text-align: justify;">
- Certainly, we think that it's a clear sign that he wants to return to this city to visit more often Bacon's collections. Also, there's a movie about the life of this author, titled: ‘<b>Love is the devil</b>’. So it's obvious that he still loves his last club, the Liverpool FC.</div>
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<div>
<br />
- What have in common the <b>Liverpool </b>team and the devil?</div>
<div>
<br /></div>
<div>
<div style="text-align: justify;">
- Well, the authentic team called 'red devils' is Manchester United. But Liverpool FC is also related to them. Just before playing a match against the Anderlecht team, the <b>coach </b>Bill Shankly had the idea of dressing his players completely in red, to make them <b>look like devils </b>to <b>frighten more</b> their opponents. Thus they changed forever their uniform, and since then the club is known by the name of ‘<b>the reds</b>’. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQaJSNJrH1nBrIyfX6DzCK2J4WyMgBc6cGtSGFUH4lgyUOQo9-YK4uR1T8WlnirJY0SYV0n2FwwfezsHjKAEE1XSXhBRYJZQLyXMjjx7WEA7Q0gRf2Gc7mbskh-_uwWQF59x-lhfaTaK0/s1600/22+Cambio+equipaci%C3%B3n+Liverpool+FC+-+Changing+uniform+Liverpool+FC+-+Canvi+equipaci%C3%B3+Liverpool+FC.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Uniform change of Liverpool FC" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQaJSNJrH1nBrIyfX6DzCK2J4WyMgBc6cGtSGFUH4lgyUOQo9-YK4uR1T8WlnirJY0SYV0n2FwwfezsHjKAEE1XSXhBRYJZQLyXMjjx7WEA7Q0gRf2Gc7mbskh-_uwWQF59x-lhfaTaK0/s1600/22+Cambio+equipaci%C3%B3n+Liverpool+FC+-+Changing+uniform+Liverpool+FC+-+Canvi+equipaci%C3%B3+Liverpool+FC.jpg" height="270" title="Uniform change of Liverpool FC" width="400" /></a></div>
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<div style="text-align: justify;">
- I understand. But the thing I can't comprehend is the meaning of the circles on the numbers: 2 <b>filled circles</b> on the fives, and an <b>empty </b>circle on the one...</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBSXVB6ecB9IgErYTYJvA5PlPxuoVb42Kx2P0kyiT8toXRo8CTMfMjjbnYvAOefhtv-L1IRIQuoNO5k9kGt2-lLZG-8udli34iu92Fr5KpTS2pTLdyMq-Ej4g_lz2bNEApNsOtHmukNUs/s1600/4+1+-+515+Un+jugador+menos+el+15+de+mayo+-+One+player+less+on+May+15+-+Un+jugador+menys+el+15+de+maig.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="One less player on May 15." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBSXVB6ecB9IgErYTYJvA5PlPxuoVb42Kx2P0kyiT8toXRo8CTMfMjjbnYvAOefhtv-L1IRIQuoNO5k9kGt2-lLZG-8udli34iu92Fr5KpTS2pTLdyMq-Ej4g_lz2bNEApNsOtHmukNUs/s1600/4+1+-+515+Un+jugador+menos+el+15+de+mayo+-+One+player+less+on+May+15+-+Un+jugador+menys+el+15+de+maig.jpg" height="193" title="One less player on May 15." width="640" /></a><br />
<br /></div>
<div>
<div style="text-align: justify;">
- Or maybe yes. I think that this is the key to understand what's all this about: Francis Bacon, evil numbers, devils...</div>
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<div>
<div style="text-align: justify;">
- Have you noticed the <b>layout </b>of the players on the top of the card? There are five visible faces, one crossed out face, and another five visible faces, right?</div>
</div>
<div>
<br /></div>
<div>
- That's true.<br />
<br /></div>
<div>
<div style="text-align: justify;">
- Just the same layout that he repeats with the figures of number <b>515 </b>and with the circles set over the numbers 5, 1 and 5. What if we draw 5 full circles, an empty circle, and 5 other full circles, and we compare them with the players' photos?</div>
</div>
<div>
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOT4qPfdF3RImQQrg3xoa6tKV9g_oj61v-9UU1KvekhQz4nkuT3Ckql0tBAUFOSaBzMdoffMX8dVBAYf1OfblGhebxCofB9qfAPnC36EpQEbxgEcYVhGjSiDMl1u4zCmkMV0zfoI0n0VU/s1600/24+C%C3%ADrculos+y+jugadores+-+Circles+and+players+-+Cercles+i+jugadors.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Correspondence of circles and players." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOT4qPfdF3RImQQrg3xoa6tKV9g_oj61v-9UU1KvekhQz4nkuT3Ckql0tBAUFOSaBzMdoffMX8dVBAYf1OfblGhebxCofB9qfAPnC36EpQEbxgEcYVhGjSiDMl1u4zCmkMV0zfoI0n0VU/s1600/24+C%C3%ADrculos+y+jugadores+-+Circles+and+players+-+Cercles+i+jugadors.jpg" height="204" title="Correspondence of circles and players." width="640" /></a><br />
<br /></div>
<div>
- What a strange coincidence!</div>
<div>
<br /></div>
<div>
<div style="text-align: justify;">
- Well, I said before that Francis Bacon was an outstanding philosopher. But he was also the inventor of the <b>binary encoding system</b>, not in its present notation, with ones and zeros, but with the letters A and B, known by the name of <b>Bacon's Cipher</b>, described in his book <i>The Advancement of Learning</i>.</div>
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<div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggFS7Eveff11APGGG3C-axwzPkyHHaqs640_C8ysgvI8d0TvAHXzWDrUSaQInvFU43r7zt7jfPgAruq7WngJEevP-nVjvmy0JG1DdrTJESPWiSD5oGIgRUJXC9Rm96XV2UYqwcVy-wS30/s1600/25+C%C3%B3digo+de+Francis+Bacon+-+Francis+Bacon's%2Bcode%2B-%2BCodi%2Bde%2BFrancis%2BBacon.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: justify;"><img alt="Francis Bacon's Cypher." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggFS7Eveff11APGGG3C-axwzPkyHHaqs640_C8ysgvI8d0TvAHXzWDrUSaQInvFU43r7zt7jfPgAruq7WngJEevP-nVjvmy0JG1DdrTJESPWiSD5oGIgRUJXC9Rm96XV2UYqwcVy-wS30/s1600/25+C%C3%B3digo+de+Francis+Bacon+-+Francis+Bacon's%2Bcode%2B-%2BCodi%2Bde%2BFrancis%2BBacon.jpg" height="320" title="Francis Bacon Cypher." width="190" /></a><br />
<div style="text-align: justify;">
- You surely know that the <b>binary </b> system is a <b>system of numeration</b> in which numbers are written only using two types of figures: ones and zeros. We can say that all our today world, with the boom of the digital technology, consists nowadays of an infinite sequence of ones and zeros.</div>
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<div style="text-align: justify;">
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<div style="text-align: justify;">
- And that's what Suarez has made in this message: a <b>binary greetings card</b>. Let's transcribe the <b>number 2015</b>, which corresponds to the new year, to its expression in the <b>binary system</b>: </div>
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<div style="text-align: center;">
<span style="color: red; font-size: 130%;"><b>2015 <sub style="font-size: 60%;">10</sub> = 11111011111 <sub style="font-size: 60%;">2</sub></b></span></div>
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- What do you think? Haven't you seen before this distribution of ones and zeros: a row of 5 ones, a zero, and another row of 5 ones again?</div>
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<div style="text-align: justify;">
- That's right. With the photos of his teammates, and with number 515, Luis is trying to <b>represent the new year</b> <b>2015</b>.</div>
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<div style="text-align: justify;">
- In addition, it's a very nice binary number, because it's a <b>palindromic binary number</b> (you can read it the same forwards as backwards), the largest you can write with only 11 digits. The next one will have 12, and will be the 100000000001 (number 2049 in decimal system).</div>
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<div>
- All right. And does it mean the number one before the 515?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjco2r8rveFPRHeN1oAoo6vU2M7m5zqWQiCJZs4xCvB3iIfgDZIt8h_m5y9Jqd_ajeXzJS_TTHFE1AdXtk-dlC3qYPtZb2dbvWnow0BCQPmfb-5pBNOrXppB2UQYL_yEWHcilY9DS7fdX0/s1600/26+Football+and+happiness.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Football and happiness." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjco2r8rveFPRHeN1oAoo6vU2M7m5zqWQiCJZs4xCvB3iIfgDZIt8h_m5y9Jqd_ajeXzJS_TTHFE1AdXtk-dlC3qYPtZb2dbvWnow0BCQPmfb-5pBNOrXppB2UQYL_yEWHcilY9DS7fdX0/s1600/26+Football+and+happiness.gif" height="128" title="Football and happiness." width="320" /></a></div>
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<div style="text-align: justify;">
- For number one, as everybody who have read my <a href="http://mathsball.blogspot.com.es/2013/07/football-and-happiness.html" target="_blank">story about the happy numbers</a> knows, is a happy number. In fact, it's the happiest number, because all happy numbers converge at number 1.</div>
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- Therefore, <b>1 – 515</b> means in <b>mathematical language</b> <b>Happy 2015</b>!</div>
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<div>
- So, this is only <b> a mathematical greetings card</b> for the <b>New Year</b>, isn't it?</div>
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- Exactly. </div>
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<div style="text-align: justify;">
- Ok, but how can we be sure that he doesn't wants to <b>go back to Liverpool</b>? Why else he would put so many <b>red </b>demons in the card?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0-aoa2xgUGkQEA66QN8kd32-qtLoTcKcBP75XEiQfLHyobvrSmctj0k-vn6kSbQwE6eZ4QLuPQhZhRMx5DtvGLcUxwYQTaY52yw2OlFGZYaUqXwSS6u2QUZjNepjb6VI1iz2tgKRP_Dc/s1600/6+Diablos+rojos+-+Red+devils+-+Diables+vermells.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="Diablos rojos." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0-aoa2xgUGkQEA66QN8kd32-qtLoTcKcBP75XEiQfLHyobvrSmctj0k-vn6kSbQwE6eZ4QLuPQhZhRMx5DtvGLcUxwYQTaY52yw2OlFGZYaUqXwSS6u2QUZjNepjb6VI1iz2tgKRP_Dc/s1600/6+Diablos+rojos+-+Red+devils+-+Diables+vermells.gif" height="173" title="Diablos rojos." width="200" /></a></div>
- Well, he might have a good memory of his former team, and he has made it a little homage. But we must look at the text he has written: some letters are written in <b>uppercase</b>, and other in <b>lowercase</b>, without any apparent order, right?</div>
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<div>
- Yes, it's true, but we have given it no importance.</div>
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<div>
<div style="text-align: justify;">
- Yes it has, a lot. Before I mentioned <b>Bacon's Cipher</b>. With it, the philosopher replaced the characters with <b>sequences of 5 letters</b>, specifically <b>A</b>s and<b> B</b>s, <b>to encrypt </b>confidential documents. Later, <b>Leibniz </b>devised a detailed study on binary systems, using ones and zeros, thus establishing the foundations of the <b>current binary numbering system</b>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwDgpPl9lw9nJgi27gdZGvWbZcgSNJzBqOyo2WAqDWXgf-HwCn6NePU-tg8CSwTDoJ3_p-gI_cMWMKkvfje_FWZ-XJ-novHZeky9pJfgQ3gCaRg-WIbwCcibQI6i23NEWXPLfRLV_JtBc/s1600/27+Bacon's%2Bcypher%2Bcode.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Bacon's cypher." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwDgpPl9lw9nJgi27gdZGvWbZcgSNJzBqOyo2WAqDWXgf-HwCn6NePU-tg8CSwTDoJ3_p-gI_cMWMKkvfje_FWZ-XJ-novHZeky9pJfgQ3gCaRg-WIbwCcibQI6i23NEWXPLfRLV_JtBc/s1600/27+Bacon's%2Bcypher%2Bcode.jpg" height="289" title="Bacon's cypher." width="640" /></a></div>
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<div style="text-align: justify;">
- For example, if we want to <b>encode </b>the word '<b>hello</b>' with the Bacon's Cipher, first we replace each letter by its code of 5 letters with As and Bs. Then we think on a <b>phrase</b>, with the <b>same length as the formed series</b> of As and Bs (it will have 5 times more characters than the original text). In our case, I've chosen the phrase of 25 characters (excluding the spaces) '<b>I like very much country music</b>'.</div>
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<div style="text-align: justify;">
- Now we form <b>groups of 5 letters</b>, which will match each one with the code A or B of the series. If the first letter of the series is an A, we will write the corresponding first letter of our invented text in capital letters, and if it's a B, then we write it in lowercase.</div>
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<div style="text-align: justify;">
- We do the same with the remaining letters, in order, and finally we <b>separate </b> again <b>the words </b>of the resulting text, to mislead our 'enemy':</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZA8lUMOdPkQ6qehRKXUwCgzg_5JOx_aaCFy5vkRL4YIVeMnmSVQC37HX0RjrWRtV2TzpSJyPm2twxLvRW9udRRfVC1kP3gz29kT8R0EILbqcsYbWDvgrxfK4dQupLQOUlEBoQiTe25Lo/s1600/28+Text+encrypted+with+Bacon's%2Bcipher%2Bcode%2B2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Text cyphered with Bacon's Cypher." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZA8lUMOdPkQ6qehRKXUwCgzg_5JOx_aaCFy5vkRL4YIVeMnmSVQC37HX0RjrWRtV2TzpSJyPm2twxLvRW9udRRfVC1kP3gz29kT8R0EILbqcsYbWDvgrxfK4dQupLQOUlEBoQiTe25Lo/s1600/28+Text+encrypted+with+Bacon's%2Bcipher%2Bcode%2B2.jpg" height="452" title="Text cyphered with Bacon's Cyipher." width="640" /></a></div>
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<div style="text-align: justify;">
- This way, we've converted the word '<b>Hello</b>' into the sentence '<b>I Like VErY MUcH cOUnTrY MusiC</b>'.</div>
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<div>
<div style="text-align: justify;">
- We've set that the uppercase letters correspond to an A, and lowercase letters to a B, but we could also use other methods to differentiate the characters, such as using different fonts of letters, or different colors...</div>
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<br /></div>
<div>
<div style="text-align: justify;">
- Now let's apply this <b>steganographic method</b> of Bacon's Cipher to the sentence of Luis Suarez. For that, we separate the letters of the text in groups of five letter, and <span style="text-align: start;">write an A at the position where there's a capital letter, and a B for lowercase. The exclamation sign will be considered as lowercase, because it's very tiny.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5a_oRA3bbHqPgYVB6crqD04gkaX9ImBPYVoN6DzFPErRcgl2fSZvmSQvRwpzfAbg6S9OfykHlm_JrEVIhbxsPUTDJOqSM60CJLYuD45yw7ZrC6HguNW_DmWSXMgo3TflEQKuB45crtBs/s1600/31+Mensaje+desencriptado+-+Message+decrypted+-+Missatge+desencriptat.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Encoded text with Bacon's Cypher." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5a_oRA3bbHqPgYVB6crqD04gkaX9ImBPYVoN6DzFPErRcgl2fSZvmSQvRwpzfAbg6S9OfykHlm_JrEVIhbxsPUTDJOqSM60CJLYuD45yw7ZrC6HguNW_DmWSXMgo3TflEQKuB45crtBs/s1600/31+Mensaje+desencriptado+-+Message+decrypted+-+Missatge+desencriptat.jpg" height="276" title="Encoded text with Bacon's Cypher." width="640" /></a></div>
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<div style="text-align: justify;">
- What have we got? <b>Love FCB</b>! I think you have lost your cover page...</div>
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<div>
<br />
- Don't matter. We just have to <b>change the headlines</b>:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQdQn-SasolVxImfUy4c8BLo6EUPZH6Ijuszu0m4gnp3VG4Q3AIHDVnj3lYFXkEU1hop2-ED8OmTbS55ANO2nS2g9bHKIpZheHI5SHDgKd7B0ECXIIug79v-H8vBaLctOJBTFr4wNwKK0/s1600/28+bis+Luis+Su%C3%A1rez+back+to+Liverpool.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQdQn-SasolVxImfUy4c8BLo6EUPZH6Ijuszu0m4gnp3VG4Q3AIHDVnj3lYFXkEU1hop2-ED8OmTbS55ANO2nS2g9bHKIpZheHI5SHDgKd7B0ECXIIug79v-H8vBaLctOJBTFr4wNwKK0/s1600/28+bis+Luis+Su%C3%A1rez+back+to+Liverpool.jpg" height="320" width="250" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjeFhQCL0f75HwN51l-lhwSArNi91XoMqHaFTqfTnU1r_H4NNtEGBDRpEU_Tjv7bY0XubDG8GjxtEbChk_jPLXRQASqCy9CFaazwsyVql6EkS79ytl0y_T7JGdwLbr7KueBOprn6BgtKTE/s1600/28+Headlines+of+the+Liverpool+Post.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="New cover of Liverpool Post." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjeFhQCL0f75HwN51l-lhwSArNi91XoMqHaFTqfTnU1r_H4NNtEGBDRpEU_Tjv7bY0XubDG8GjxtEbChk_jPLXRQASqCy9CFaazwsyVql6EkS79ytl0y_T7JGdwLbr7KueBOprn6BgtKTE/s1600/28+Headlines+of+the+Liverpool+Post.jpg" height="320" title="New cover of Liverpool Post." width="249" /></a></div>
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<div>
- Many thanks for all, Joe!</div>
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<div style="text-align: justify;">
- Thank you for thinking of me to solve the <b>problem</b>. And I wish you all a <b><span style="font-size: large;">HAPPY 13*5*31</span></b> !</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRIJ6QR5OXgSyFmnM-BhV0hQULjYBnmaavRu9LRq0VV65qnHFz4cPOFKrets4knXD5IVNmHRXUaPqCfv8vtpe4OOVN5SFK8UYcKRbp0miVGs55HYAhyMl-jpQyfLIooB9GHwxxCmfF_h8/s1600/arbolito+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="I wish you a Happy New Year 2015." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRIJ6QR5OXgSyFmnM-BhV0hQULjYBnmaavRu9LRq0VV65qnHFz4cPOFKrets4knXD5IVNmHRXUaPqCfv8vtpe4OOVN5SFK8UYcKRbp0miVGs55HYAhyMl-jpQyfLIooB9GHwxxCmfF_h8/s1600/arbolito+2.jpg" height="360" title="I wish you a Happy New Year 2015." width="640" /></a></div>
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If you're interested in <b>learning more </b>about the topics of this story, you can visit any of these wonderful articles: <a href="http://www.sciencedirect.com/science/article/pii/0898122189902654" target="_blank">A symmetric number in Dante</a>, <a href="http://enigmathemeunmasked.blogspot.com.es/2013/10/elgars-515-ciphers.html" target="_blank">Elgar's 515 cyphers</a>, <a href="http://www.wondersandmarvels.com/2012/12/shakespeares-secrets-a-hidden-cipher-in-literatures-greatest-works.html" target="_blank">Shakespeare's secrets: a hidden cipher in literature's greatest works?</a>, <a href="http://www.dcode.fr/bacon-cipher" target="_blank">Bacon Cipher - Decoder, encoder, decrypt, encrypt</a>, <a href="http://www.cs.cas.cz/portal/AlgoMath/NumberTheory/Arithmetics/NumeralSystems/PositionalNumeralSystems/BinarySystem.htm" target="_blank">Binary system</a>, <a href="https://mothereff.in/bacon" target="_blank">Bacon's cipher encoder/decoder</a>, <a href="http://www.cabinetmagazine.org/issues/40/sherman.php" target="_blank">How to make anything signify anything</a>.</div>
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And don't forget to take a walk by the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">Carnival of Mathematics</span></a>. There you'll find lots of excellent math posts that you'll surely like too.
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WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-15985683748660059442014-10-27T14:47:00.003+00:002014-10-28T23:05:46.489+00:00Mathematical Halloween<!-- Page resume -->
<span style="display: none;">Mathematics often inspire fear to many people. And sometimes even rightly. So, in a game that takes place in Transylvania we discover some very special and mysterious figures: the vampire numbers.
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhz7FykioHSVrRIIJ_l4w_j3g9UtS1bFQi12seI9Otq5mO3MfhYSoctAL0vn1QqfqUe3UarzdMO5eEz8YuRXael6t_RAaa7_jGs47owHwZh0VgzxUu8sda4Wg9D1g50o4P1R9RSqChHm2c/s1600/header+transylvania.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhz7FykioHSVrRIIJ_l4w_j3g9UtS1bFQi12seI9Otq5mO3MfhYSoctAL0vn1QqfqUe3UarzdMO5eEz8YuRXael6t_RAaa7_jGs47owHwZh0VgzxUu8sda4Wg9D1g50o4P1R9RSqChHm2c/s1600/header+transylvania.jpg" /></a>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSJ9-Bn3IfNjMqnBCjVZWL6hEZygOmg3IL1pFlM-RYe_-35zl4_hcBCuzmC0icbn6pYBsvx9vtnIDGh_8iB0y1T31aAA_EzUBoVnLf3hLznXETSlAQQ4fN8nMib7i0gm8LRyZHHEWeBYw/s1600/Vallas+publicitarias+11.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSJ9-Bn3IfNjMqnBCjVZWL6hEZygOmg3IL1pFlM-RYe_-35zl4_hcBCuzmC0icbn6pYBsvx9vtnIDGh_8iB0y1T31aAA_EzUBoVnLf3hLznXETSlAQQ4fN8nMib7i0gm8LRyZHHEWeBYw/s1600/Vallas+publicitarias+11.jpg" /></a>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfQc2MhgAaaSgsagq_Nc3llRON8ieVovKt35QH4p80Lv5TJoWg2xkKv_43zm9dKnPWsiyZMbrzIN-55WGIG7g9zTcGhb4TOPEuxjRmGQvshh9UYDMRZ8XjRfWwB_iGfLE-G8hDDqKV1pc/s1600/picasion.com_041bb0d8415fb78e3b0c4b1faaa61679.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfQc2MhgAaaSgsagq_Nc3llRON8ieVovKt35QH4p80Lv5TJoWg2xkKv_43zm9dKnPWsiyZMbrzIN-55WGIG7g9zTcGhb4TOPEuxjRmGQvshh9UYDMRZ8XjRfWwB_iGfLE-G8hDDqKV1pc/s1600/picasion.com_041bb0d8415fb78e3b0c4b1faaa61679.gif" /></a>
</span>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhz7FykioHSVrRIIJ_l4w_j3g9UtS1bFQi12seI9Otq5mO3MfhYSoctAL0vn1QqfqUe3UarzdMO5eEz8YuRXael6t_RAaa7_jGs47owHwZh0VgzxUu8sda4Wg9D1g50o4P1R9RSqChHm2c/s1600/header+transylvania.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Mathematical Halloween: vampire numbers" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhz7FykioHSVrRIIJ_l4w_j3g9UtS1bFQi12seI9Otq5mO3MfhYSoctAL0vn1QqfqUe3UarzdMO5eEz8YuRXael6t_RAaa7_jGs47owHwZh0VgzxUu8sda4Wg9D1g50o4P1R9RSqChHm2c/s1600/header+transylvania.jpg" height="444" title="Mathematical Halloween: vampire numbers" width="640" /></a></div>
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<div class="marino" style="text-align: justify;">
<span style="display: none;">(This post participes on the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">111th edition</span></a> of the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="http://boolesrings.org/krautzberger/" target="_blank"><span class="negrita unidas">Boole's Rings</span></a>.)</span></div>
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<span style="font-family: Verdana, sans-serif;">FIRST HALF</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfQc2MhgAaaSgsagq_Nc3llRON8ieVovKt35QH4p80Lv5TJoWg2xkKv_43zm9dKnPWsiyZMbrzIN-55WGIG7g9zTcGhb4TOPEuxjRmGQvshh9UYDMRZ8XjRfWwB_iGfLE-G8hDDqKV1pc/s1600/picasion.com_041bb0d8415fb78e3b0c4b1faaa61679.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Terrifying numbers" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfQc2MhgAaaSgsagq_Nc3llRON8ieVovKt35QH4p80Lv5TJoWg2xkKv_43zm9dKnPWsiyZMbrzIN-55WGIG7g9zTcGhb4TOPEuxjRmGQvshh9UYDMRZ8XjRfWwB_iGfLE-G8hDDqKV1pc/s1600/picasion.com_041bb0d8415fb78e3b0c4b1faaa61679.gif" title="Terrifying numbers" /></a>
Never a few numbers had caused so much <b><span style="color: red; font-size: 120%;">terror</span></b>.</div>
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Today's <b>Halloween</b>, and an interesting match of the Champions League is about to be played between the teams of CFR Cluj and Manchester United FC.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9mGHkTtlvok2ncKJSijUZ9n0lgxQrLgR46iU_ACiFZ-wjmwzVllFupWSKWx3536t5jIBv8eCbjoGwJHZbtbpuNi6cioYs_zQl_kUMlCl_eQKCWYdveSIv0y6GowgFQA-Chjfb2ptZgVk/s1600/Escudo+del+CFR+Cluj.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="CFR Cluj badge" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9mGHkTtlvok2ncKJSijUZ9n0lgxQrLgR46iU_ACiFZ-wjmwzVllFupWSKWx3536t5jIBv8eCbjoGwJHZbtbpuNi6cioYs_zQl_kUMlCl_eQKCWYdveSIv0y6GowgFQA-Chjfb2ptZgVk/s1600/Escudo+del+CFR+Cluj.gif" height="200" title="CFR Cluj badge" width="175" /></a><span style="font-size: x-large;">vs.</span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_rr2UyfC3HI3U3BfptlywheYFdKz7RA4mr3gfpe6tiRm32QsLaz8W1A72k57iJGWNzq6-vV1uuLYSh1eIn7RFr8EPHnRcNTRDk3cQVcS6aIPawpXUp9_chIrGJQf64yn_tHq7Onr6VG0/s1600/escudo+del+manchester+united.gif" imageanchor="1" style="font-size: xx-large; margin-left: 1em; margin-right: 1em;"><img alt="Manchester United FC badge" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_rr2UyfC3HI3U3BfptlywheYFdKz7RA4mr3gfpe6tiRm32QsLaz8W1A72k57iJGWNzq6-vV1uuLYSh1eIn7RFr8EPHnRcNTRDk3cQVcS6aIPawpXUp9_chIrGJQf64yn_tHq7Onr6VG0/s1600/escudo+del+manchester+united.gif" height="200" title="Manchester United FC badge" width="200" /></a><span style="font-size: x-large;"></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGlbEDZwBFPc2eHZGRJ7GCklpLGYPNwKhr6757ncRwa1oEimHYZ_8ZfrqCRZ8CYfE-CDl3cTMbB7mvvJQOrYO12SiWB3A15zfTULkY6Q67jBeiGzbgTogUGTgsDo8kz2b1c6fdyeeMh1s/s1600/stadium+constantin+radulescu+2.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Dr. Constantin Rădulescu stadium of Cluj CFR" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGlbEDZwBFPc2eHZGRJ7GCklpLGYPNwKhr6757ncRwa1oEimHYZ_8ZfrqCRZ8CYfE-CDl3cTMbB7mvvJQOrYO12SiWB3A15zfTULkY6Q67jBeiGzbgTogUGTgsDo8kz2b1c6fdyeeMh1s/s1600/stadium+constantin+radulescu+2.JPG" height="150" title="Dr. Constantin Rădulescu stadium of Cluj CFR" width="200" /></a></div>
The bus of the English team has arrived to the stadium a couple of hours before the kick off.<br />
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The players have put their tracksuits on and have come to the pitch to do the warm-up exercises.</div>
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It was in that moment when <a href="http://www.juanmata8.com/index.php?idioma=en" target="_blank"><b>Juan Mata</b></a> has fled in panic into the changing room, and has taken refuge in it. He says he doesn't want to go back to the field again.</div>
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The Manchester United coach, Louis van Gaal, is very worried because Mata is an essential piece of the team. He doesn't know what to do, nor the reason of Mata's behaviour.</div>
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It seems that the player returned to the dressing room when he saw some advertisements with bloodied numbers.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSJ9-Bn3IfNjMqnBCjVZWL6hEZygOmg3IL1pFlM-RYe_-35zl4_hcBCuzmC0icbn6pYBsvx9vtnIDGh_8iB0y1T31aAA_EzUBoVnLf3hLznXETSlAQQ4fN8nMib7i0gm8LRyZHHEWeBYw/s1600/Vallas+publicitarias+11.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Commercials with bloodied numbers" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSJ9-Bn3IfNjMqnBCjVZWL6hEZygOmg3IL1pFlM-RYe_-35zl4_hcBCuzmC0icbn6pYBsvx9vtnIDGh_8iB0y1T31aAA_EzUBoVnLf3hLznXETSlAQQ4fN8nMib7i0gm8LRyZHHEWeBYw/s1600/Vallas+publicitarias+11.jpg" height="227" title="Commercials with bloodied numbers" width="400" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhejI4zBtU_dNpHiY7OtsGsAxHKhhtbFtHMsA3peToWCcAv2pS65A5cB-JwCmHHCFbngA5BWQEgvt1T5hih-tn3sULsBMYuWrNCYdOIAAp5qpo1MVyMHIt8AbYeGfToHl6J0c4BkcCVqHs/s1600/Juan+Mata+Manchester.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Juan Mata, Manchester United player" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhejI4zBtU_dNpHiY7OtsGsAxHKhhtbFtHMsA3peToWCcAv2pS65A5cB-JwCmHHCFbngA5BWQEgvt1T5hih-tn3sULsBMYuWrNCYdOIAAp5qpo1MVyMHIt8AbYeGfToHl6J0c4BkcCVqHs/s1600/Juan+Mata+Manchester.jpg" height="200" title="Juan Mata, Manchester United player" width="171" /></a></div>
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Knowing that Juan Mata is fond of <b>Mathematics</b>, Louis decides to call his friend <b>Joe Vitruvius </b>with his last generation phone, to see if he can find any solution.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj81hXaPhtOtMsoeaYt8SvbcIyXVYQs2gujEyIl4baovg70Ijnc81u06AjWfMqCLgPT7AFtVwGQ5vQsAaRNqyWvCW5qb9nCY8jdjvUbjrx7vKifxinwGKdu9L7YEw_CxeqEP3POj9_Jp2w/s1600/van+gaal.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Louis van Gaal, the Manchester United coach" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj81hXaPhtOtMsoeaYt8SvbcIyXVYQs2gujEyIl4baovg70Ijnc81u06AjWfMqCLgPT7AFtVwGQ5vQsAaRNqyWvCW5qb9nCY8jdjvUbjrx7vKifxinwGKdu9L7YEw_CxeqEP3POj9_Jp2w/s1600/van+gaal.jpg" height="200" title="Louis van Gaal, the Manchester United coach" width="145" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYgE7xNI4kpoBY4P9cUH0HvbHr5w_IYvv50bltB98elVuJ066ZdUVHCA2BNxuDLnd6Mk9Ud0ly_ZS0dLE3Jn37rTrfPSGvfw40wlBa26Fr_NruabwCl_Bz79xgtKJXpA4APDBH4vNReEs/s1600/telefono.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Last generation mobile" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYgE7xNI4kpoBY4P9cUH0HvbHr5w_IYvv50bltB98elVuJ066ZdUVHCA2BNxuDLnd6Mk9Ud0ly_ZS0dLE3Jn37rTrfPSGvfw40wlBa26Fr_NruabwCl_Bz79xgtKJXpA4APDBH4vNReEs/s1600/telefono.png" height="150" title="Last generation mobile" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPiJBpMB583JFR55R-W_MVzhi0xgc1BDMSjrA8gYvomzX4NTbUpB9egHml0QFBwEI7UJeMv867GJ1_iDlQJXmNDybt65Cia0pg5XA_BYFGDSH1vfqzOR_mP5MT2uWQBwVDB8geCGWco_g/s1600/pepevitruvio.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Joe Vitruvius" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPiJBpMB583JFR55R-W_MVzhi0xgc1BDMSjrA8gYvomzX4NTbUpB9egHml0QFBwEI7UJeMv867GJ1_iDlQJXmNDybt65Cia0pg5XA_BYFGDSH1vfqzOR_mP5MT2uWQBwVDB8geCGWco_g/s1600/pepevitruvio.jpg" height="200" title="Joe Vitruvius" width="97" /></a> </div>
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What do you think is happening?<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" height="252" width="640" /></span></a></div>
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<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
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Van Gaal phones Joe Vitruvius, and tells him all the details of what happened.</div>
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<div style="text-align: justify;">
- So, where did you tell me you play?</div>
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<div style="text-align: justify;">
- We're in Cluj. It's the capital of <b>Transylvania</b>, in Romania, a very beautiful Carpathian town.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8EMXk64rTiPOnS1jz3XhjoVokcEtI2SQBfnUKev8I52QD-xWSP7kFcqOXOsNUHZ5OwlQObQM5fcgqUzJneyoegHq5YiCduvAxVr-5dpj4UjsBIMhEQu2ZDtMCBR1WXeSW2Bwn8prB0sM/s1600/cluj+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Historical center of Cluj" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8EMXk64rTiPOnS1jz3XhjoVokcEtI2SQBfnUKev8I52QD-xWSP7kFcqOXOsNUHZ5OwlQObQM5fcgqUzJneyoegHq5YiCduvAxVr-5dpj4UjsBIMhEQu2ZDtMCBR1WXeSW2Bwn8prB0sM/s1600/cluj+1.jpg" height="123" title="Historical center of Cluj" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCtvMDHVjjQO3yuO8jldapAyj3t2FgTrzBEHvyD8ZVkQrZW1HVUVZXb_qdQYLAIcxmRYcN5Bxc3PQkWc6NqwFby9yKSk-djHqdwD3Ecm_Ms90MtdRlAHxcZepTUx0QObPpNggenw8tcnI/s1600/cluj+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Historical center of Cluj" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCtvMDHVjjQO3yuO8jldapAyj3t2FgTrzBEHvyD8ZVkQrZW1HVUVZXb_qdQYLAIcxmRYcN5Bxc3PQkWc6NqwFby9yKSk-djHqdwD3Ecm_Ms90MtdRlAHxcZepTUx0QObPpNggenw8tcnI/s1600/cluj+2.jpg" height="150" title="Historical center of Cluj" width="200" /></a></div>
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- I see. And, what time is it there now?<br />
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- It's half past eight. Just fifteen minutes to start the game, and though it's cold, the sky is clear and the moon shines perfectly.</div>
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- Is there a full moon?<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhF-9_AEQAeFYFNvwM6Z3nugVW7dqHO2tLRX1Spk3YfhgcVXKRaLBuui8-EcSfeYbrf7ACUvIWVS7n2PTsXUcZv-hVtPwtyUO2sK-XpKIanf6L8nuOsHkMvyyilB72pGj8pHoVVqStEgmE/s1600/luna+en+estadio.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Full moon over Cluj's stadium" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhF-9_AEQAeFYFNvwM6Z3nugVW7dqHO2tLRX1Spk3YfhgcVXKRaLBuui8-EcSfeYbrf7ACUvIWVS7n2PTsXUcZv-hVtPwtyUO2sK-XpKIanf6L8nuOsHkMvyyilB72pGj8pHoVVqStEgmE/s1600/luna+en+estadio.JPG" height="233" title="Full moon over Cluj's stadium" width="320" /></a></div>
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- Yes, today it's a full moon.<br />
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- And what happened when you came to the pitch to train?</div>
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<div style="text-align: justify;">
- Well, I had noticed that Juan was a little nervous. But when he saw those billboards, he began to run to the changing room.</div>
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- Which billboards?<br />
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<div style="text-align: justify;">
- Some billboards with red numbers, as if they were written with blood.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjl9MBqpYd9u7Wk5PIgvND4q3vISwBbYdOOr18GuCk7yWi31bNJC0V5mR05VOD0X-UEvBIerKrIF6fa-oluzpQUxEMwv-qxgk-_RH8Gd5Xvpr0oqzDn77ksbd4AblfXQejqAPTRv2Gr-Ms/s1600/Vallas+publicitarias+12.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Billboards with bloody numbers" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjl9MBqpYd9u7Wk5PIgvND4q3vISwBbYdOOr18GuCk7yWi31bNJC0V5mR05VOD0X-UEvBIerKrIF6fa-oluzpQUxEMwv-qxgk-_RH8Gd5Xvpr0oqzDn77ksbd4AblfXQejqAPTRv2Gr-Ms/s1600/Vallas+publicitarias+12.jpg" height="111" title="Billboards with bloody numbers" width="400" /></a></div>
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- Were there only numbers?<br />
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<div style="text-align: justify;">
- Yes. I though that they were some telephone numbers of a local on-line car insurance company, or something like this</div>
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- OK. Do you remember what numbers were they?<br />
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<div style="text-align: justify;">
- No. But wait a moment, I took several photos of the warm-up session with my mobile. Look, here I've got them. I can see, among others, the following numbers:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTiz8_MqZ09WZrWqlgrgnC-wNU7ZjymKAGFqTpB8ouIcySJn0F586X_AvpCCxRGpg5BnABuepXCm2GJNEbgfWkr8INwQnLfMy_BKQETPIE0niqJ1rdwJKbWyD9GsrRLmH4W9mvHeoFmck/s1600/Sin+t%C3%ADtulo.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Vampire numbers" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTiz8_MqZ09WZrWqlgrgnC-wNU7ZjymKAGFqTpB8ouIcySJn0F586X_AvpCCxRGpg5BnABuepXCm2GJNEbgfWkr8INwQnLfMy_BKQETPIE0niqJ1rdwJKbWyD9GsrRLmH4W9mvHeoFmck/s1600/Sin+t%C3%ADtulo.gif" height="55" title="Vampire numbers" width="400" /></a></div>
<div style="text-align: justify;">
- Hmm... I think I have an idea of what happens. Let me have a look at the wonderful website of the <a href="https://oeis.org/?language=spanish" target="_blank"><b>OEIS</b></a>... Yes, that's what I though.</div>
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- What happens, then?<br />
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<div style="text-align: justify;">
- It's clearly obvious. Everybody knows that Juan Mata is a big fan of <b>Mathematics</b>. And that he's also a little superstitious. The Cluj's managers also know this fact, and they know how important is his participation in the game of Manchester United. That's why they've put those ads on the panels of the stadium.</div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijKD5LdNm63LWtJWlLgeoiLvjplC4Qu4Z4LTfqd4QguuyrR1RK1kKm3I1bt6Hku81BwbjCrF9IycPwPs8tuxjj3t127JOlrIx9p7ibYtEH5Xwe6rvVrrjH6N97S1waYsgZfGcWw281fr4/s1600/transilvania.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Map of Transylvania" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijKD5LdNm63LWtJWlLgeoiLvjplC4Qu4Z4LTfqd4QguuyrR1RK1kKm3I1bt6Hku81BwbjCrF9IycPwPs8tuxjj3t127JOlrIx9p7ibYtEH5Xwe6rvVrrjH6N97S1waYsgZfGcWw281fr4/s1600/transilvania.jpg" height="222" title="Map of Transylvania" width="320" /></a><br />
- I don't understand.<br />
<br />
<div style="text-align: justify;">
- All right. You are in <b>Transylvania</b>, it's night, <b>full moon</b>, and <b>Halloween</b>. A really sinister combination, and very disturbing for someone so superstitious as Juan Mata. But they needed to frighten him a bit more.</div>
<br />
<div style="text-align: justify;">
- And they've succeeded with a few simple numbers?</div>
<br />
<div style="text-align: justify;">
- Yes. They could have put into the pitch someone dressed like a <b>vampire</b>, but it would have been too obvious. They have devised something more subtle.</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrG5emi9Vjzn7ARCuX5qp-Irf3nL1phZpjmm2kL7kNMbe-wOYW_TM2Cg-DKD_nFHaaDHhOnjYFqir6ClApx8hypArvEaRDARqwd28gqNNHqonvBM6wZ7D6541LlqP-HcxPZMRGBiLqQug/s1600/vampiro.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Transylvanian vampire" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrG5emi9Vjzn7ARCuX5qp-Irf3nL1phZpjmm2kL7kNMbe-wOYW_TM2Cg-DKD_nFHaaDHhOnjYFqir6ClApx8hypArvEaRDARqwd28gqNNHqonvBM6wZ7D6541LlqP-HcxPZMRGBiLqQug/s1600/vampiro.gif" height="200" title="Transylvanian vampire" width="125" /></a></div>
<div style="text-align: justify;">
- Why are these numbers so special? I don't see them very sinister...</div>
<br />
<div style="text-align: justify;">
- Yes, they are so dark. These numbers are known as <b><span style="color: red;">vampire numbers</span></b>.</div>
<br />
- Huh? Why?<br />
<br />
<div style="text-align: justify;">
- Well, this name was suggested by the great mathematician <a href="http://www.pickover.com/" target="_blank">Clifford A. Pickover</a>. Vampire numbers are numbers that satisfy some properties:</div>
<br />
1.- They must have an even number of digits.<br />
<br />
<div style="text-align: justify;">
<a href="https://cdn.thinglink.me/api/image/582694678797746176/1024/10/scaletowidth#tl-582694678797746176;1043138249'" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" class="alwaysThinglink" height="276" src="https://cdn.thinglink.me/api/image/582694678797746176/1024/10/scaletowidth#tl-582694678797746176;1043138249'" style="max-width: 100%;" width="400" /></a>2.- They're the result of the product of 2 numbers, which we call <b>fangs</b>.<br />
<br /></div>
<div style="text-align: justify;">
3.- They have the same digits as their fangs.</div>
<br />
<div style="text-align: justify;">
4.- Not all their fangs end in zero.</div>
<br />
<script async="" charset="utf-8" src="//cdn.thinglink.me/jse/embed.js"></script>
- I haven't understood anything.<br />
<br />
<div style="text-align: justify;">
- I'll explain it to you with an example: the number 1395. This number has 4 digits, and can be obtained multiplying numbers 15 and 93, that will be its fangs.</div>
<br />
<div style="text-align: center;">
<span style="color: red; font-size: 130%;"><b>15 · 93 = 1395</b></span></div>
<br />
<div style="text-align: justify;">
- You can see that the vampire number has the same digits as its fangs <span class="unidas">(1, 3, 5 and 9)</span>, and that all of its fangs don't end in zero.</div>
<br />
- That's true.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfQc2MhgAaaSgsagq_Nc3llRON8ieVovKt35QH4p80Lv5TJoWg2xkKv_43zm9dKnPWsiyZMbrzIN-55WGIG7g9zTcGhb4TOPEuxjRmGQvshh9UYDMRZ8XjRfWwB_iGfLE-G8hDDqKV1pc/s1600/picasion.com_041bb0d8415fb78e3b0c4b1faaa61679.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfQc2MhgAaaSgsagq_Nc3llRON8ieVovKt35QH4p80Lv5TJoWg2xkKv_43zm9dKnPWsiyZMbrzIN-55WGIG7g9zTcGhb4TOPEuxjRmGQvshh9UYDMRZ8XjRfWwB_iGfLE-G8hDDqKV1pc/s1600/picasion.com_041bb0d8415fb78e3b0c4b1faaa61679.gif" /></a></div>
<br />
<div style="text-align: justify;">
- Now look at number 125460. This vampire number can be decomposed into these products:</div>
<br />
<div style="text-align: center;">
<span style="color: red; font-size: 130%;"><b>204 · 615 = 246 · 510 = 125460</b></span><br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
</div>
- Wow, it has 2 pairs of fangs!<br />
<br />
<div style="text-align: justify;">
- Yes, and there're still some numbers with more fangs. Thus, let's look at this one, with 3 pairs of fangs:<br />
<br />
<div style="text-align: center;">
<b><span style="color: red; font-size: 115%;">1620 · 8073 = 1863 · 7020 = 2070 · 6318 = 13078260</span></b></div>
<br />
- There's even a 70-digit number, that has 100.025 pairs of fangs...<br />
<br />
- There're also pseudovampire numbers, whose fangs don't need to have an even number of digits:<br />
<br />
<div style="text-align: center;">
<span style="color: red; font-size: 115%;"><b>8 · 86 = 688</b></span></div>
<br />
- Or prime vampire numbers, those that can be decomposed into a product of prime fangs, like this one:<br />
<br />
<div style="text-align: center;">
<span style="color: red; font-size: 115%;"><b>167 · 701 = 117067</b></span></div>
</div>
<div style="text-align: justify;">
<br />
- So all those numbers on the billboards were <b>vampire numbers</b>, weren't they?</div>
<br />
- That's right.<br />
<br />
- Now I understand why Juan was so scared.<br />
<br />
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNQvuKEJjzT4qpVuayoztFD6o1hI2qe_5sBlcRaVorjBJeWSwTiQWFXwggANaMmN52BK2oTKCLJ67J72-nPXyMBqKgOrAEOtBIjIbtU8u9B21d9GFRHtvdO3E-MBQ9o7TVnnYdONqUCMw/s1600/Juan+Mata+collar.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNQvuKEJjzT4qpVuayoztFD6o1hI2qe_5sBlcRaVorjBJeWSwTiQWFXwggANaMmN52BK2oTKCLJ67J72-nPXyMBqKgOrAEOtBIjIbtU8u9B21d9GFRHtvdO3E-MBQ9o7TVnnYdONqUCMw/s1600/Juan+Mata+collar.jpg" height="200" width="171" /></a>
- Yes. I think you should talk to the Cluj's manager and tell him that we have discovered their ploy for intimidating Juan Mata, and then try to reassure the player.</div>
<br />
<div style="text-align: justify;">
- Perfect. That's what I'll do. But I don't know if I'll be able to calm Juan. He was very nervous.</div>
<br />
<div style="text-align: justify;">
- Good. I know that FIFA forbids playing with rings, bracelets, watches and other objects that could injure the opponents. But maybe there's no rule about playing with a <b>garlic collar</b> on the neck...</div>
<br />
- I'm going to go over the rules. Thank you very much, Joe.<br />
<br />
- It has been a pleasure. And good luck in the match!<br />
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If you're interested in <b>learning more </b>about this topic of the <b>Will Rogers phenomenon</b> and about the <b>game theory</b>, you can visit any of this magnific articles: <a href="http://primerecords.dk/vampires/" target="_blank">Vampire numbers</a>, <a href="http://coffeeghost.net/2011/07/19/vampire-numbers-visualized/" target="_blank">Vampire numbers visualized</a>, <a href="http://mathworld.wolfram.com/VampireNumber.html" target="_blank">Vampire number</a>, <a href="http://www.numberphile.com/videos/vampire_numbers.html" target="_blank">Vampire numbers Numberphile</a>, <a href="http://ezmdd.com/vamp/vmp.htm" target="_blank">Vampire numbers for Dummies</a>,</div>
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WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0Cluj-Napoca 400000, Rumanía46.766667 23.58333300000003946.592698 23.26060950000004 46.940636 23.906056500000037tag:blogger.com,1999:blog-8244131673148079224.post-64770562838230048002014-09-30T16:03:00.000+01:002014-10-05T22:07:08.342+01:00An impossible transfer<!-- Page resume -->
<span style="display: none;">A football agent burdened by failing to make any transfer during the summer, suddenly comes across a unique opportunity. The problem is that the acquiring team and the selling team have seemingly opposing interests. Will he be able to please both sides? The Will Rogers phenomenon and the game theory.
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<div style="text-align: justify;">
It's been an awful summer for Klaus Händler.</div>
<br />
<div style="text-align: justify;">
<b>Klaus </b>is a football <b>agent</b>, and during the last months none of the transfers that he has negotiated has come to a successful conclusion.</div>
<br />
<div style="text-align: justify;">
Some clubs he works with have <b>no money</b> for signing new players because of the economic crisis, and moreover, some of the players he represents are injured.</div>
<br />
<div style="text-align: justify;">
He's thinking he'll have to <b>sell </b>some of his dearest <b>Lamborghini</b> cars to afford the expenses of his comfortable life style during this winter, while, suddenly, the phone rings.</div>
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<div style="text-align: justify;">
It's Josef, chairman of the <b><a href="http://www.schalke04.de/en/" target="_blank">FC Schalke 04</a></b>:</div>
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<div style="text-align: justify;">
-<span class="Apple-tab-span" style="white-space: pre;"> </span>Klaus, just now we've got 5 <b>forwards </b>in our team, and we want to <b>transfer </b>one of them. Could you take charge of it?</div>
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<div style="text-align: justify;">
Just after hanging up the phone, it rings again. This time it's Michael, chairman of the <b><a href="http://www.schalke04.de/en/" target="_blank">Bayer 04 Leverkusen</a></b>:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWf8ycYh4jcguJrzIR3FWjtAj3ntqDn_-yvheBABNMbKJY8b9WIPjeM4bpnyWEFE0iUd3e_CgP_H1xs-Dcjf17Zwjvl7JeTMWCFm-NYBEUx0r4tJ26C6IgxGH2foq3ZuZVkK3tuH4j-2o/s1600/Bayer+04+Leverkusen+logo.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Bayer 04 Leverkussen badge" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWf8ycYh4jcguJrzIR3FWjtAj3ntqDn_-yvheBABNMbKJY8b9WIPjeM4bpnyWEFE0iUd3e_CgP_H1xs-Dcjf17Zwjvl7JeTMWCFm-NYBEUx0r4tJ26C6IgxGH2foq3ZuZVkK3tuH4j-2o/s1600/Bayer+04+Leverkusen+logo.gif" height="199" title="Bayer 04 Leverkussen badge" width="320" /></a></div>
<div style="text-align: justify;">
-<span class="Apple-tab-span" style="white-space: pre;"> </span>Klaus, how are you? I call you because today is the transfer deadline day, and we want to <b>sign </b>one additional <b>forward</b>...</div>
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<div style="text-align: justify;">
It looks like Klaus won't have to sell his luxury cars...</div>
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<div style="text-align: justify;">
But there's a problem. Both teams are interested in <b>increasing </b>the <b>average goal-scoring</b> ability of their forward line and, at the same time, <b>reducing </b>the <b>average wage</b> of their players.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgehHY6-IBssAuCIquyCoNt0oJ4fcICMJZDvHT2L5jayvgLI9k17DzCr9fc8cBzcdVobqOPpNyjyadiGxieRgTjvs0M7TPUhwNRVY_7ziCJsg3SvXBZUt9jqnreHmq99KhtrOua620wN4/s1600/moregoalslessbitcoins.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Increasing the average scoring ability and reducing the average wages at a time" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgehHY6-IBssAuCIquyCoNt0oJ4fcICMJZDvHT2L5jayvgLI9k17DzCr9fc8cBzcdVobqOPpNyjyadiGxieRgTjvs0M7TPUhwNRVY_7ziCJsg3SvXBZUt9jqnreHmq99KhtrOua620wN4/s1600/moregoalslessbitcoins.gif" height="137" title="Increasing the average scoring ability and reducing the average wages at a time" width="640" /></a></div>
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<div style="text-align: justify;">
Klaus phones some teams, to see if it's possible any agreement, but all the teams have completed their playing staffs.</div>
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<div style="text-align: justify;">
It seems clear that it's not possible to transfer a forward from Schalke to Bayer, because of the <b>imposed conditions </b>by both teams. So, what can Klaus do? Could we help him in some way?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" height="252" width="640" /></span></a></div>
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<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp9sSUqj-drZfAcmY4FIE0EH70gbRJYN0_Dq-blrjVy4MlM9bx9BZIRYAAz9Iaqg5Qclt82Nt86dJdsyKYOgNUxF1_9XJtL-Em71mQtkt675U72kOQzvZFUnvbz2UKdp3gYP0aoWovxGE/s1600/pepevitruvio.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Joe Vitruvius" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp9sSUqj-drZfAcmY4FIE0EH70gbRJYN0_Dq-blrjVy4MlM9bx9BZIRYAAz9Iaqg5Qclt82Nt86dJdsyKYOgNUxF1_9XJtL-Em71mQtkt675U72kOQzvZFUnvbz2UKdp3gYP0aoWovxGE/s1600/pepevitruvio.jpg" height="200" title="Joe Vitruvius" width="97" /></a><br />
<div style="text-align: justify;">
Klaus has decided to call his friend <a href="http://matematicasyfutbol.blogspot.com.es/2014/06/asesino-matematico-combinatoria-triangulo-pascal.html" target="_blank"><b>Joe Vitruvius</b></a>, to see if he can help with this apparently unsolvable <b>problem</b>.</div>
<br />
<div style="text-align: justify;">
- Joe, I think that if we <b>transfer </b>one player from Schalke 04 to Bayer Leverkussen, the goal-scoring average of Bayer will <b>rise</b> at the same rate as Schalke's average <b>decreases</b>. And, in same way, the average wage of Bayer will go up at the same rate as the Schalke's average goes down.</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9Nf_3tQ4nrFVUd2pLA2vgtMoErSgcWGx3kEWxTiOh4vozDyF51gjRCh7YpOSnQknV2nps8t3Msp_j6XBNUmTCX7l2QRzL7gMl4KfTT-84vSNAyEWazmFCAmmqLbS9G1a2cuzi7oMb5fk/s1600/columpio.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Everything a team loses, the other wins it" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9Nf_3tQ4nrFVUd2pLA2vgtMoErSgcWGx3kEWxTiOh4vozDyF51gjRCh7YpOSnQknV2nps8t3Msp_j6XBNUmTCX7l2QRzL7gMl4KfTT-84vSNAyEWazmFCAmmqLbS9G1a2cuzi7oMb5fk/s1600/columpio.gif" height="215" title="Everything a team loses, the other wins it" width="400" /></a></div>
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<div style="text-align: justify;">
- So, you think that everything a team gets, the other loses it. That's what we know as <b><a href="http://en.wikipedia.org/wiki/Zero-sum_game" target="_blank">zero-sum game</a></b> in <b><a href="http://en.wikipedia.org/wiki/Game_theory" target="_blank">mathematical game theory</a></b>. But it's not always like this. As in real life, we can find situations in which all parts win or lose (<b><a href="http://en.wikipedia.org/wiki/Zero-sum_game#Non-zero-sum" target="_blank">non-zero-sum games</a></b>).</div>
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<div style="text-align: justify;">
- Yes, but I don't think this is one of the times it works.</div>
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<div style="text-align: justify;">
- Well, we can try it. You told me that both teams want to <b>increase </b>the average goal-scoring ability of their forward line, and<b> at the same time, reduce </b>their average wage, didn't you?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgehHY6-IBssAuCIquyCoNt0oJ4fcICMJZDvHT2L5jayvgLI9k17DzCr9fc8cBzcdVobqOPpNyjyadiGxieRgTjvs0M7TPUhwNRVY_7ziCJsg3SvXBZUt9jqnreHmq99KhtrOua620wN4/s1600/moregoalslessbitcoins.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Increase the average goal-scoring level and reduce the average wage at the same time." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgehHY6-IBssAuCIquyCoNt0oJ4fcICMJZDvHT2L5jayvgLI9k17DzCr9fc8cBzcdVobqOPpNyjyadiGxieRgTjvs0M7TPUhwNRVY_7ziCJsg3SvXBZUt9jqnreHmq99KhtrOua620wN4/s1600/moregoalslessbitcoins.gif" height="137" title="Increase the average goal-scoring level and reduce the average wage at the same time." width="640" /></a><br />
<br />
- Yes, that's right.<br />
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<div style="text-align: justify;">
- if we talk in absolute <b>values</b>, there's no way of doing it. But as we talk about <b>averages</b>, it may be possible. Can you give me the data of goals and salaries of the forwards of the two teams?</div>
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- Yes, of course. Here you are.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEUwZLgBRMY4IA6sr9vdfKKw-L03nvPim8kT8Y3Df0hmiI0yLAx68UOO7ks3M01yoOrv0C5XBT1hNhyphenhyphent_vQz9OWEwjlH0zq_FqQ7btEIGDRvLmCgXiPlSNg-pANpWHVY0abSPejsfzrKE/s1600/average5.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Salaries and goals of Schalke 04's forwards" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEUwZLgBRMY4IA6sr9vdfKKw-L03nvPim8kT8Y3Df0hmiI0yLAx68UOO7ks3M01yoOrv0C5XBT1hNhyphenhyphent_vQz9OWEwjlH0zq_FqQ7btEIGDRvLmCgXiPlSNg-pANpWHVY0abSPejsfzrKE/s1600/average5.JPG" title="Salaries and goals of Schalke 04's forwards" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjC3s8NRNZ9h5KssIdU_htrX1OlQpXef5GeK8h_mvuBKYXEAX1iETblrJ9izh5D9qDwmQ15mU6EgK0YaFBZyYjMDG_wsoGAnYhGVNdB7vtY1yp5u7nCEcv16Y1PjSJWDawnMwFLXbfskik/s1600/average7.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Salaries and goals of Bayer 04 Leverkussen's forwards" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjC3s8NRNZ9h5KssIdU_htrX1OlQpXef5GeK8h_mvuBKYXEAX1iETblrJ9izh5D9qDwmQ15mU6EgK0YaFBZyYjMDG_wsoGAnYhGVNdB7vtY1yp5u7nCEcv16Y1PjSJWDawnMwFLXbfskik/s1600/average7.JPG" title="Salaries and goals of Bayer 04 Leverkussen's forwards" /></a></div>
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<div style="text-align: justify;">
- Well, let's <b>calculate the means </b>of goals and wages of both forward lines.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIWMtVLE3UntlMK5GhyF7d58RuqXNjfYhiq83NJ0Ky2CJxf70WcZc-0p0DKk-NaQIP5LHfe3llPsndLJaZLJmj45rOc2pyq_7yFLgqjZ8ahXyJxnZ8Nn1BR73w4i3ognq7SiTaceWZXAI/s1600/average1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Data and averages of wages and goals of the Schalke 04's forwards" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIWMtVLE3UntlMK5GhyF7d58RuqXNjfYhiq83NJ0Ky2CJxf70WcZc-0p0DKk-NaQIP5LHfe3llPsndLJaZLJmj45rOc2pyq_7yFLgqjZ8ahXyJxnZ8Nn1BR73w4i3ognq7SiTaceWZXAI/s1600/average1.JPG" title="Data and averages of wages and goals of the Schalke 04's forwards" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiO9tc5wlprrTkp3_YQK5l0gfa8jJNcQg4Hb4wjXPXUtY9C5TaI-WDxlhMf3Uo-wplo2cNiVR7eWCfL7-TXF0kTTA-Fo9eubdHeg7iBBD8IVuLHOUOZjxQSRwIDzZI_H_0zvc-0TH-wAII/s1600/average3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Data and averages of wages and goals of the Bayer 04 Leverkussen's forwards" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiO9tc5wlprrTkp3_YQK5l0gfa8jJNcQg4Hb4wjXPXUtY9C5TaI-WDxlhMf3Uo-wplo2cNiVR7eWCfL7-TXF0kTTA-Fo9eubdHeg7iBBD8IVuLHOUOZjxQSRwIDzZI_H_0zvc-0TH-wAII/s1600/average3.JPG" title="Data and averages of wages and goals of the Bayer 04 Leverkussen's forwards" /></a></div>
<div style="text-align: justify;">
- Now we can verify if there's a solution for your problem. We only have to apply the <b><a href="http://en.wikipedia.org/wiki/Will_Rogers_phenomenon" target="_blank">Will Rogers phenomenon</a></b>.</div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUpBO9Bmb-fs3wELP0QiezAIKV19LhdVmxOJqTmr3xudxo8thf0zh4cSHITqi7U8N4eSCAHNKkjhOpk1ysayur4YDXBf2vF0tk-FxshuiKvmGkVtmYaHdpEpf_vhiAnHZA9eVCBTn1Cds/s1600/willrogers.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Will Rogers, an American humorist" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUpBO9Bmb-fs3wELP0QiezAIKV19LhdVmxOJqTmr3xudxo8thf0zh4cSHITqi7U8N4eSCAHNKkjhOpk1ysayur4YDXBf2vF0tk-FxshuiKvmGkVtmYaHdpEpf_vhiAnHZA9eVCBTn1Cds/s1600/willrogers.jpg" title="Will Rogers, an American humorist" /></a><br />
- And... what's that about?<br />
<br />
<div style="text-align: justify;">
- This paradox is related to a comment that <a href="http://en.wikipedia.org/wiki/Will_Rogers" target="_blank"><b>Will Rogers</b></a>, an American artist, said in the early 20th century: “When the Okies left Oklahoma and moved to California, they <b>raised</b> the <b>average intelligence level</b> in <b>both states</b>".</div>
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<div style="text-align: justify;">
- He meant that when the citizens from <b>Oklahoma </b>move to California are the <b>less intelligent</b> persons of their state, even so they're <b>more intelligent </b>than <b>California</b> inhabitants, right?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcx5E4aQHnxUaczTD1C_RmStbihamdhYD-Hs5-LpKLohVJCXWKoHOC1_i1Fp2hwGTEx3r7Rgy_Uu5h6nu9pAYtEZPYTf9uoazOieB-r3KiqUl9uG59W6QQsYINv8I8rH99GAV2S_MEZ9o/s1600/oklahomatocalifornia.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Moving from Oklahoma to California" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcx5E4aQHnxUaczTD1C_RmStbihamdhYD-Hs5-LpKLohVJCXWKoHOC1_i1Fp2hwGTEx3r7Rgy_Uu5h6nu9pAYtEZPYTf9uoazOieB-r3KiqUl9uG59W6QQsYINv8I8rH99GAV2S_MEZ9o/s1600/oklahomatocalifornia.jpg" height="247" title="Moving from Oklahoma to California" width="400" /></a></div>
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<div style="text-align: justify;">
- That's right. This way, any inhabitant from Oklahoma whose intelligence is under the state average, that moves to California, makes that the <b>average intelligence levels</b> of both states experience an <b>increment</b>.</div>
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<div style="text-align: justify;">
- And how can we <b>apply </b>it to our case?</div>
<br />
<div style="text-align: justify;">
- Well. Let's look at the Schalke 04 player, the Peruvian <a href="http://en.wikipedia.org/wiki/Jefferson_Farf%C3%A1n" target="_blank">Jefferson Farfán</a>. His goal-scoring ability is smaller than the team's average, but his salary is higher than his teammates average. However, he scores more goals than Bayer Leverkussen's forwards, and earns less money than them.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkjT_lCxhGWebjvmvaTmcAjyVBmJ8OmMOsBN_5srnMTHw0OvYzn694RuIH9ImaNuYUPyBTEbH7ng09j16-tzhvm98zckugapRsr1j5Exm450x_JBWIm6zUH_ltjYxrGmMXmao5ylGcS7c/s1600/average10.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Comparison of the salary, and goal-scoring capacity of Jefferson Farfán vs. Schalke 04 levels" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkjT_lCxhGWebjvmvaTmcAjyVBmJ8OmMOsBN_5srnMTHw0OvYzn694RuIH9ImaNuYUPyBTEbH7ng09j16-tzhvm98zckugapRsr1j5Exm450x_JBWIm6zUH_ltjYxrGmMXmao5ylGcS7c/s1600/average10.jpg" title="Comparison of the salary, and goal-scoring capacity of Jefferson Farfán vs. Schalke 04 levels" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj95ak32PYylEHP-wY6V9Xu59EypEfudXOPA1WwViMbX1OkDcs7uUCf0Cl1-17W6MB_L_I7GGFWtSrEMsR8X5UttWT18kDIn8KnNcWmTgjsEBfS-_UUaAr7sbAaaug9yPJVp-I8XO5PWtY/s1600/average9.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Comparison of the salary, and goal-scoring capacity of Jefferson Farfán vs. Schalke 04 Bayer Leverkussen levels" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj95ak32PYylEHP-wY6V9Xu59EypEfudXOPA1WwViMbX1OkDcs7uUCf0Cl1-17W6MB_L_I7GGFWtSrEMsR8X5UttWT18kDIn8KnNcWmTgjsEBfS-_UUaAr7sbAaaug9yPJVp-I8XO5PWtY/s1600/average9.jpg" title="Comparison of the salary, and goal-scoring capacity of Jefferson Farfán vs. Schalke 04 Bayer Leverkussen levels" /></a></div>
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<div style="text-align: justify;">
- I understand. If we <b>transfer </b>the player from one team to the other, assuming that he will earn the same, and that he will score the same goals, both teams will improve their statistics.</div>
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<div style="text-align: justify;">
- That's right. Let's see how it works on the table.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4iDo428AWA_dqfYSIBNaR5p7zkTdqeoR51mA_Q41yeGkElg_S_xznMH5IWw-_npfYLlN3whz0FXDz3T4-lpNhyarFpO9qT1ubJiTP4_dpCP3Vu-1x5mgrK7A483h34aAHeGaztFqq7Es/s1600/average2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Average wages and goal-scorings of Schalke 04's forwards after the transfer of Jefferson Farfán" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4iDo428AWA_dqfYSIBNaR5p7zkTdqeoR51mA_Q41yeGkElg_S_xznMH5IWw-_npfYLlN3whz0FXDz3T4-lpNhyarFpO9qT1ubJiTP4_dpCP3Vu-1x5mgrK7A483h34aAHeGaztFqq7Es/s1600/average2.JPG" title="Average wages and goal-scorings of Schalke 04's forwards after the transfer of Jefferson Farfán" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWmOicrjX9OcpcmywbQnONfNYf10Jd6MT16H05DEBK-7y-48nt4p46Jv6lzfik0ngJNFvAedrizn7n7dVHaqkd_eLnPczG8XChDRNTZKAeqtvko3D52kOqJo4FN75riSSW51tDVOe61_k/s1600/average4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Average wages and goal-scorings of Bayer 1904 Leverkussen's forwards after the transfer of Jefferson Farfán" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWmOicrjX9OcpcmywbQnONfNYf10Jd6MT16H05DEBK-7y-48nt4p46Jv6lzfik0ngJNFvAedrizn7n7dVHaqkd_eLnPczG8XChDRNTZKAeqtvko3D52kOqJo4FN75riSSW51tDVOe61_k/s1600/average4.JPG" title="Average wages and goal-scorings of Bayer 1904 Leverkussen's forwards after the transfer of Jefferson Farfán" /></a></div>
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- Indeed, both teams have improved their average goal-scoring level, and have reduced their average wages. And does this happen always?</div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBWpspEwZBarkq3oeI0Gv1s0EarrdJ29QW4Yr_xfQOKQzH75C_ziBkGntvEq-nudO8-niJYOzqw1kdW0yo0Yu4ivOI6B5A2HlBPS4voe_7aC74dGa4qnQnDhY6xMCB1hHytkk-xgc2LY4/s1600/comida-alemana.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Succulent and tipical German food" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBWpspEwZBarkq3oeI0Gv1s0EarrdJ29QW4Yr_xfQOKQzH75C_ziBkGntvEq-nudO8-niJYOzqw1kdW0yo0Yu4ivOI6B5A2HlBPS4voe_7aC74dGa4qnQnDhY6xMCB1hHytkk-xgc2LY4/s1600/comida-alemana.jpg" height="150" title="Succulent and tipical German food" width="200" /></a><br />
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-<span class="Apple-tab-span" style="white-space: pre;"> </span>No, not ever. But sometimes it does, like in this case.</div>
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-<span class="Apple-tab-span" style="white-space: pre;"> </span>Well, you've just solved my economics this winter. I would have to invite you to a <b>good meal</b>…</div>
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-<span class="Apple-tab-span" style="white-space: pre;"> </span>I think this will cost you more than a simple meal. I prefer we do the following <b>deal</b>: you'll <b>pay </b>me in <b>function </b>of the <b>goals</b> that Jefferson scores this season. If he scores only one goal, you'll pay me 2 bitcoins. If he scores 2, you'll double the amount, 4 bitcoins. If he scores 3 goals, I will receive the double, that is, 8 <a href="http://en.wikipedia.org/wiki/Bitcoin" target="_blank">bitcoins</a>. And so on.</div>
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<div style="text-align: justify;">
-<span class="Apple-tab-span" style="white-space: pre;"> </span>That seems a fair deal. I agree.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMO2oEoQRILAjsLKDNKeUZwokmXavGvtzj_thX4NFufMamDgV5yc8xdAdRKQeKJ0CkKpS_5UZ7lhdoYQ6dMyXi1X1moIB9LwKQzMdX4CNR9fmTQk2kCkoUjawk6Ot2UxT_mQFaXOdZot8/s1600/golfarfan2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Jefferson Farfán scoring a goal for Bayer Leverkussen." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMO2oEoQRILAjsLKDNKeUZwokmXavGvtzj_thX4NFufMamDgV5yc8xdAdRKQeKJ0CkKpS_5UZ7lhdoYQ6dMyXi1X1moIB9LwKQzMdX4CNR9fmTQk2kCkoUjawk6Ot2UxT_mQFaXOdZot8/s1600/golfarfan2.jpg" height="278" title="Jefferson Farfán scoring a goal for Bayer Leverkussen." width="400" /></a></div>
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The transfer was done. Schalke sold one of its excedent players, Bayer signed an <b>extraordinary forward</b>, and Klaus didn't have to get rid of his precious Lamborghini automobiles. At least during this winter, because Farfán scored <b>25 goles</b> for his new team…</div>
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<div style="text-align: justify;">
Can you calculate how many bitcoins Joe Vitruvius received from Klaus Händler at the end of the football season?</div>
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If you're interested in <b>learning more </b>about this topic of the <b>Will Rogers phenomenon</b> and about the <b>game theory</b>, you can visit any of this magnific articles: <a href="http://io9.com/the-will-rogers-phenomenon-lets-you-save-lives-by-doi-1443177486" target="_blank">The "Will Rogers Phenomenon" lets you save lives by doing nothing</a>, <a href="http://pem-tuklas.blogspot.com.es/2011/12/will-rogers-phenomenon.html" target="_blank">The Will Rogers phenomenon</a>, <a href="http://www.science4all.org/le-nguyen-hoang/game-theory/" target="_blank">Game Theory and the Nash Equilibrium</a>, <a href="http://www.science4all.org/le-nguyen-hoang/advanced-game-theory-overview/" target="_blank">Evolutionary Game Theory</a>, <a href="http://www.science4all.org/le-nguyen-hoang/advanced-game-theory-overview/" target="_blank">Advanced Game Theory Overview</a>,</div>
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<!-- End of main article -->WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-20705778884354555412014-06-24T15:29:00.001+01:002014-07-13T15:58:44.768+01:00The mathematical murderer<!-- Page resume -->
<span style="display: none;">A player has been killed in the locker room at halftime of a match of the World Cup Brazil 2014. A series of mathematical clues lead you to solve the mystery, with the help of inspector Larry Flanagan and official Joe Vitruvius, that will introduce the fascinating world of Combinatorics. Do you dare to solve the case?
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<br />
A player has been killed in the locker room at halftime of a match of the World Cup Brazil 2014. A series of mathematical clues leads you to solve the mystery, with the help of inspector Larry Flanagan and official Joe Vitruvius, that will introduce the fascinating world of Combinatorics.
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<br />
Do you dare to solve the case?
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<!-- End of main article -->
<!-- Links to other languages -->
<a href="http://matematicasyfutbol.blogspot.com/2014/06/asesino-matematico-combinatoria-triangulo-pascal.html" style="float: right; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq2wq5rjLaANT4l-sjeQg9UTchvqOSy36d_j8_VYlCaL-ZkzhY0ta22s-gCo946Ss-8f_rkY9hK-ZExg36uhE2Ajp7lX3OTLKNnvSaTVgKiu3C0Sm4zRwAFLwytg5TaAyhdjurO7FQngc/s1600/versionespanyola.gif" height="60" width="60" /></a>
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(This post participes on the <a href="http://www.theoremoftheday.org/SpecialEvents/CoM112.html" target="_blank"><span class="negrita">112th edition</span></a> of the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="http://www.theoremoftheday.org/" target="_blank"><span class="negrita unidas">Theorem of the Day</span></a>.)</div>
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<span style="color: red; font-family: Courier New, Courier, monospace; font-size: 200%;">The mathematical murderer</span></h2>
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Nobody could believe it. They were waiting for him to begin the second half of the match, but he didn’t go out to the pitch. Finally, the physical trainer went down to the changing room to see what was happening, and found him there, lying on the floor, with a dagger in his chest.</div>
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It was <b>Drakulic</b>, the center forward and captain of the national team of <b>Transylvania</b>, that had brilliantly qualified for the first time for a final tournament of a World Cup, that this year Brasil holds.</div>
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None of the players remembered who had been the last one to leave the locker room, because they were very upset by the talk of their coach, Pep Tourinho, as they were losing at halftime by 2 goals to 0 with the national team of Western Micronesia.</div>
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The organizers of the Championship wanted the case to be solved quickly and discreetly, so they requested the <b>Police</b> for their best officials, so as to solve the case as soon as possible.</div>
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In few minutes, inspector <b>Larry Flanagan</b>, and agent <b>Joe Vitruvius</b>, from the Mathematics Department of the Police, arrived to the stadium. </div>
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Why were they chosen? Well, Larry Flanagan (F) was the most brilliant inspector of the police. But he wasn’t very good at Mathematics. </div>
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The collaboration of an official of the <b>Mathematics Department</b> seemed essential because, stuck on the dagger, there were three scraps of the today’s newspaper, all of them related to the match they were going to play, and with some mathematical notations written in them.</div>
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Furthermore, the news clippings didn’t have the traditional rectangular shape, but they were cut in a <b>triangular shape</b>, and the dagger went through them by their center.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvTCbxE3FSumHCWyG5f-4KJT3pXrAUvlBSR_2NH6U_Gl1LxY1_wx8VAZgyOpcP-2pemFq0N5QvKVihQ6p3sYv70HEdc7MnJDj1ChOHVgbi_yYD66wZDavgLiY4k4b0NcZ1dcaRT7giSlQ/s1600/triangle+with+dagger.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvTCbxE3FSumHCWyG5f-4KJT3pXrAUvlBSR_2NH6U_Gl1LxY1_wx8VAZgyOpcP-2pemFq0N5QvKVihQ6p3sYv70HEdc7MnJDj1ChOHVgbi_yYD66wZDavgLiY4k4b0NcZ1dcaRT7giSlQ/s1600/triangle+with+dagger.jpg" height="414" width="640" /></a></div>
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Everything made them think that the criminal was a person who liked Mathematics, and that, she/he had left some clues whose resolution will lead them to the assassin.</div>
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Therefore, Joe Vitruvius (V) contribution seemed fundamental to solve the case, because he was an expert in the resolution of such <b>enigmas</b>.</div>
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The inspector got into work. He took the scraps, and read the first one: ‘For tonight’s match, our national coach hesitates <b>which pair</b> of central defenders <b>will pick</b> among the <b>4 available</b> players’. On the upper part of the scrap, there was a mysterious note in red: <span class="unidas" style="color: red;"><b>
(4 2)</b></span>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNRC6l9ouogFhxjUWYd9b0AsHiFhtN1EN8HeCJt9LeZ81ZtDkcdjnRusLDxn5yzjljxs4QyZiiR2yOd71WYS1uuczlBDk1d_REiRFAxRdBqZ1y0guXH1dNBv5W7yZl0h5lmzA5KCYo7TQ/s1600/eng.+triangle+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNRC6l9ouogFhxjUWYd9b0AsHiFhtN1EN8HeCJt9LeZ81ZtDkcdjnRusLDxn5yzjljxs4QyZiiR2yOd71WYS1uuczlBDk1d_REiRFAxRdBqZ1y0guXH1dNBv5W7yZl0h5lmzA5KCYo7TQ/s1600/eng.+triangle+1.jpg" height="448" width="640" /></a></div>
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F: I don’t understand the relation between this news of the defenders and the murdered player, who is a forward. Moreover, what will these numbers in red mean, 4 and 2, written one below the other, and in brackets? –inspector Larry Flanagan said.</div>
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<span style="color: #073763;">
V: It could be a mathematical notation, a <b>binomial coefficient</b>, which represents the combinations of 2 elements taken from a set of 4 items. In this case, it can indicate the number of possible <b>combinations</b> of 2 central defenders that the coach can pick if he has 4 players available to play in this position. As we would say in our Department, combinations (without repetition) are the number of subsets of k distinct elements that we can extract from a set of n elements. – Joe Vitruvius answered.</span></div>
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F: I see. If we have 4 players, whom we’ll call A, B, C and D (Transylvanian names are really complicated, so we’ll better use these letters), the coach may choose the following pairs of players: AB, BC, CD, AC, BD, and AD. This is, a total of <b>6 different pairs</b>, right?</div>
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V: Right. This will be that way if we don’t mind what position they play. That is, it doesn’t matter if each player plays on the right or on the left of the pitch. In other case, we’d have a total of 12 combinations, because we should count the same pairs, but changing their order: BA, CB, DC, CA, DB, and DA. This way, we’d talk about <b>partial permutations</b> (without repetition) of 2 elements taken from a set of 4 items, and not about ‘combinations’. </span></div>
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F: Well, it seems that the first clue leads us to number <span style="color: blue; font-size: 200%;">6</span>. Let’s see the next scrap: ‘Indignation in Brașov. After the good season of the Sporting of Brașov, the coach will only field <b>2 of the 7 players</b> of the team, that he took with him to Brazil’. Also in this clipping somebody has written a 7 and a 2, in brackets and one above the other.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1cD07jRjgh4LQByIn-mHjioqREq9DoSTpE7gBc7-Qh2mFNnD-FipdBCB1UhLAnqTXwvwIg6lkG1MJYqUKSB8Oge8nwNzyXw1BShPDPAKB3gOlQmS5KzJuNl2syGgCG2rqu7NUeIcssdk/s1600/eng.+triangle+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1cD07jRjgh4LQByIn-mHjioqREq9DoSTpE7gBc7-Qh2mFNnD-FipdBCB1UhLAnqTXwvwIg6lkG1MJYqUKSB8Oge8nwNzyXw1BShPDPAKB3gOlQmS5KzJuNl2syGgCG2rqu7NUeIcssdk/s1600/eng.+triangle+2.jpg" height="430" width="640" /></a></div>
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F: Don’t tell me, Joe. I think this time I’ve found it by myself: the notation <span class="unidas" style="color: red;"><b class="unidas">(7 2)</b></span> represents the number of possible combinations of 2 elements taken from a set of 7 players, that the coach can pick. This is, the number of possible different pairs of players from Sporting that the coach may choose to play the match. Am I wrong? </div>
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V: No, you aren’t, inspector Larry. I see that you learn very quickly.</span></div>
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F: Ok, let’s see the pairs we can get. If we have 7 players, whom we’ll call A, B, C, D, E, F and G, their <b>combinations</b> will be: AB, FE, AC, CD, AD, BD, CF... </div>
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F: Oops! I think I’ve lost. Now it’s no so easy to calculate them... I’m going to start again, in a more ordered way: AB, AC, AD, AE....</div>
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After a while, the inspector comes to the result:</div>
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<span style="color: red; font-size: 150%;"><b>AB AC AD AE AF AG BC BD BE BF BG CD CE CF CG DE DF DG EF EG FG</b></span><br />
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F: I think we have <b><span style="color: blue; font-size: 200%;">21</span></b> pairs, but I’m not absolutely sure that I haven’t forgotten any…</div>
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V: Yes, that’s right. Nevertheless, in this case, it would be better to use Mathematics. The formula for calculating the <b>combinations</b> of k elements taken from a group of n elements, is as follows:</span><br />
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F: Where does this formula come from?</div>
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V: Well, let’s think on the first player that the coach can choose. Clearly he has 7 possible candidates, the 7 players from Sporting. When he’s going to choose the second, he only has 6 players to pick, right? Therefore, for each of the 7 players that can be chosen initially, there will be 6 possible playmates. So we have <span class="unidas">7 x 6 = <b>42</b></span><b> possible pairs</b>:</span></div>
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<span style="color: red; font-size: 150%;"><b>AB AC AD AE AF AG BA BC BD BE BF BG CA CB CD CE CF CG DA DB DC DE DF DG EA EB EC ED EF EG FA FB FC FD FE FG</b></span><br />
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V: But in this case we get again the <b>partial permutations</b> of 2 elements taken from a set of 7 elements, as we have repeated pairs: AB and BA, for instance. And we said that we don’t care the position in which they play. What could we do to remove them from our formula? </span></div>
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F: I can’t imagine how...<br />
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V: We should divide the result by the number of possible ways we can find the same combination of players, but in a different order. It’s what, in our Department, we call ‘<b>permutations</b>’ (without repetition): the number of distinct ordinations we can make in a set of elements, using all the numbers, but not repeating them. </span> </div>
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F: And how do we calculate the number of permutations?<br />
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V: It’s quite simple: If we have k elements, we have to think that any of them can be the first one. Then, there will be (k-1) elements that can be in second place (as we can’t repeat them). There will be (k-2) elements available for the third place. And so on. The total of different ways we can sort k elements will be </span></div>
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P(k) = k • (k-1) • (k-2) • … • 3 • 2 • 1</div>
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V: In our Department we don’t like wasting time when writing such long formulas, so we invented a secret key for this product: ‘<b>factorial</b> of k’, that we write: <span style="font-size: 200%;">k!</span> So we can reduce our formula: </span></div>
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P(k) = k!</div>
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<span style="color: #073763;">V: For our case with sets of 2 players, we’ll have that <span class="unidas">k! = 2! = 2 • 1 = 2</span></span></div>
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V: So now we have the result for our problem. To obtain the number of combinations of 2 elements taken from a set of 7, we have to divide the number of variations of 2 elements taken from a set of 7 items by the permutations of 2 items:</span></div>
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F: That’s what I said at the beginning, without all these formulas!</div>
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V: It’s true, but you spent a lot of time to get the result. And there were only 7 players. What would happen if you have to <b>choose</b>, for example, <b>5 players from</b> the <b>23</b> selected? Would you write all possible combinations?</span></div>
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F: Hummm… Ok. Continue your explanation, please.</div>
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V: Now, let’s modify slightly the precedent formula, by multiplying the numerator and the denominator by 5!</span></div>
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<span style="color: #073763;">V: We’ve arrived to the initial formula, can you see it?</span></div>
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F: Correct. But I’m a little tired about all these figures, <b>combinations</b>, <b>permutations</b>, <b>partial permutations</b>… I think we’d better take the finger prints from the dagger, and we’ll solve the case more quickly.</div>
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<span style="color: #073763;">
V: Inspector, you know that assassins tend to wear gloves in these cases… Moreover, because of the celebration of the World Cup, the Laboratory is understaffed. And we know that they’re not so quickly in their analysis as we wanted.</span></div>
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F: All right. Let’s go on with our investigation. We take the third scrap: ‘Today everybody expects a hard fight in midfield, so the coach says he will <b>choose 5 of the 7 midfield players</b> he has brought for the World Cup.’ Above this news we can read another note <span class="unidas">(7 5)</span>.</div>
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F: Well, now we have a combination of 5 players taken from a total of 7, haven’t we? Let’s calculate it with the formula you have taught to me.</div>
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<span style="color: #073763;">V: It’s also </span> <span style="color: blue; font-size: 200%;"><b>21</b></span>.<br />
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F: How did you calculate it so quickly?<br />
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<span style="color: #073763;">V: I haven’t done any calculation, indeed. We have that</span></div>
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<span style="color: #073763;">V: In general terms, </span><br />
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<span style="text-align: justify;">F: And why do we get the same result?</span><br />
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<span style="color: #073763;">
V: Well, you have to think that we can express in other terms the problem of calculating the number of possible combinations of 5 players if we have a total of 7 players. We can think about the 2 players that remain on the team bench each time the coach picks 5 players to enter the pitch.</span> </div>
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<div style="text-align: center;">
<span style="font-size: 150%;"><span style="color: blue;">ABCDE</span> – <span style="color: red;">FG</span></span></div>
<div style="text-align: center;">
<span style="font-size: 150%;"><span style="color: blue;">ABCDF </span>– <span style="color: red;">EG</span></span></div>
<div style="text-align: center;">
<span style="font-size: 150%;"><span style="color: blue;">ABCEF </span>– <span style="color: red;">DG</span></span></div>
<div style="text-align: center;">
<span style="font-size: 150%;"><span style="color: blue;">ABCEG </span>– <span style="color: red;">DF</span></span></div>
<div style="text-align: center;">
<span style="font-size: 150%;">…</span></div>
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<div style="text-align: justify;">
<span style="color: #073763;">
V: That way, for each group of 5 players selected to play we have a set of 2 players not selected. There will be as many distinct groups of 5 line-up players than groups of 2 players that stay on the bench.</span></div>
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<div style="text-align: justify;">
<span style="color: #073763;">
V: So, instead of calculating how many <b>groups of 5 starter players</b> we can form, we can see how many <b>groups of 2 reserve players</b> we can get, if we have a total of 7 players, since the solutions will be the same. And as in the previous clipping we’ve already calculated that there were 21 possible combinations for this case, now we don’t need to calculate it again.</span></div>
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<div style="text-align: justify;">
F: Perfect. Now we only have to fit the pieces we have. The results of the 3 mysterious parentheses<span class="unidas" style="font-size: 150%; font-weight: bold;"> (4 2)</span>,<span class="unidas" style="font-size: 150%; font-weight: bold;">(7 2) </span>and <b><span style="font-size: 150%;">(7 5)</span></b> turned out to be numbers <b><span class="unidas" style="font-size: 150%;">6</span></b>, <span style="font-size: 150%;"><b>21</b></span> and <b><span style="font-size: 150%;">21</span></b>. I think that if the dagger went through the 3 scraps by their centre, maybe we should find the arithmetical ‘center’ of the obtained numbers, that is, their <b>average</b>:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6ucNStfCrhQIrygtNg1PH5AsF13fGUnoyJ96Hxm_rCH60KA5TtHYmqgrTmGyNy_QNmVq-W0JIKf6i5NaYT362ckoNGN94-2LA8P715sBMsrsRL2S5_NzOKorkqh4qHebDh8QbArnCUpM/s1600/16.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6ucNStfCrhQIrygtNg1PH5AsF13fGUnoyJ96Hxm_rCH60KA5TtHYmqgrTmGyNy_QNmVq-W0JIKf6i5NaYT362ckoNGN94-2LA8P715sBMsrsRL2S5_NzOKorkqh4qHebDh8QbArnCUpM/s1600/16.gif" /></a></div>
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<span style="text-align: justify;">
F: Waw, </span><b style="text-align: justify;"><span style="font-size: 200%;">16!</span></b><span style="text-align: justify;"> Just the number that the murdered player wears on his back! </span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIqcRzbfM2kUPYw69AuCbVhoRb3k70WGXVsgGc-sNqZMmMB9TLnYTv3Z_WxMhQXr1FxZwO-rz1K_5Qd-oCRrK3E1uuEftoIa-uqNlPZAgRM7aVFx6pkU_aBPiwEwW3ATjoIZYwKPWKCAc/s1600/16+drakulic.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIqcRzbfM2kUPYw69AuCbVhoRb3k70WGXVsgGc-sNqZMmMB9TLnYTv3Z_WxMhQXr1FxZwO-rz1K_5Qd-oCRrK3E1uuEftoIa-uqNlPZAgRM7aVFx6pkU_aBPiwEwW3ATjoIZYwKPWKCAc/s1600/16+drakulic.jpg" height="640" width="416" /></a></div>
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<span style="text-align: justify;">
F: It seems that the murderer is a fan of <b>Mathematics</b> and of <b>logic</b> problems, and he’s going to make it difficult to us. And now, Joe, what can we do?</span><br />
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<div style="text-align: justify;">
<span style="color: #073763;">
V: There’s a data that we haven’t used yet in our investigation, perhaps the most obvious: the scraps have a <b>triangular shape</b>.</span></div>
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<div style="text-align: justify;">
<span style="color: #073763;">
V: If the murderer is a fond of Mathematics, and also if all the problems on the scraps are related to the discipline of <b>Combinatorics</b>, no doubt all leads us to the <b>Pascal’s triangle</b>. Do you know this triangle, Flanagan?</span></div>
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F: No, I don’t know what it is.<br />
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<div style="text-align: justify;">
<span style="color: #073763;">
V: That’s normal. This triangle, as the cleverest criminals, adopts a lot of distinct personalities. For example, it’s also known as <b>Tartaglia’s triangle</b>. Anyway, and returning to our case, I’ll tell you that it’s a triangle very easy to draw up. First we write an 1 in the superior vertex, and then, above it, successively and in diagonal, we write the result of adding the 2 numbers above it. If there’s no number, we add a zero.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5a-WuN19iu245HfkItUKFH4m2bmDhKcevPtfCKg2UPmOWa-0p4lvqv5qT2DFqucRkvfUGZxBckq3mkuGV8RShfvtJfMZE0sEV3c4Ss60u4mMlr_8zFC6BA9X6YB51_x9xeVVIZC4Ebh0/s1600/triangulo+de+pascal+3+4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5a-WuN19iu245HfkItUKFH4m2bmDhKcevPtfCKg2UPmOWa-0p4lvqv5qT2DFqucRkvfUGZxBckq3mkuGV8RShfvtJfMZE0sEV3c4Ss60u4mMlr_8zFC6BA9X6YB51_x9xeVVIZC4Ebh0/s1600/triangulo+de+pascal+3+4.jpg" height="640" width="512" /></a><br />
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<div style="text-align: justify;">
<span style="color: #073763;">
V: Now, we’re going to point, in this Pascal triangle, the results obtained. Look, inspector. In the first case (4 2), we take the fifth row (we leave out the first lonely 1) and we go to the third number, so we obtain the value 6. The same result as we get with our formula.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjod60dmEvHs7Dx2IN3oRuZKyt8rqokXkHzJYI63kexpscaXlHMrYEEJMj6f-4qp2PHySRAvrA6TDdrL1YK7O4zoIOFQUpifOXxUSmO1XCnGCvhFqclKfsgkedCxlispoXvL5Y4upUSUT8/s1600/combinatoria+(4+2).jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjod60dmEvHs7Dx2IN3oRuZKyt8rqokXkHzJYI63kexpscaXlHMrYEEJMj6f-4qp2PHySRAvrA6TDdrL1YK7O4zoIOFQUpifOXxUSmO1XCnGCvhFqclKfsgkedCxlispoXvL5Y4upUSUT8/s1600/combinatoria+(4+2).jpg" height="640" width="512" /></a><br />
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<div style="text-align: justify;">
F: Why have you pointed put the third number on the fifth row, and not the second number on the fourth row?</div>
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<div style="text-align: justify;">
<span style="color: #073763;">
V: Because the apex of the triangle corresponds to <b>binomial coefficient</b> <span class="unidas">(0 0)</span>. And, on each row n, the first number represents the combination <span class="unidas">(n 0)</span>. Thus, on our fifth line <span class="unidas">(4 0)</span> is equal to 1, <span class="unidas">(4 1)</span> is equal to 4, <span class="unidas">(4 2)</span> is equal to 6, and so on. Agreed?</span></div>
<br />
F: Yes. I didn’t think on number 0.<br />
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<div style="text-align: justify;">
<span style="color: #073763;">
V: Now we’re going to do the same with <span class="unidas">(7 2)</span> and <span class="unidas">(7 5)</span>. You can check that the 3 numbers make an awesome triangle, with a similar shape as the news clippings. </span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdbGuu0YYJGlK2J-55O9bTR5UuAcoPuefsxUVLfRCxiGz4hHoiahxTIndCOz_6Bz0SMjL1VVLJ3Az1g3Bc_RLJ1-J3UQlcwQKLQeRGchIQcIAuShNj9vGHsz704PI3CC2uAygDF2YgmKE/s1600/triangulo+de+pascal+3+de+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdbGuu0YYJGlK2J-55O9bTR5UuAcoPuefsxUVLfRCxiGz4hHoiahxTIndCOz_6Bz0SMjL1VVLJ3Az1g3Bc_RLJ1-J3UQlcwQKLQeRGchIQcIAuShNj9vGHsz704PI3CC2uAygDF2YgmKE/s1600/triangulo+de+pascal+3+de+3.jpg" height="640" width="512" /></a><br />
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<div style="text-align: justify;">
F: And as the triangle is <b>symmetrical</b>, now I understand why <span class="unidas">(7 2)</span> is equal to <span class="unidas">(7 5)</span>.</div>
<div style="text-align: justify;">
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<div style="text-align: justify;">
<span style="color: #073763;">
V: Fabulous. Now, we only have to point out the number which is located at the <b>geometrical center</b> of the triangle we have drawn, because the dagger crossed the papers through their center. And we get number </span> <span style="color: red; font-size: 200%;">20</span>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7ZA0xjxHfv1ZR9MeWpzsmrG4dWIVA11cv-xhlgNwSYHKmDfPrHb8Rr3ozYbGZz4LEgQgFGlvwyWEvKK3EixX6dX8Bb1MLrNffyI1RJHR5yuTfm5Kio3DQ9fncpCB1AeYnq6S39hV_tEc/s1600/triangulo+de+pascal+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7ZA0xjxHfv1ZR9MeWpzsmrG4dWIVA11cv-xhlgNwSYHKmDfPrHb8Rr3ozYbGZz4LEgQgFGlvwyWEvKK3EixX6dX8Bb1MLrNffyI1RJHR5yuTfm5Kio3DQ9fncpCB1AeYnq6S39hV_tEc/s1600/triangulo+de+pascal+3.jpg" height="640" width="512" /></a><br />
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<div style="text-align: justify;">
F: Which player has this number? Go and arrest him! – Inspector Larry Flanagan cried.</div>
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<div style="text-align: justify;">
It was <b>Vladimiric</b>. The police didn’t take much time to catch him inside the stadium, and brought him to the locker room.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpPCN0hrWuQyp81kFq84vDC6oJWsYqr5_nIuZzAcwuDzzo9SbB8A5g9u3aEFZZ61TciH1W9NfkTsoTKMpDuOI0-NcG3znPfNzeQDU-Vo2HOgBMXQAwMUbIAmR15iYG0ZyAuL6TidPinrE/s1600/vladimiric.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpPCN0hrWuQyp81kFq84vDC6oJWsYqr5_nIuZzAcwuDzzo9SbB8A5g9u3aEFZZ61TciH1W9NfkTsoTKMpDuOI0-NcG3znPfNzeQDU-Vo2HOgBMXQAwMUbIAmR15iYG0ZyAuL6TidPinrE/s1600/vladimiric.jpg" height="640" width="512" /></a><br />
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<div style="text-align: justify;">
There, Vladimiric confessed his crime. He was Drakulic’s best friend since childhood. Drakulic had overcome him in every aspect of life: he had better academical records, was luckier with girls, had more friends and played football better. In fact, Drakulic was the captain of the national team of Transylvania, and Vladimiric was a reserve player, he played only a few minutes per match. But none of these things had been the cause that has lead to the fatal destiny. </div>
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<div style="text-align: justify;">
F: And then, why have you done it? - Larry Flanagan asked him.</div>
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<div style="text-align: justify;">
Well, Vladimiric said, there was only a pair of things in which Drakulic had never surpassed me: Mathematics and Logic problems. At least until today... When we were coming on the bus from the team hotel to the stadium, we made a bet on who was going to solve sooner the sudoku on the newspaper. And for the first time in our lives, Drakulic has been faster than me. I couldn’t bear it...<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw5QylyPnfLAxeIkBTf2P68ZEkJ_AzCLA_c2FOitoz9EElbzTAmplR_oI8NJKyAHCgIaLPLzTavj1p_GbqE-Ym5SWyqqLYIJQ_3CuDGGJ6s6MliTWb6TT9vcB5FrlYWv65a7ouasF_zSk/s1600/sudoku+3.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiw5QylyPnfLAxeIkBTf2P68ZEkJ_AzCLA_c2FOitoz9EElbzTAmplR_oI8NJKyAHCgIaLPLzTavj1p_GbqE-Ym5SWyqqLYIJQ_3CuDGGJ6s6MliTWb6TT9vcB5FrlYWv65a7ouasF_zSk/s1600/sudoku+3.gif" /></a></div>
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<span style="color: #20124d; display: none; font-family: Verdana, sans-serif;">Si te gustó esta historia, puedes votar por ella en <a href="http://www.meneame.net/story/sorprendente-caso-coche-extra" target="_blank">menéame</a> y <a href="http://divoblogger.com/story?id=41970" target="_blank">divoblogger</a>. Muchas gracias.</span>WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0Pernambuco, Brasil-8.8137173 -36.954107000000022-16.8625493 -47.281255500000022 -0.76488529999999955 -26.626958500000022tag:blogger.com,1999:blog-8244131673148079224.post-18395800980679896202014-06-09T18:58:00.002+01:002014-06-11T14:44:58.334+01:00Trees and googols at World Cup 2014<!-- Page resume -->
<span style="display: none;">A new mathematical cocktail whose stars are the World Cup in Brazil, the different systems of competition, the number of Wedderburn-Etherington and tree topology.
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(This post participes on the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">111th edition</span></a> of the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="http://boolesrings.org/krautzberger/" target="_blank"><span class="negrita unidas">Boole's Rings</span></a>.)</div>
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<span style="font-family: Verdana, sans-serif;">FIRST HALF</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjXaCL3jSgerecsQ99MimWpsLzfvKejImplT-FDPcGBHt5WRyWdxxCo350s_VGkhGPuhVI_WiJgzGJGc1QD68G9AwySNTmDOmwYOnzfWE6E26zKjgf9cEEmYVuUZa1-S_L3MrL_Br5l7o/s1600/im1+Brasil+2014.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="The World Cup Brazil 2014 in Matifutbol" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjXaCL3jSgerecsQ99MimWpsLzfvKejImplT-FDPcGBHt5WRyWdxxCo350s_VGkhGPuhVI_WiJgzGJGc1QD68G9AwySNTmDOmwYOnzfWE6E26zKjgf9cEEmYVuUZa1-S_L3MrL_Br5l7o/s1600/im1+Brasil+2014.jpg" height="320" title="The World Cup Brazil 2014 in Matifutbol" width="292" /></a>There’s a great commotion few days before the start of the <b>Football World Cup Brazil 2014</b>. The organizing committee has decided to introduce an <b>amendment at the last minute</b>.</div>
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<div style="text-align: justify;">
They want to add <b>more countries</b> to t he 32 nationsalready qualified for the final tournament. There’re many national teams that have been left out of the championship.<br />
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Given the huge stir generated by this World Championship, and the disappointment of many countries for failing to qualify, World Cup organizers have decided to increase the number of participants.</div>
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The <b>competition system</b> initially established for the 32 countries was as follows: a first stage based on a <b>league </b>format with 8 groups of 4 countries, the first 2 teams of which qualify for next round.</div>
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The 16 teams qualified will meet on 3 <b>knockout stages</b> till the final, where the winner will take the World Cup.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihDZdqbB5S1x1PA0K32STT6CFmMsFUcj6rm1ZWDFk3tTmD-ff6oBrGtwWnCLoXewehMtS8K8HrWG29qcIzeuXdjfYoCTzCJd4bUq3dgsnTlxVSKr47rIUHM0LEOrlLDNUksc2DCYn52wk/s1600/im2+WorldCupBrazil2014.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Match plan 2014 World Championship" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihDZdqbB5S1x1PA0K32STT6CFmMsFUcj6rm1ZWDFk3tTmD-ff6oBrGtwWnCLoXewehMtS8K8HrWG29qcIzeuXdjfYoCTzCJd4bUq3dgsnTlxVSKr47rIUHM0LEOrlLDNUksc2DCYn52wk/s1600/im2+WorldCupBrazil2014.jpg" height="449" title="Match plan 2014 World Championship" width="640" /></a></div>
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However, if they want to add more countries, and achieve a not over-extended championship, they must necessarily adopt a <b>new competition format, </b>discarding the initial leagues and performing a knockout format from the beginning.</div>
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Moreover, given the availability of the football stadiums and the logistical problems that may result from this modification, they have determined to bring the 64 matches originally listed with the old system down to <b>60 games</b>.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpyVTOqMHuvkTxIocii1oCE_wWeQvi_X-eSnbqpT3_DJcXBjW6Q5e8_5kENpa-yuSCqazqZ_b-9QgeUdUAODQQ-_ax4l6ek5nceTvSdxCUDMczmfbqc9lRYZDoS_sLnybeeoPthsMj8OI/s1600/im3+matematicos.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="Group of Math experts" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpyVTOqMHuvkTxIocii1oCE_wWeQvi_X-eSnbqpT3_DJcXBjW6Q5e8_5kENpa-yuSCqazqZ_b-9QgeUdUAODQQ-_ax4l6ek5nceTvSdxCUDMczmfbqc9lRYZDoS_sLnybeeoPthsMj8OI/s1600/im3+matematicos.jpg" height="265" title="Group of Math experts" width="400" /></a>The Local Organizing Committee has called a meeting with several experts from the Department of <b>Applied Mathematics</b> of the Ministry of Education.<br />
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At this meeting, the Committee chairman exposes to the other attendees the couple of issues that arise:</div>
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- The first issue consists on to determine <b>the maximum number of teams</b> that can participate if a total of 60 matches are held. We want the highest number of countries to come, because, first, we will have more visits of tourists from the participating countries, and second, we will have more income from television rights of those who choose to watch it at home.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaOam7a109n4DXSrzm6aSnX1x7kcGUWdqdVroq4Wlgm0leOe6FbWayDrBxtgsc85TvrWlHlaWCcgBXbKkPOB3ynj_CbEIK8wTU5QkjRXPT4i_qdtUttRLeyICOQEa2cgHqkCN6DXMSRx8/s1600/im7+habitacionconpan.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="The Organizing Committee and mathematical experts discussing how to organize the World Cup" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaOam7a109n4DXSrzm6aSnX1x7kcGUWdqdVroq4Wlgm0leOe6FbWayDrBxtgsc85TvrWlHlaWCcgBXbKkPOB3ynj_CbEIK8wTU5QkjRXPT4i_qdtUttRLeyICOQEa2cgHqkCN6DXMSRx8/s1600/im7+habitacionconpan.jpg" height="374" title="The Organizing Committee and mathematical experts discussing how to organize the World Cup" width="640" /></a></div>
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- The second one is to analyze <b>how many different ways</b> we can arrange the tournament with knockout matches.</div>
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- What do you mean exactly?</div>
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- I mean that the championship can be played in many ways. For example, we can establish that there are <b>8 seeded teams</b>, and make the other countries compete until there're only 8 teams to meet them in round of 16. Thus, if we represent the matches to dispute as small green rectangular stadiums, and teams with a blueish soccer clothing, our competition diagram looks like this: </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-DnhOrv2GdCipCtBqcsa_R6OYXN2gdaxsTbGPTa-s3crZClifPeWoQm1MQMYOHxgjZwPjCsYhO6rb28Amx5T7yHuxiDY5Mxjxk4X9PXGDwnLwbA5fxCAGgmNP98U8QMH-gkwqS3E-6CE/s1600/im4+8+cabezas+de+serie+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme with 8 seeded teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-DnhOrv2GdCipCtBqcsa_R6OYXN2gdaxsTbGPTa-s3crZClifPeWoQm1MQMYOHxgjZwPjCsYhO6rb28Amx5T7yHuxiDY5Mxjxk4X9PXGDwnLwbA5fxCAGgmNP98U8QMH-gkwqS3E-6CE/s1600/im4+8+cabezas+de+serie+2.jpg" height="168" title="Scheme with 8 seeded teams" width="640" /></a></div>
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- Or we can choose only <b>4 seeded teams</b> and the rest are eliminated until there're only 4 national teams to face them.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvGfU2GBre9FvWrHqAy6Fs6oCgvoVWwV9xwTe_BaVwackUMj2vQQSX01GuE_QehwBdnadNbu4Qpv4uWqnTR8zeC9e6BanoIu3klm7OMxhAEfHkuihxmurlhFQwwojGWd0oaZ4zEQb5UnA/s1600/im5+4+cabezas+de+serie+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme with 4 seeded teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvGfU2GBre9FvWrHqAy6Fs6oCgvoVWwV9xwTe_BaVwackUMj2vQQSX01GuE_QehwBdnadNbu4Qpv4uWqnTR8zeC9e6BanoIu3klm7OMxhAEfHkuihxmurlhFQwwojGWd0oaZ4zEQb5UnA/s1600/im5+4+cabezas+de+serie+2.jpg" height="176" title="Scheme with 4 seeded teams" width="640" /></a></div>
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- Or we set just <b>1 seeded team</b> (the host country, of course), and the rest are eliminated until there is only one to play the final with us. Anyway, all the variations you can think of.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9zWSMNHDzBlm0D78MoIQTiIXF1MA1j0olbAHuc430yl4_n3dbI9cLr4pEId_fkSTepWuvNZfr3IeuzAZ8eRQXqVlOcqOlVcGUGJ70soHWjCCRhLMWx9owUmPWqYhAI4Yxhit2Gf3q4zw/s1600/im6+1+cabeza+de+serie+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme with one only seeded team" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9zWSMNHDzBlm0D78MoIQTiIXF1MA1j0olbAHuc430yl4_n3dbI9cLr4pEId_fkSTepWuvNZfr3IeuzAZ8eRQXqVlOcqOlVcGUGJ70soHWjCCRhLMWx9owUmPWqYhAI4Yxhit2Gf3q4zw/s1600/im6+1+cabeza+de+serie+2.jpg" height="224" title="Scheme with one only seeded team" width="640" /></a></div>
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- I understand. But... it won’t be possible... It's a lot of work... for so little time.</div>
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- I don’t want excuses. Come on, let's get to work, and we’ll meet here again tomorrow.</div>
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The head of the Applied Mathematics Department is very concerned about these questions. Can you help him?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" height="252" width="640" /></span></a></div>
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<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
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The following day, the mathematical experts meet again the members of the Organizing Committee.</div>
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- We'll try to solve the first question. What do you think about it? What’s the <b>maximum number of countries</b> we can call if 60 matches are held?</div>
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- Well, I think the best will be to make a match plan, as in tennis tournaments. We’ll represent <b>the final</b> match with a small soccer field, and we’ll draw 2 lines from it, to the two teams that will play the match. We have <b>1 match</b>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEin1b2YOT4c8ZlhePrjEPgRNBTPFf9sYEQGIYUzYUecisOd9ZEh-E1SpueleENik31P_qenLbLTDPzi3H_sQ7qqlcQbX83rOJxVXeUVeQKG53P0zQRkB8f5BnEekwF91YoIOka2pfLPYwI/s1600/im8+final2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme of a championshipo with 2 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEin1b2YOT4c8ZlhePrjEPgRNBTPFf9sYEQGIYUzYUecisOd9ZEh-E1SpueleENik31P_qenLbLTDPzi3H_sQ7qqlcQbX83rOJxVXeUVeQKG53P0zQRkB8f5BnEekwF91YoIOka2pfLPYwI/s1600/im8+final2.jpg" height="174" title="Scheme of a championshipo with 2 teams" width="320" /></a></div>
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- Now we replace the shirts representing the teams with 2 pitches symbolizing the 2 <b>semifinal matches</b>. From each of them we draw another 2 lines, with the teams who play the matches. And if we add these 2 new matches to the one of the final, we already have <b>3 matches</b> in total.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNRSWqIMYJWp-4ebl0ln3KRrdTcnlUEEJjYsFdGg8kZTm_Yg7THcKnAQVXacMyfyowd5E7d1J3qaLRbIImJqtr-wDkPnphCOSDsfK8F-tGeZocAxFXhuplT0B45hjr8GiOb9iKy42QwQA/s1600/im9+semifinales2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme of a championship with 4 teams and 2 semifinals" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNRSWqIMYJWp-4ebl0ln3KRrdTcnlUEEJjYsFdGg8kZTm_Yg7THcKnAQVXacMyfyowd5E7d1J3qaLRbIImJqtr-wDkPnphCOSDsfK8F-tGeZocAxFXhuplT0B45hjr8GiOb9iKy42QwQA/s1600/im9+semifinales2.jpg" height="180" title="Scheme of a championship with 4 teams and 2 semifinals" width="320" /></a></div>
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- And we go on with the 4 <b>quarter-final</b> matches, the 8 <b>round of 16</b> matches and the 16<b> round of 32</b> matches. We now have a total of 1+2+4+8+16 = <b>31 matches</b>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLcEkHeh76q0_dx9Rcg2s8ubXo92N1Ml5Z9Xc9MhnBG3VYmP-MVKkwsLtjJGiOZoz9IndgD2FAd-GhYJelzsXbFzLTCbSSeKe26ogV6Z3EmtbF4fZIWp3uCpFc1Iw0P_DGLlFhtl3sn90/s1600/im10+dieciseisavos2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme of a championship with 32 teams and a direct elimination system" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLcEkHeh76q0_dx9Rcg2s8ubXo92N1Ml5Z9Xc9MhnBG3VYmP-MVKkwsLtjJGiOZoz9IndgD2FAd-GhYJelzsXbFzLTCbSSeKe26ogV6Z3EmtbF4fZIWp3uCpFc1Iw0P_DGLlFhtl3sn90/s1600/im10+dieciseisavos2.jpg" height="148" title="Scheme of a championship with 32 teams and a direct elimination system" width="640" /></a></div>
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- We still need 29 more games to complete the 60 matches we want to be played in the championship. So we can’t hold all the 32 matches of <b>round of 64</b>, since we would have a total of 31 + 32 = 63 games. You can count the soccer pitches in the scheme, to check it.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfUyxcTpPBrr-Za_YqhZm1V0-_Hzpo8G8kffSzV-xhbEXRZVhMWM9ROsFkXKr6H3sF1C02XnzyPnLq-Y5yasEvy8ycRPjf9Dfo2ofQTsDEZX22XxHU-mCcrHYXkPd-i-8Gcx_1gRAPaeA/s1600/im11+treintaidosavos+completos+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme of a championship of 64 teams and a direct elimination system" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfUyxcTpPBrr-Za_YqhZm1V0-_Hzpo8G8kffSzV-xhbEXRZVhMWM9ROsFkXKr6H3sF1C02XnzyPnLq-Y5yasEvy8ycRPjf9Dfo2ofQTsDEZX22XxHU-mCcrHYXkPd-i-8Gcx_1gRAPaeA/s1600/im11+treintaidosavos+completos+2.jpg" height="169" title="Scheme of a championship of 64 teams and a direct elimination system" width="640" /></a></div>
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- 3 matches are left over. So we remove 3 matches, and there’ll be 3 teams directly qualified for the round of 32.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGQ3Y-GrQIIB4WBG2IGDdJSNFOwMdyPF5_vBtvz4JaHKtEv0FSI3IHzKTXhT1swMMQOy7CepxyUcJFfNy8Wawj-C-SPJ4Arhke7DLT_Z9FKCiojm-oi749et5VMTglA7CrtlSweD37fQk/s1600/im12+treintaidosavos+incompletos+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Uncomplete round of 64, with 3 teams directly qualified for next round" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGQ3Y-GrQIIB4WBG2IGDdJSNFOwMdyPF5_vBtvz4JaHKtEv0FSI3IHzKTXhT1swMMQOy7CepxyUcJFfNy8Wawj-C-SPJ4Arhke7DLT_Z9FKCiojm-oi749et5VMTglA7CrtlSweD37fQk/s1600/im12+treintaidosavos+incompletos+2.jpg" height="169" title="Uncomplete round of 64, with 3 teams directly qualified for next round" width="640" /></a></div>
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- Now let's count the teams participating in the tournament: we must count all the shirts hanging like leaves at the ends of each branch.</div>
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- It's true, the diagram of the tournament looks like a tree. Well, after counting them, I get 61, that is, we can hold up to <b>61 countries</b>, right?</div>
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- That's right, whenever the tournament is arranged with this <b>competition scheme</b>. But what about if we manage another distinct trees? If we want to have 8 top seeded teams, or if we want to have 4 seeded teams, always respecting the number of 60 games, the trees would be:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgi4nnH-9GWtCeA-zLWhaK5bno0CDOurpmSia68WK4c2tB36q4b7WF2_k_V3hwi7troieHIrQmYYYACzNvN7aKZaXBFr6CjB_ANf2MciW4yjD5dtz76q4hA7WBMfjp_OcLvXJw6pEzTRyY/s1600/im13+8+cabezas+de+serie+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Competition scheme with 61 countries and 8 seeded teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgi4nnH-9GWtCeA-zLWhaK5bno0CDOurpmSia68WK4c2tB36q4b7WF2_k_V3hwi7troieHIrQmYYYACzNvN7aKZaXBFr6CjB_ANf2MciW4yjD5dtz76q4hA7WBMfjp_OcLvXJw6pEzTRyY/s1600/im13+8+cabezas+de+serie+2.jpg" height="259" title="Competition scheme with 61 countries and 8 seeded teams" width="640" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz-0yOzx7J2sVHVFrzmssFz9Y9lQhai5HYncOYGmX0ih2spcrpB5n7lSk_CEabRO4QiXI7FIEfaktc43HPITudZLpJXurIRmvj7x5r94UoVKisdqd7WM4aOvVDIp3uu8kuL2jA5ufylTc/s1600/im14+4+cabezas+de+serie+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Competition scheme with 61 countries and 4 seeded teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz-0yOzx7J2sVHVFrzmssFz9Y9lQhai5HYncOYGmX0ih2spcrpB5n7lSk_CEabRO4QiXI7FIEfaktc43HPITudZLpJXurIRmvj7x5r94UoVKisdqd7WM4aOvVDIp3uu8kuL2jA5ufylTc/s1600/im14+4+cabezas+de+serie+2.jpg" height="208" title="Competition scheme with 61 countries and 4 seeded teams" width="640" /></a></div>
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- What now? Will be there <b>more or less countries</b> this way? And what about if we think on other different trees?<br />
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- We can see that in these two examples we get also 61 countries, but I don't know what would happen to the rest of <b>trees </b>we can design... </div>
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- We’ll always get 61 countries, and we’ll use to a curious procedure to prove it.</div>
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- Let's think about the following: in all knockout games, there will always be a loser, that no longer will play more games. We just have to think that in 60 matches there will be 60 ‘losers’ countries, whom we have to add the final winner. Therefore, there will be 61 countries, regardless of the form the <b>competition schedule</b> takes.</div>
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-We can look at it in another way: if in every game we have represented a football pitch, we state that the team that wins the match and moves to the next phase occupies the <b>darker half</b>, while the <b>loser </b>is set in the <b>lighter </b>green <b>part</b>, we can see that there are 60 clearer parts corresponding to the 60 eliminated teams, to which it must be added the team that wins the tournament, who has always been on the dark area of its matches. So we have a total of <b>61 teams</b>. And it doesn’t matter the 'tree' of the competition that we form.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_4z-FwoDsk6layUMLkFvzBeP52eNmHTtXzaPz3S8rAr6bKi_z68X9mr-h7rYsPXYdwrthahwEGSzWrof97eVGqPhjfzdxTx7qlfCb9Hl48wNaSUXf8VRcRXvzbbbhWRlRk-3gcU6VdE0/s1600/im15+treintaidosavos+incompletos+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Competition scheme with 60 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_4z-FwoDsk6layUMLkFvzBeP52eNmHTtXzaPz3S8rAr6bKi_z68X9mr-h7rYsPXYdwrthahwEGSzWrof97eVGqPhjfzdxTx7qlfCb9Hl48wNaSUXf8VRcRXvzbbbhWRlRk-3gcU6VdE0/s1600/im15+treintaidosavos+incompletos+2.jpg" height="168" title="Competition scheme with 60 teams" width="640" /></a></div>
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- It's true. So we can say that the <b>first question</b> of finding the maximum number of countries is <b>solved</b>.</div>
<br />
<div style="text-align: justify;">
- Now let's study the second question: <b>how many different ways</b> we can arrange the tournament. In other words, we have to evaluate the distinct trees that we can design to <b>organize the championship </b>with 61 countries.</div>
<br />
<div style="text-align: justify;">
- Well, it doesn't seem very difficult. Let's start with a championship <b>without teams</b>. With zero participants, we could not organize <b>any championship</b>, right? </div>
<br />
<div style="text-align: justify;">
- Now consider a <b>one-team</b> championship. It's very easy. There’is only <b>one way</b> to do it: just give the trophy to the one team, and that's it.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOjIwwtbreWnEeu1m0y_viRPFwGNisfoTtx_ooeJxAvRHFG49bKuTX1MtNIMObYDhvElJJmQfM443KaYiLb7lXh66huUSRbbMG79j9ZavXM5F9dD7OluhnJsAqLgIr5L-l1HjoYFvBhQQ/s1600/im16+1+equipo.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Championship with one team" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOjIwwtbreWnEeu1m0y_viRPFwGNisfoTtx_ooeJxAvRHFG49bKuTX1MtNIMObYDhvElJJmQfM443KaYiLb7lXh66huUSRbbMG79j9ZavXM5F9dD7OluhnJsAqLgIr5L-l1HjoYFvBhQQ/s1600/im16+1+equipo.jpg" height="200" title="Championship with one team" width="178" /></a></div>
<br />
<div style="text-align: justify;">
- With <b>2 teams</b>, both would play the final. There would only be <b>one way</b> to organize it.</div>
<br />
<div style="text-align: justify;">
- Correct.</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0zfXR6N_SXEHpVqcMn6CJ5M6ePiBZY-8VPN9kJnj3zw-07iS8cWrvyrOIFnADrVr_yBY8By3UQhqN6l-1YQMPe7JAG_fqjF4NG8vENapxz10RTuYW4b1InyKlhUb80ooJz1ZQu5Fszjo/s1600/im17+final2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Championshipo with 2 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0zfXR6N_SXEHpVqcMn6CJ5M6ePiBZY-8VPN9kJnj3zw-07iS8cWrvyrOIFnADrVr_yBY8By3UQhqN6l-1YQMPe7JAG_fqjF4NG8vENapxz10RTuYW4b1InyKlhUb80ooJz1ZQu5Fszjo/s1600/im17+final2.jpg" height="175" title="Championshipo with 2 teams" width="320" /></a></div>
<br />
<div style="text-align: justify;">
- With <b>3 teams</b>, there’s also only <b>one </b>possible <b>way </b>to arrange the event: first we bring together 2 teams, and then the winner plays with the third team, right?</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5QWc-jRjIE0Qj69B0004alBTvPdJrILNYGliCbC9wQv6mzkzLtYfxhmjw5A9bE2IfHl1Iy88iY7LtMtDtq_Odc4z_CyowtrTncEE5yOIJEWg0Tuz-ZzBLY7tfZMgIbv2lVn3YWjVSzSI/s1600/im18+3+equipos+1+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Championship scheme with 3 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5QWc-jRjIE0Qj69B0004alBTvPdJrILNYGliCbC9wQv6mzkzLtYfxhmjw5A9bE2IfHl1Iy88iY7LtMtDtq_Odc4z_CyowtrTncEE5yOIJEWg0Tuz-ZzBLY7tfZMgIbv2lVn3YWjVSzSI/s1600/im18+3+equipos+1+2.jpg" height="200" title="Championship scheme with 3 teams" width="320" /></a></div>
<br />
<div style="text-align: justify;">
- Indeed. But if instead of bringing together the first and the second team, we make the first and the third team face each other, or the second and the third, then we have two additional arrangements, right?</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9muj0vP4GkbtUv5YBaVnNPQrV6N2Vt3_aIraSmh1dd_LhdLyhUkVU_cg_URZk1Eu60yDZm-8FvYqBZ7hpDd2TUj35k3JHYd_7Xc73e6b81X9rihGaWTjb_MXhznMkv9Do_0wcKv_p83I/s1600/im19+3+equipos+1+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Two different ways of organising a championshipo with 3 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9muj0vP4GkbtUv5YBaVnNPQrV6N2Vt3_aIraSmh1dd_LhdLyhUkVU_cg_URZk1Eu60yDZm-8FvYqBZ7hpDd2TUj35k3JHYd_7Xc73e6b81X9rihGaWTjb_MXhznMkv9Do_0wcKv_p83I/s1600/im19+3+equipos+1+2.jpg" height="130" title="Two different ways of organising a championshipo with 3 teams" width="400" /></a></div>
<br />
<div style="text-align: justify;">
- No. It’s true that the teams would be different, but the pattern of competition would be the same: at first, 2 teams play a match, and the winner plays with the remaining team. For now, what we want is to find out how many distinct match schedules we can make and decide for one of them. Then we'll see how to put each of the national teams within that scheme, okay? So we’ll go on painting all teams with the same color.</div>
<br />
<div style="text-align: justify;">
- Perfect. Now we'll work with 4 teams. The most usual is to make them confront each other on 2 semifinals, and the winners play the final.</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-xSwxw8MOwAX91aSQoqMC53XYiKxoD7YV41kKHjPn6MZJot6AXe_owAsKXCGpImHc9BAihpf_vMyKUm0kiIOi_jUbTjFgWDvUW9BjKoTZcsaLtX5tO0j4I818-81Fc0MBzJx-WcJAxN8/s1600/im20+4+equipos+1+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Competition scheme with 4 teams and 2 semifinals" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-xSwxw8MOwAX91aSQoqMC53XYiKxoD7YV41kKHjPn6MZJot6AXe_owAsKXCGpImHc9BAihpf_vMyKUm0kiIOi_jUbTjFgWDvUW9BjKoTZcsaLtX5tO0j4I818-81Fc0MBzJx-WcJAxN8/s1600/im20+4+equipos+1+2.jpg" height="174" title="Competition scheme with 4 teams and 2 semifinals" width="320" /></a></div>
<br />
- But there’s another way to deal with them: eliminating one by one.<br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEV7jYFsDVKgllfqlLA1bCiiMhSSKuap567G-_DU1xO0VINz4VlqDlMkxCcsweNfRA8NjK0BlxsI3QKhEIoAyS_htwRhb5witGgKBfEKl_hTlOm2lcjeL0bfye02QdXOuL5IvtHPUp0eo/s1600/im21+4+equipos+1+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Competition scheme with 4 teams and no semifinals" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEV7jYFsDVKgllfqlLA1bCiiMhSSKuap567G-_DU1xO0VINz4VlqDlMkxCcsweNfRA8NjK0BlxsI3QKhEIoAyS_htwRhb5witGgKBfEKl_hTlOm2lcjeL0bfye02QdXOuL5IvtHPUp0eo/s1600/im21+4+equipos+1+2.jpg" height="238" title="Competition scheme with 4 teams and no semifinals" width="320" /></a></div>
<br />
<span style="text-align: justify;">- So with <b>4 teams</b>, we have <b>2 different trees</b>. Let's see with<b> 5</b> teams: there are <b>3 possible schemes</b>.</span><br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhF_VaYkawhjUFQi6WPg2uMS0dH7XuW5sdyzTVCJxodvZIDwk5QaJZa2wmtq7-dIItcEa7aOjY9hn5dNeIPUNfoCLy9gmQWipFHrLHwdjvrlwNCzwmB7NdcutwOWCFzIJIqJFNbSmQPuZs/s1600/im22+5+equipos+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 5 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhF_VaYkawhjUFQi6WPg2uMS0dH7XuW5sdyzTVCJxodvZIDwk5QaJZa2wmtq7-dIItcEa7aOjY9hn5dNeIPUNfoCLy9gmQWipFHrLHwdjvrlwNCzwmB7NdcutwOWCFzIJIqJFNbSmQPuZs/s1600/im22+5+equipos+1.jpg" height="126" title="Possible competition scheme for a tournament with 5 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhChhwQXEKxyRtsyX9G5h8kpv5LDFVBCU9UDGsddFcGSbCM0Sxm7v_ln66zL_3Y9E31ZKghdPdfPsg300_Enud8uODGv4OZpjPnriSYDMRzmyfPFdFvBproAMD6eIdCompSTW5pmfibBxE/s1600/im23+5+equipos+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 5 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhChhwQXEKxyRtsyX9G5h8kpv5LDFVBCU9UDGsddFcGSbCM0Sxm7v_ln66zL_3Y9E31ZKghdPdfPsg300_Enud8uODGv4OZpjPnriSYDMRzmyfPFdFvBproAMD6eIdCompSTW5pmfibBxE/s1600/im23+5+equipos+2.jpg" height="141" title="Possible competition scheme for a tournament with 5 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidClEGE7c_Q10TbirfMesOwHIh6cVknfaKjbRAgGEJPXnLx_mFCZA08wPTXC9ZCmHOQbGJ0e4zIxpU5Puy2oBpDONhV3Jqnx0ugm85VBSMh5AbkLaWsuiUPR1wYdUkP1M_hxLXza-WyNI/s1600/im24+5+equipos+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 5 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidClEGE7c_Q10TbirfMesOwHIh6cVknfaKjbRAgGEJPXnLx_mFCZA08wPTXC9ZCmHOQbGJ0e4zIxpU5Puy2oBpDONhV3Jqnx0ugm85VBSMh5AbkLaWsuiUPR1wYdUkP1M_hxLXza-WyNI/s1600/im24+5+equipos+3.jpg" height="153" title="Possible competition scheme for a tournament with 5 teams" width="160" /></a></div>
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<br /></div>
- And with <b>6</b> teams, I can think of<b> 6</b> possible ways.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoorgf1YCiu5LpliRLmy6lY9qbrKmvDJsq1DV3K8oFTvNfnAL7EtIRmv8u02t7bW_VFL6OaKSKEJ41pgZNysCbnp6dJ308drP6yGJO97GzjidBa9hMVJ8FarhRaEnD7VEU9qADRQL4dzA/s1600/im25+6+equipos+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoorgf1YCiu5LpliRLmy6lY9qbrKmvDJsq1DV3K8oFTvNfnAL7EtIRmv8u02t7bW_VFL6OaKSKEJ41pgZNysCbnp6dJ308drP6yGJO97GzjidBa9hMVJ8FarhRaEnD7VEU9qADRQL4dzA/s1600/im25+6+equipos+1.jpg" height="171" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOURpJbZgXyr-PJXLnlq-nlg4WUtU_68BaV-5e7x7KQ4CyDNw7OcG6G5WKOhRwMH4SaiufQAnNq63FnKR3KwD8gP_mdj-h_TzPJqxA_DKFu3gDG2xcNb8Ytfd2nDVe1_giiK7JvLfrEV4/s1600/im26+6+equipos+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOURpJbZgXyr-PJXLnlq-nlg4WUtU_68BaV-5e7x7KQ4CyDNw7OcG6G5WKOhRwMH4SaiufQAnNq63FnKR3KwD8gP_mdj-h_TzPJqxA_DKFu3gDG2xcNb8Ytfd2nDVe1_giiK7JvLfrEV4/s1600/im26+6+equipos+2.jpg" height="153" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfGvbitasyIkJy2XNPwPmJas-bJYlEKuAx1ZaQRr3Ts36CE8FkZvJPIq6Re19lqoqfUOie7PptpIEQtZhvOXIRS8xbXVr5OqlSUEO6FVTvZxCGQNKkDu5S8GQPf-BL4x8dj0q32KxynKQ/s1600/im29+6+equipos+5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfGvbitasyIkJy2XNPwPmJas-bJYlEKuAx1ZaQRr3Ts36CE8FkZvJPIq6Re19lqoqfUOie7PptpIEQtZhvOXIRS8xbXVr5OqlSUEO6FVTvZxCGQNKkDu5S8GQPf-BL4x8dj0q32KxynKQ/s1600/im29+6+equipos+5.jpg" height="140" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjaGnfV4tWsb9xrw35vXCVpNtXiMu3yQNKocwom4v9kJK_pd75CTYvZnfbxK9sHchfbxuqFoVVJf1vs9zOyfzaZRvX9SQylDMaZ1Z-gZpw-1dOoEcZ_eFruwcd4fSURgnHlu-_BhlMKFgw/s1600/im27+6+equipos+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjaGnfV4tWsb9xrw35vXCVpNtXiMu3yQNKocwom4v9kJK_pd75CTYvZnfbxK9sHchfbxuqFoVVJf1vs9zOyfzaZRvX9SQylDMaZ1Z-gZpw-1dOoEcZ_eFruwcd4fSURgnHlu-_BhlMKFgw/s1600/im27+6+equipos+3.jpg" height="158" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtOYKv_1YNEwCp3m-QdxsQdR57QR1LsWhWcnQznTfivj3Nnp-rXyoxDycBcv9mwOio7Iy712SJOgAYrXXns3LsSsubMijmwNdfHTWaeKGPBcni-_UTrxTjrPCyDPzKeJXu8PyAN-cobUk/s1600/im28+6+equipos+4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtOYKv_1YNEwCp3m-QdxsQdR57QR1LsWhWcnQznTfivj3Nnp-rXyoxDycBcv9mwOio7Iy712SJOgAYrXXns3LsSsubMijmwNdfHTWaeKGPBcni-_UTrxTjrPCyDPzKeJXu8PyAN-cobUk/s1600/im28+6+equipos+4.jpg" height="160" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgYb1ELT-CLjS5oPr4Sj0np73gEs_DeXDgKMyoX3m2qjzT6WGjEI6ylKKSfbFh7LOZaDyMV4SJ-Gagdz4z0CijvFVZIJVxH-XEokfIGDy5mE4-L5wY5xgidAmKRzV4SIYRcWlSTqiaSPI/s1600/im30+6+equipos+6.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgYb1ELT-CLjS5oPr4Sj0np73gEs_DeXDgKMyoX3m2qjzT6WGjEI6ylKKSfbFh7LOZaDyMV4SJ-Gagdz4z0CijvFVZIJVxH-XEokfIGDy5mE4-L5wY5xgidAmKRzV4SIYRcWlSTqiaSPI/s1600/im30+6+equipos+6.jpg" height="125" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a></div>
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<br /></div>
<div style="text-align: justify;">
- With <b>7</b> teams we can form<b> 11</b> different trees. With <b>8</b> we build <b>23</b> trees, and with<b> 9</b> we can design <b>46</b> trees. But it's increasingly difficult to draw the different options. I think this way we won't get anywhere... </div>
<br />
<div style="text-align: justify;">
- Maybe we should find a formula to calculate how many distinct trees can be formed with 61 competitors. To do this, we'll study the <b>series </b>we have so far, to see if we can deduce how it follows.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCPb8KeR1R8KDzd5cNu2lmhJ1evVNgjOU9nHABxkaFvbrGBRGV2rZCxorNGWvY37kCoxOXbYMu3OvY1xk9TvKxyq1kT0YcRkovgVEiDMFFL2jRnFJuzPyyWIqFKc5RXGCx3lLk6fJQgCA/s1600/im31+serie+espa%25C3%25B1ol.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible different competition models, regarding on the number of participants" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCPb8KeR1R8KDzd5cNu2lmhJ1evVNgjOU9nHABxkaFvbrGBRGV2rZCxorNGWvY37kCoxOXbYMu3OvY1xk9TvKxyq1kT0YcRkovgVEiDMFFL2jRnFJuzPyyWIqFKc5RXGCx3lLk6fJQgCA/s1600/im31+serie+espa%25C3%25B1ol.jpg" title="Possible different competition models, regarding on the number of participants" /></a></div>
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<br /></div>
<div style="text-align: justify;">
- It doesn’t seem that the numbers we’ve obtained obey any <b>mathematical sequence</b>. Maybe we should search on internet for a solution to our problem.</div>
<br />
<div style="text-align: justify;">
- Yes, but the first thing we have to do is to define exactly what we're looking for.</div>
<br />
<div style="text-align: justify;">
<a href="http://es.wikipedia.org/wiki/%C3%81rbol_(teor%C3%ADa_de_grafos)" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" target="_blank"><img alt="Trees on Wikipedia" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNCQMZYLkh2J8yVfXZ8A1FXYD_NUzEDmW3IQqmM5K6ijyZOQRKkGSz9GrZWtmsgA3zl9Q4zt7IQ4IEIftMQSnMQnXQAXSkJV7FStxWEiI7xMCB_Fb8GpmPN_6sGThRy5nt3uAstiWruzM/s1600/im32+arbol.jpg" title="Trees on Wikipedia" /></a>- Well, these trees we're drawing are called '<b>graphs</b>' in mathematical terms. And as they have a hierarchical form, which simulates a tree (or a root), they’re also known by the name of ‘<b>trees</b>’ in Mathematics and Computer science.</div>
<br />
<div style="text-align: justify;">
- In addition, these trees are very special because from all of those little pitches (which we call <b>nodes</b>) representing the football matches, we start 2 lines (<b>branches </b>or children). There can’t be a game in which more than 2 teams meet, so we talk about ‘<b>binary trees</b>’. And no match can be played with only one team, so therefore, we say they’re '<b>strictly binary trees</b>’.</div>
<br />
<div style="text-align: justify;">
- On the other hand, we said before that it doesn’t matter the order of the different branches, as we're looking at the structure of the tree, and it doesn't care who plays the matches. So we're talking about ‘<b>strictly unordered binary trees</b>’.</div>
<br />
<div style="text-align: justify;">
- We just have to use the internet browser, write 'how many strictly unordered binary trees can be formed', and wait for the results...</div>
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisr0XhQckUduQF0imUlsTEACs4n86EtSHcJWWmca2c_hvn3rO1GbPUVPQr75F0_49N1bK8tFF8x71hYObAk6pBWStJUG_36dC6Rf-tqyGqZdmVXWR28hRs95tdg4sosNER_CfYE8E8LG4/s1600/im33+habitacionconpan.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="The mathematical experts and the Organising Committee are going to search on internet a solution for the problem" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisr0XhQckUduQF0imUlsTEACs4n86EtSHcJWWmca2c_hvn3rO1GbPUVPQr75F0_49N1bK8tFF8x71hYObAk6pBWStJUG_36dC6Rf-tqyGqZdmVXWR28hRs95tdg4sosNER_CfYE8E8LG4/s1600/im33+habitacionconpan.jpg" height="374" title="The mathematical experts and the Organising Committee are going to search on internet a solution for the problem" width="640" /></a><br />
<br />
<div style="text-align: justify;">
- Well, there aren’t many results. But I think none of them seems to fit what we're looking for. Perhaps we are not approaching the problem correctly.</div>
<br />
<div style="text-align: justify;">
- Maybe it's better to resume again the series of trees that we had previously calculated:</div>
<br />
<div style="text-align: center;">
<span style="font-size: x-large;">0 1 1 1 2 3 6 11 23 46 98</span></div>
<br />
<div style="text-align: justify;">
- And we introduce it on this very useful website, which recognizes all kinds of series from its terms:</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://oeis.org/" target="_blank"><img alt="OEIS. The On-Line Encyclopedia of Integer Sequences" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVkQWfEGk7JWhbRpEgXqlX_25NwhzeWLRUEbBEhm-WWaB9Q1387PFryWrOcCNWWYkIBV2FrPXtLTOjt5T_UvwyvnRmHIn-z3fy7-ma-LCKUl5MnitNq0FAMPcdhk0jBSHHeFRZqVkqQw8/s1600/im34+web+de+series.JPG" height="326" title="OEIS. The On-Line Encyclopedia of Integer Sequences" width="400" /></a></div>
<br />
<br />
<div style="text-align: justify;">
<a href="https://oeis.org/A001190/b001190.txt" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" target="_blank"><img alt="First 200 terms ot the series A001190 of Wedderburn-Etherington" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxUkDb4MQvkx5zP4VOxXSasGlj84sBeY_fXP0-Vmi6SYFvtVbjR-iYHIfHGV4OrLmoAIqdM_v1_TeycrStZ2eTSEkoqepeO3xVux4AYsfEV0hS-qb9htPd2E8PH_c-rKjbBJaFDSmjAXs/s1600/im35+serie+completa.JPG" height="200" title="First 200 terms ot the series A001190 of Wedderburn-Etherington" width="71" /></a>- Aha! Here we've got the solution: it’s <a href="https://oeis.org/A001190" target="_blank">series number A001190</a>, called the <a href="http://en.wikipedia.org/wiki/Wedderburn%E2%80%93Etherington_number" target="_blank">Wedderburn-Etherington</a> series. We can see that within the utilities described on the web, there’s the one concerned with our problem: the number of distinct ways of organizing a single-elimination tournament for n players (with the players name left blank), that's it, the amount of <b>possible different schedules</b> that can be set for a number n of participants.</div>
<br />
<div style="text-align: justify;">
- Now we just have to click the link where <b>T. D. Noe</b> calculates a table with <b>the first 200 terms of the series</b>, and pick the result for 61 teams:</div>
<br />
<div style="text-align: center;">
<span style="text-align: justify;"><span style="font-size: x-large;">844206159208807054529</span></span></div>
<br />
<div style="text-align: justify;">
- What a number! Let's put the commas, for to have an accurate idea of its magnitude.</div>
<br />
<div style="text-align: center;">
<span style="font-size: x-large;">844,206,159,208,807,054,529</span></div>
<br />
<div style="text-align: justify;">
- Over 844 trillion different trees! Even in the Amazon rainforest there aren’t so many trees!</div>
<br />
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDXfMszpbw2OKJgvy7xjEt8fj8nzIW_8MbJWkqjnheFH2XfiCzKdMLKf0GEECEvh-EohM_zgBf6eaDlDcSz4IM-ti7gOt2KEOcPz88jH45NKUVnU2VdCwad-LcV3s4PYuB1NTCJKCau5A/s1600/im36+habitantes+de+brasil.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Approximately Brazilian population in 2013" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDXfMszpbw2OKJgvy7xjEt8fj8nzIW_8MbJWkqjnheFH2XfiCzKdMLKf0GEECEvh-EohM_zgBf6eaDlDcSz4IM-ti7gOt2KEOcPz88jH45NKUVnU2VdCwad-LcV3s4PYuB1NTCJKCau5A/s1600/im36+habitantes+de+brasil.jpg" height="320" title="Approximately Brazilian population in 2013" width="320" /></a>- It looks like a pretty big number. And we said we wanted to study all the cases to see which system we adopt, right? I think we’ll have to work 24 hours a day.</div>
<br />
<div style="text-align: justify;">
- Well, let's think a bit. If one of us devotes 1 minute to each 'tree', and he/she doesn’t eat , sleep, nor rest, he/she needs a total of 160,617,610,995,447 years to finish the work.</div>
<br />
<div style="text-align: justify;">
- We should put to work on it all the Department.</div>
<br />
<div style="text-align: justify;">
- Nah, even if all Brazilians (over 201 million people), worked in this task it would take 7,989,625 years to complete it.</div>
<br />
<div style="text-align: justify;">
<span style="text-align: justify;">- Well, nowadays that almost everyone likes football, how about we ask the help of the entire <a href="http://www.worldometers.info/world-population/" target="_blank">world population</a>?</span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWb-T43VgMF5tFNVBKrBRAn6XKVu_3RIuBlteeMkA_6n92SZveZVFNF4-GPuDjcxNoWpigY7ZieYE0FRt7gJV2YiSRzt2_jw2EoMYIkbJjmVk1UkQJAR7dtQVXE4DaEjAXTLpvvH0gcvw/s1600/im37+mapamundi2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="World population" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWb-T43VgMF5tFNVBKrBRAn6XKVu_3RIuBlteeMkA_6n92SZveZVFNF4-GPuDjcxNoWpigY7ZieYE0FRt7gJV2YiSRzt2_jw2EoMYIkbJjmVk1UkQJAR7dtQVXE4DaEjAXTLpvvH0gcvw/s1600/im37+mapamundi2.jpg" height="292" title="World population" width="640" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<div style="text-align: justify;">
- In that case, it would take only <b>221,987 years</b> to study for 1 minute each of the possible ways of organizing our 61-teams tournament.<br />
<br />
- By the way, I assume you remember when I mentioned that we weren't interested in <b>differentiating </b>the various <b>teams</b>, and therefore we focused only on the structure of the championship.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9muj0vP4GkbtUv5YBaVnNPQrV6N2Vt3_aIraSmh1dd_LhdLyhUkVU_cg_URZk1Eu60yDZm-8FvYqBZ7hpDd2TUj35k3JHYd_7Xc73e6b81X9rihGaWTjb_MXhznMkv9Do_0wcKv_p83I/s1600/im19+3+equipos+1+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Two different ways of organising a 3-teams championship" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9muj0vP4GkbtUv5YBaVnNPQrV6N2Vt3_aIraSmh1dd_LhdLyhUkVU_cg_URZk1Eu60yDZm-8FvYqBZ7hpDd2TUj35k3JHYd_7Xc73e6b81X9rihGaWTjb_MXhznMkv9Do_0wcKv_p83I/s1600/im19+3+equipos+1+2.jpg" height="130" title="Two different ways of organising a 3-teams championship" width="400" /></a></div>
<br />
- Yes, but I didn’t quite understand why.<br />
<br />
- Let’s revisit the issue for one moment. We have 61 teams, so we want to know the number of different ways we can sort them. In Mathematics, we talk about the <b>permutations </b>that can occur in a set of 61 elements.<br />
<br />
- That’s right. And it’s easy to calculate. Any of the 61 national teams can be the first one to be chosen. For each one of them, we will pick another team among the remaining 60 teams. For each of these combinations of 2 teams, we have 59 teams to choose the third country. And so on.<br />
<br />
- Therefore, the total number of permutations of the 61 countries will be:<br />
<br />
<div style="text-align: center;">
<span style="font-size: x-large;">61 · 60 · 59 · 58 · ..... · 3 · 2 ·1 = 61!</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- Indeed, for brevity we use the expression '<b>factor of</b> 61’, represented by the symbol of admiration. And if we calculate the product, we get the following result:</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<div style="text-align: center;">
<span style="font-size: x-large;">61! = 5.0758 · 10 <sup>83</sup></span></div>
</div>
</div>
<br />
<div style="text-align: justify;">
- That’s a number five followed by 83 more figures. If you thought that the number of trees we saw before looked huge (<span style="font-size: large;">8,4421 · 10 <sup>20</sup></span>), imagine how it will be this new number. And if we multiply the number of trees by the number of forms we can fit the 61 teams in them, we get a lovely number :<br />
<br />
<div style="text-align: center;">
<span style="font-size: x-large;">4.2850 · 10 <sup>104</sup> = 42,850 · 10 <sup>100</sup></span></div>
<br />
- Forty-two thousand eight hundred and fifty <b>googols </b>of different ways of organizing our championship! We must remember that a googol is greater than the number of hydrogen atoms in the <b>known universe</b>...<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuW0jcD3zVDBEp74QYZY4iDEG8vQpth5yAsjhxX_cbZHiS9gTG8TtCm42ykxHMey6r3WnJPk1u5_-tN5Tcih2LmyJkPHlVdZ2R-wDDrYvH5oerrkZ5SX4okdA8Db0Hb3jlQDHcZCW8S2Q/s1600/universo.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Microwave background radiation of the Universe" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuW0jcD3zVDBEp74QYZY4iDEG8vQpth5yAsjhxX_cbZHiS9gTG8TtCm42ykxHMey6r3WnJPk1u5_-tN5Tcih2LmyJkPHlVdZ2R-wDDrYvH5oerrkZ5SX4okdA8Db0Hb3jlQDHcZCW8S2Q/s1600/universo.jpg" height="369" title="Microwave background radiation of the Universe" width="640" /></a></div>
<br />
- Uff . Then, I think we’d better use the <b>traditional method</b>: 29 matches of round of 64, freeing three countries in this first round, and full direct eliminatory rounds since round of 32.</div>
<br />
- Brazil, and two others.<br />
<br />
- Of course!<br />
<br />
<div style="text-align: justify;">
- Come on then! Let’s decide now which new countries we’re going to invite for completing the schedule of 61 teams. And we should think of organising something special for the opening ceremony: <a href="http://www.matifutbol.com/en/balls.html" target="_blank">What about giving away some balls?</a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMoQuXhBcpWD9AGqdDH6Le6O2U9WlrrHdEIkezcabbRPVaj716yaDtsXhTywXmgIB8zTSYFjChEpSGkQVnrkFB-obGvILkB0786rZHG40gdC9QjK-wYJvGjuczmavTzNJhoitUEirpl68/s1600/maracanaconbalones.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMoQuXhBcpWD9AGqdDH6Le6O2U9WlrrHdEIkezcabbRPVaj716yaDtsXhTywXmgIB8zTSYFjChEpSGkQVnrkFB-obGvILkB0786rZHG40gdC9QjK-wYJvGjuczmavTzNJhoitUEirpl68/s1600/maracanaconbalones.jpg" /></a></div>
<br />
<div style="text-align: justify;">
- Right. But first of all, if anyone is interested in this topic of <a href="http://en.wikipedia.org/wiki/Tree_structure" target="_blank">tree structures</a>, here you have some links to deepen on it: <a href="http://en.wikipedia.org/wiki/Tree_(graph_theory)" target="_blank">Tree (graph theory)</a>, <a href="http://webhome.cs.uvic.ca/~ruskey/Theses/GangLiMScThesis.pdf" target="_blank">Generation of Rooted Trees and Free Trees</a>, <a href="http://en.wikipedia.org/wiki/Binary_tree" target="_blank">Binary Tree</a>.</div>
<br />
<div style="text-align: justify;">
- And here you have other links, if you liked the story and want to share it with your friends.</div>
<br />
- Goodbye. And may the best win!<br />
<div class="separator" style="clear: both; text-align: center;">
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<span style="color: #20124d; display: none; font-family: Verdana, sans-serif;">Si te gustó esta historia, puedes votar por ella en <a href="https://www.blogger.com/xxx%20poner%20enlace%20a%20men%C3%A9ame%20xxx%20http://www.meneame.net/story/sorprendente-caso-coche-extra" target="_blank">menéame</a> y <a href="https://www.blogger.com/xxx%20poner%20enlace%20a%20divoblogger%20xxx%20http://divoblogger.com/story?id=41970" target="_blank">divoblogger</a>. Muchas gracias.</span>WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-73780403131591232412014-06-07T16:22:00.000+01:002014-06-11T14:45:18.513+01:00Trees and googols at World Cup 2014<!-- Page resume -->
<span style="display: none;">A new mathematical cocktail whose stars are the World Cup in Brazil, the different systems of competition, the number of Wedderburn-Etherington and tree topology.
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaOam7a109n4DXSrzm6aSnX1x7kcGUWdqdVroq4Wlgm0leOe6FbWayDrBxtgsc85TvrWlHlaWCcgBXbKkPOB3ynj_CbEIK8wTU5QkjRXPT4i_qdtUttRLeyICOQEa2cgHqkCN6DXMSRx8/s1600/im7+habitacionconpan.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaOam7a109n4DXSrzm6aSnX1x7kcGUWdqdVroq4Wlgm0leOe6FbWayDrBxtgsc85TvrWlHlaWCcgBXbKkPOB3ynj_CbEIK8wTU5QkjRXPT4i_qdtUttRLeyICOQEa2cgHqkCN6DXMSRx8/s1600/im7+habitacionconpan.jpg" /></a>
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(This post participes on the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">111th edition</span></a> of the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="http://boolesrings.org/krautzberger/" target="_blank"><span class="negrita unidas">Boole's Rings</span></a>.)</div>
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<span style="font-family: Verdana, sans-serif;">FIRST HALF</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjXaCL3jSgerecsQ99MimWpsLzfvKejImplT-FDPcGBHt5WRyWdxxCo350s_VGkhGPuhVI_WiJgzGJGc1QD68G9AwySNTmDOmwYOnzfWE6E26zKjgf9cEEmYVuUZa1-S_L3MrL_Br5l7o/s1600/im1+Brasil+2014.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img alt="The World Cup Brazil 2014 in Matifutbol" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjXaCL3jSgerecsQ99MimWpsLzfvKejImplT-FDPcGBHt5WRyWdxxCo350s_VGkhGPuhVI_WiJgzGJGc1QD68G9AwySNTmDOmwYOnzfWE6E26zKjgf9cEEmYVuUZa1-S_L3MrL_Br5l7o/s1600/im1+Brasil+2014.jpg" height="320" title="The World Cup Brazil 2014 in Matifutbol" width="292" /></a>There’s a great commotion few days before the start of the <b>Football World Cup Brazil 2014</b>. The organizing committee has decided to introduce an <b>amendment at the last minute</b>.</div>
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They want to add <b>more countries</b> to t he 32 nationsalready qualified for the final tournament. There’re many national teams that have been left out of the championship.<br />
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Given the huge stir generated by this World Championship, and the disappointment of many countries for failing to qualify, World Cup organizers have decided to increase the number of participants.</div>
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The <b>competition system</b> initially established for the 32 countries was as follows: a first stage based on a <b>league </b>format with 8 groups of 4 countries, the first 2 teams of which qualify for next round.</div>
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The 16 teams qualified will meet on 3 <b>knockout stages</b> till the final, where the winner will take the World Cup.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihDZdqbB5S1x1PA0K32STT6CFmMsFUcj6rm1ZWDFk3tTmD-ff6oBrGtwWnCLoXewehMtS8K8HrWG29qcIzeuXdjfYoCTzCJd4bUq3dgsnTlxVSKr47rIUHM0LEOrlLDNUksc2DCYn52wk/s1600/im2+WorldCupBrazil2014.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Match plan 2014 World Championship" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihDZdqbB5S1x1PA0K32STT6CFmMsFUcj6rm1ZWDFk3tTmD-ff6oBrGtwWnCLoXewehMtS8K8HrWG29qcIzeuXdjfYoCTzCJd4bUq3dgsnTlxVSKr47rIUHM0LEOrlLDNUksc2DCYn52wk/s1600/im2+WorldCupBrazil2014.jpg" height="449" title="Match plan 2014 World Championship" width="640" /></a></div>
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However, if they want to add more countries, and achieve a not over-extended championship, they must necessarily adopt a <b>new competition format, </b>discarding the initial leagues and performing a knockout format from the beginning.</div>
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Moreover, given the availability of the football stadiums and the logistical problems that may result from this modification, they have determined to bring the 64 matches originally listed with the old system down to <b>60 games</b>.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpyVTOqMHuvkTxIocii1oCE_wWeQvi_X-eSnbqpT3_DJcXBjW6Q5e8_5kENpa-yuSCqazqZ_b-9QgeUdUAODQQ-_ax4l6ek5nceTvSdxCUDMczmfbqc9lRYZDoS_sLnybeeoPthsMj8OI/s1600/im3+matematicos.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="Group of Math experts" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpyVTOqMHuvkTxIocii1oCE_wWeQvi_X-eSnbqpT3_DJcXBjW6Q5e8_5kENpa-yuSCqazqZ_b-9QgeUdUAODQQ-_ax4l6ek5nceTvSdxCUDMczmfbqc9lRYZDoS_sLnybeeoPthsMj8OI/s1600/im3+matematicos.jpg" height="265" title="Group of Math experts" width="400" /></a>The Local Organizing Committee has called a meeting with several experts from the Department of <b>Applied Mathematics</b> of the Ministry of Education.<br />
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At this meeting, the Committee chairman exposes to the other attendees the couple of issues that arise:</div>
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- The first issue consists on to determine <b>the maximum number of teams</b> that can participate if a total of 60 matches are held. We want the highest number of countries to come, because, first, we will have more visits of tourists from the participating countries, and second, we will have more income from television rights of those who choose to watch it at home.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaOam7a109n4DXSrzm6aSnX1x7kcGUWdqdVroq4Wlgm0leOe6FbWayDrBxtgsc85TvrWlHlaWCcgBXbKkPOB3ynj_CbEIK8wTU5QkjRXPT4i_qdtUttRLeyICOQEa2cgHqkCN6DXMSRx8/s1600/im7+habitacionconpan.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="The Organizing Committee and mathematical experts discussing how to organize the World Cup" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaOam7a109n4DXSrzm6aSnX1x7kcGUWdqdVroq4Wlgm0leOe6FbWayDrBxtgsc85TvrWlHlaWCcgBXbKkPOB3ynj_CbEIK8wTU5QkjRXPT4i_qdtUttRLeyICOQEa2cgHqkCN6DXMSRx8/s1600/im7+habitacionconpan.jpg" height="374" title="The Organizing Committee and mathematical experts discussing how to organize the World Cup" width="640" /></a></div>
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- The second one is to analyze <b>how many different ways</b> we can arrange the tournament with knockout matches.</div>
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- What do you mean exactly?</div>
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- I mean that the championship can be played in many ways. For example, we can establish that there are <b>8 seeded teams</b>, and make the other countries compete until there're only 8 teams to meet them in round of 16. Thus, if we represent the matches to dispute as small green rectangular stadiums, and teams with a blueish soccer clothing, our competition diagram looks like this: </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-DnhOrv2GdCipCtBqcsa_R6OYXN2gdaxsTbGPTa-s3crZClifPeWoQm1MQMYOHxgjZwPjCsYhO6rb28Amx5T7yHuxiDY5Mxjxk4X9PXGDwnLwbA5fxCAGgmNP98U8QMH-gkwqS3E-6CE/s1600/im4+8+cabezas+de+serie+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme with 8 seeded teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-DnhOrv2GdCipCtBqcsa_R6OYXN2gdaxsTbGPTa-s3crZClifPeWoQm1MQMYOHxgjZwPjCsYhO6rb28Amx5T7yHuxiDY5Mxjxk4X9PXGDwnLwbA5fxCAGgmNP98U8QMH-gkwqS3E-6CE/s1600/im4+8+cabezas+de+serie+2.jpg" height="168" title="Scheme with 8 seeded teams" width="640" /></a></div>
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- Or we can choose only <b>4 seeded teams</b> and the rest are eliminated until there're only 4 national teams to face them.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvGfU2GBre9FvWrHqAy6Fs6oCgvoVWwV9xwTe_BaVwackUMj2vQQSX01GuE_QehwBdnadNbu4Qpv4uWqnTR8zeC9e6BanoIu3klm7OMxhAEfHkuihxmurlhFQwwojGWd0oaZ4zEQb5UnA/s1600/im5+4+cabezas+de+serie+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme with 4 seeded teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvGfU2GBre9FvWrHqAy6Fs6oCgvoVWwV9xwTe_BaVwackUMj2vQQSX01GuE_QehwBdnadNbu4Qpv4uWqnTR8zeC9e6BanoIu3klm7OMxhAEfHkuihxmurlhFQwwojGWd0oaZ4zEQb5UnA/s1600/im5+4+cabezas+de+serie+2.jpg" height="176" title="Scheme with 4 seeded teams" width="640" /></a></div>
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- Or we set just <b>1 seeded team</b> (the host country, of course), and the rest are eliminated until there is only one to play the final with us. Anyway, all the variations you can think of.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9zWSMNHDzBlm0D78MoIQTiIXF1MA1j0olbAHuc430yl4_n3dbI9cLr4pEId_fkSTepWuvNZfr3IeuzAZ8eRQXqVlOcqOlVcGUGJ70soHWjCCRhLMWx9owUmPWqYhAI4Yxhit2Gf3q4zw/s1600/im6+1+cabeza+de+serie+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme with one only seeded team" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9zWSMNHDzBlm0D78MoIQTiIXF1MA1j0olbAHuc430yl4_n3dbI9cLr4pEId_fkSTepWuvNZfr3IeuzAZ8eRQXqVlOcqOlVcGUGJ70soHWjCCRhLMWx9owUmPWqYhAI4Yxhit2Gf3q4zw/s1600/im6+1+cabeza+de+serie+2.jpg" height="224" title="Scheme with one only seeded team" width="640" /></a></div>
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- I understand. But... it won’t be possible... It's a lot of work... for so little time.</div>
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- I don’t want excuses. Come on, let's get to work, and we’ll meet here again tomorrow.</div>
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The head of the Applied Mathematics Department is very concerned about these questions. Can you help him?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" height="252" width="640" /></span></a></div>
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<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
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The following day, the mathematical experts meet again the members of the Organizing Committee.</div>
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- We'll try to solve the first question. What do you think about it? What’s the <b>maximum number of countries</b> we can call if 60 matches are held?</div>
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- Well, I think the best will be to make a match plan, as in tennis tournaments. We’ll represent <b>the final</b> match with a small soccer field, and we’ll draw 2 lines from it, to the two teams that will play the match. We have <b>1 match</b>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEin1b2YOT4c8ZlhePrjEPgRNBTPFf9sYEQGIYUzYUecisOd9ZEh-E1SpueleENik31P_qenLbLTDPzi3H_sQ7qqlcQbX83rOJxVXeUVeQKG53P0zQRkB8f5BnEekwF91YoIOka2pfLPYwI/s1600/im8+final2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme of a championshipo with 2 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEin1b2YOT4c8ZlhePrjEPgRNBTPFf9sYEQGIYUzYUecisOd9ZEh-E1SpueleENik31P_qenLbLTDPzi3H_sQ7qqlcQbX83rOJxVXeUVeQKG53P0zQRkB8f5BnEekwF91YoIOka2pfLPYwI/s1600/im8+final2.jpg" height="174" title="Scheme of a championshipo with 2 teams" width="320" /></a></div>
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- Now we replace the shirts representing the teams with 2 pitches symbolizing the 2 <b>semifinal matches</b>. From each of them we draw another 2 lines, with the teams who play the matches. And if we add these 2 new matches to the one of the final, we already have <b>3 matches</b> in total.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNRSWqIMYJWp-4ebl0ln3KRrdTcnlUEEJjYsFdGg8kZTm_Yg7THcKnAQVXacMyfyowd5E7d1J3qaLRbIImJqtr-wDkPnphCOSDsfK8F-tGeZocAxFXhuplT0B45hjr8GiOb9iKy42QwQA/s1600/im9+semifinales2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme of a championship with 4 teams and 2 semifinals" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNRSWqIMYJWp-4ebl0ln3KRrdTcnlUEEJjYsFdGg8kZTm_Yg7THcKnAQVXacMyfyowd5E7d1J3qaLRbIImJqtr-wDkPnphCOSDsfK8F-tGeZocAxFXhuplT0B45hjr8GiOb9iKy42QwQA/s1600/im9+semifinales2.jpg" height="180" title="Scheme of a championship with 4 teams and 2 semifinals" width="320" /></a></div>
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- And we go on with the 4 <b>quarter-final</b> matches, the 8 <b>round of 16</b> matches and the 16<b> round of 32</b> matches. We now have a total of 1+2+4+8+16 = <b>31 matches</b>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLcEkHeh76q0_dx9Rcg2s8ubXo92N1Ml5Z9Xc9MhnBG3VYmP-MVKkwsLtjJGiOZoz9IndgD2FAd-GhYJelzsXbFzLTCbSSeKe26ogV6Z3EmtbF4fZIWp3uCpFc1Iw0P_DGLlFhtl3sn90/s1600/im10+dieciseisavos2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme of a championship with 32 teams and a direct elimination system" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLcEkHeh76q0_dx9Rcg2s8ubXo92N1Ml5Z9Xc9MhnBG3VYmP-MVKkwsLtjJGiOZoz9IndgD2FAd-GhYJelzsXbFzLTCbSSeKe26ogV6Z3EmtbF4fZIWp3uCpFc1Iw0P_DGLlFhtl3sn90/s1600/im10+dieciseisavos2.jpg" height="148" title="Scheme of a championship with 32 teams and a direct elimination system" width="640" /></a></div>
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- We still need 29 more games to complete the 60 matches we want to be played in the championship. So we can’t hold all the 32 matches of <b>round of 64</b>, since we would have a total of 31 + 32 = 63 games. You can count the soccer pitches in the scheme, to check it.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfUyxcTpPBrr-Za_YqhZm1V0-_Hzpo8G8kffSzV-xhbEXRZVhMWM9ROsFkXKr6H3sF1C02XnzyPnLq-Y5yasEvy8ycRPjf9Dfo2ofQTsDEZX22XxHU-mCcrHYXkPd-i-8Gcx_1gRAPaeA/s1600/im11+treintaidosavos+completos+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Scheme of a championship of 64 teams and a direct elimination system" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfUyxcTpPBrr-Za_YqhZm1V0-_Hzpo8G8kffSzV-xhbEXRZVhMWM9ROsFkXKr6H3sF1C02XnzyPnLq-Y5yasEvy8ycRPjf9Dfo2ofQTsDEZX22XxHU-mCcrHYXkPd-i-8Gcx_1gRAPaeA/s1600/im11+treintaidosavos+completos+2.jpg" height="169" title="Scheme of a championship of 64 teams and a direct elimination system" width="640" /></a></div>
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- 3 matches are left over. So we remove 3 matches, and there’ll be 3 teams directly qualified for the round of 32.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGQ3Y-GrQIIB4WBG2IGDdJSNFOwMdyPF5_vBtvz4JaHKtEv0FSI3IHzKTXhT1swMMQOy7CepxyUcJFfNy8Wawj-C-SPJ4Arhke7DLT_Z9FKCiojm-oi749et5VMTglA7CrtlSweD37fQk/s1600/im12+treintaidosavos+incompletos+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Uncomplete round of 64, with 3 teams directly qualified for next round" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGQ3Y-GrQIIB4WBG2IGDdJSNFOwMdyPF5_vBtvz4JaHKtEv0FSI3IHzKTXhT1swMMQOy7CepxyUcJFfNy8Wawj-C-SPJ4Arhke7DLT_Z9FKCiojm-oi749et5VMTglA7CrtlSweD37fQk/s1600/im12+treintaidosavos+incompletos+2.jpg" height="169" title="Uncomplete round of 64, with 3 teams directly qualified for next round" width="640" /></a></div>
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- Now let's count the teams participating in the tournament: we must count all the shirts hanging like leaves at the ends of each branch.</div>
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<div style="text-align: justify;">
- It's true, the diagram of the tournament looks like a tree. Well, after counting them, I get 61, that is, we can hold up to <b>61 countries</b>, right?</div>
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<div style="text-align: justify;">
- That's right, whenever the tournament is arranged with this <b>competition scheme</b>. But what about if we manage another distinct trees? If we want to have 8 top seeded teams, or if we want to have 4 seeded teams, always respecting the number of 60 games, the trees would be:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgi4nnH-9GWtCeA-zLWhaK5bno0CDOurpmSia68WK4c2tB36q4b7WF2_k_V3hwi7troieHIrQmYYYACzNvN7aKZaXBFr6CjB_ANf2MciW4yjD5dtz76q4hA7WBMfjp_OcLvXJw6pEzTRyY/s1600/im13+8+cabezas+de+serie+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Competition scheme with 61 countries and 8 seeded teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgi4nnH-9GWtCeA-zLWhaK5bno0CDOurpmSia68WK4c2tB36q4b7WF2_k_V3hwi7troieHIrQmYYYACzNvN7aKZaXBFr6CjB_ANf2MciW4yjD5dtz76q4hA7WBMfjp_OcLvXJw6pEzTRyY/s1600/im13+8+cabezas+de+serie+2.jpg" height="259" title="Competition scheme with 61 countries and 8 seeded teams" width="640" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz-0yOzx7J2sVHVFrzmssFz9Y9lQhai5HYncOYGmX0ih2spcrpB5n7lSk_CEabRO4QiXI7FIEfaktc43HPITudZLpJXurIRmvj7x5r94UoVKisdqd7WM4aOvVDIp3uu8kuL2jA5ufylTc/s1600/im14+4+cabezas+de+serie+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Competition scheme with 61 countries and 4 seeded teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz-0yOzx7J2sVHVFrzmssFz9Y9lQhai5HYncOYGmX0ih2spcrpB5n7lSk_CEabRO4QiXI7FIEfaktc43HPITudZLpJXurIRmvj7x5r94UoVKisdqd7WM4aOvVDIp3uu8kuL2jA5ufylTc/s1600/im14+4+cabezas+de+serie+2.jpg" height="208" title="Competition scheme with 61 countries and 4 seeded teams" width="640" /></a></div>
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<div style="text-align: justify;">
- What now? Will be there <b>more or less countries</b> this way? And what about if we think on other different trees?<br />
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- We can see that in these two examples we get also 61 countries, but I don't know what would happen to the rest of <b>trees </b>we can design... </div>
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<div style="text-align: justify;">
- We’ll always get 61 countries, and we’ll use to a curious procedure to prove it.</div>
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<div style="text-align: justify;">
- Let's think about the following: in all knockout games, there will always be a loser, that no longer will play more games. We just have to think that in 60 matches there will be 60 ‘losers’ countries, whom we have to add the final winner. Therefore, there will be 61 countries, regardless of the form the <b>competition schedule</b> takes.</div>
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<div style="text-align: justify;">
-We can look at it in another way: if in every game we have represented a football pitch, we state that the team that wins the match and moves to the next phase occupies the <b>darker half</b>, while the <b>loser </b>is set in the <b>lighter </b>green <b>part</b>, we can see that there are 60 clearer parts corresponding to the 60 eliminated teams, to which it must be added the team that wins the tournament, who has always been on the dark area of its matches. So we have a total of <b>61 teams</b>. And it doesn’t matter the 'tree' of the competition that we form.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_4z-FwoDsk6layUMLkFvzBeP52eNmHTtXzaPz3S8rAr6bKi_z68X9mr-h7rYsPXYdwrthahwEGSzWrof97eVGqPhjfzdxTx7qlfCb9Hl48wNaSUXf8VRcRXvzbbbhWRlRk-3gcU6VdE0/s1600/im15+treintaidosavos+incompletos+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Competition scheme with 60 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_4z-FwoDsk6layUMLkFvzBeP52eNmHTtXzaPz3S8rAr6bKi_z68X9mr-h7rYsPXYdwrthahwEGSzWrof97eVGqPhjfzdxTx7qlfCb9Hl48wNaSUXf8VRcRXvzbbbhWRlRk-3gcU6VdE0/s1600/im15+treintaidosavos+incompletos+2.jpg" height="168" title="Competition scheme with 60 teams" width="640" /></a></div>
<br />
<div style="text-align: justify;">
- It's true. So we can say that the <b>first question</b> of finding the maximum number of countries is <b>solved</b>.</div>
<br />
<div style="text-align: justify;">
- Now let's study the second question: <b>how many different ways</b> we can arrange the tournament. In other words, we have to evaluate the distinct trees that we can design to <b>organize the championship </b>with 61 countries.</div>
<br />
<div style="text-align: justify;">
- Well, it doesn't seem very difficult. Let's start with a championship <b>without teams</b>. With zero participants, we could not organize <b>any championship</b>, right? </div>
<br />
<div style="text-align: justify;">
- Now consider a <b>one-team</b> championship. It's very easy. There’is only <b>one way</b> to do it: just give the trophy to the one team, and that's it.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOjIwwtbreWnEeu1m0y_viRPFwGNisfoTtx_ooeJxAvRHFG49bKuTX1MtNIMObYDhvElJJmQfM443KaYiLb7lXh66huUSRbbMG79j9ZavXM5F9dD7OluhnJsAqLgIr5L-l1HjoYFvBhQQ/s1600/im16+1+equipo.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Championship with one team" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOjIwwtbreWnEeu1m0y_viRPFwGNisfoTtx_ooeJxAvRHFG49bKuTX1MtNIMObYDhvElJJmQfM443KaYiLb7lXh66huUSRbbMG79j9ZavXM5F9dD7OluhnJsAqLgIr5L-l1HjoYFvBhQQ/s1600/im16+1+equipo.jpg" height="200" title="Championship with one team" width="178" /></a></div>
<br />
<div style="text-align: justify;">
- With <b>2 teams</b>, both would play the final. There would only be <b>one way</b> to organize it.</div>
<br />
<div style="text-align: justify;">
- Correct.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0zfXR6N_SXEHpVqcMn6CJ5M6ePiBZY-8VPN9kJnj3zw-07iS8cWrvyrOIFnADrVr_yBY8By3UQhqN6l-1YQMPe7JAG_fqjF4NG8vENapxz10RTuYW4b1InyKlhUb80ooJz1ZQu5Fszjo/s1600/im17+final2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Championshipo with 2 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0zfXR6N_SXEHpVqcMn6CJ5M6ePiBZY-8VPN9kJnj3zw-07iS8cWrvyrOIFnADrVr_yBY8By3UQhqN6l-1YQMPe7JAG_fqjF4NG8vENapxz10RTuYW4b1InyKlhUb80ooJz1ZQu5Fszjo/s1600/im17+final2.jpg" height="175" title="Championshipo with 2 teams" width="320" /></a></div>
<br />
<div style="text-align: justify;">
- With <b>3 teams</b>, there’s also only <b>one </b>possible <b>way </b>to arrange the event: first we bring together 2 teams, and then the winner plays with the third team, right?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5QWc-jRjIE0Qj69B0004alBTvPdJrILNYGliCbC9wQv6mzkzLtYfxhmjw5A9bE2IfHl1Iy88iY7LtMtDtq_Odc4z_CyowtrTncEE5yOIJEWg0Tuz-ZzBLY7tfZMgIbv2lVn3YWjVSzSI/s1600/im18+3+equipos+1+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Championship scheme with 3 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5QWc-jRjIE0Qj69B0004alBTvPdJrILNYGliCbC9wQv6mzkzLtYfxhmjw5A9bE2IfHl1Iy88iY7LtMtDtq_Odc4z_CyowtrTncEE5yOIJEWg0Tuz-ZzBLY7tfZMgIbv2lVn3YWjVSzSI/s1600/im18+3+equipos+1+2.jpg" height="200" title="Championship scheme with 3 teams" width="320" /></a></div>
<br />
<div style="text-align: justify;">
- Indeed. But if instead of bringing together the first and the second team, we make the first and the third team face each other, or the second and the third, then we have two additional arrangements, right?</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9muj0vP4GkbtUv5YBaVnNPQrV6N2Vt3_aIraSmh1dd_LhdLyhUkVU_cg_URZk1Eu60yDZm-8FvYqBZ7hpDd2TUj35k3JHYd_7Xc73e6b81X9rihGaWTjb_MXhznMkv9Do_0wcKv_p83I/s1600/im19+3+equipos+1+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Two different ways of organising a championshipo with 3 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9muj0vP4GkbtUv5YBaVnNPQrV6N2Vt3_aIraSmh1dd_LhdLyhUkVU_cg_URZk1Eu60yDZm-8FvYqBZ7hpDd2TUj35k3JHYd_7Xc73e6b81X9rihGaWTjb_MXhznMkv9Do_0wcKv_p83I/s1600/im19+3+equipos+1+2.jpg" height="130" title="Two different ways of organising a championshipo with 3 teams" width="400" /></a></div>
<br />
<div style="text-align: justify;">
- No. It’s true that the teams would be different, but the pattern of competition would be the same: at first, 2 teams play a match, and the winner plays with the remaining team. For now, what we want is to find out how many distinct match schedules we can make and decide for one of them. Then we'll see how to put each of the national teams within that scheme, okay? So we’ll go on painting all teams with the same color.</div>
<br />
<div style="text-align: justify;">
- Perfect. Now we'll work with 4 teams. The most usual is to make them confront each other on 2 semifinals, and the winners play the final.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-xSwxw8MOwAX91aSQoqMC53XYiKxoD7YV41kKHjPn6MZJot6AXe_owAsKXCGpImHc9BAihpf_vMyKUm0kiIOi_jUbTjFgWDvUW9BjKoTZcsaLtX5tO0j4I818-81Fc0MBzJx-WcJAxN8/s1600/im20+4+equipos+1+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Competition scheme with 4 teams and 2 semifinals" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-xSwxw8MOwAX91aSQoqMC53XYiKxoD7YV41kKHjPn6MZJot6AXe_owAsKXCGpImHc9BAihpf_vMyKUm0kiIOi_jUbTjFgWDvUW9BjKoTZcsaLtX5tO0j4I818-81Fc0MBzJx-WcJAxN8/s1600/im20+4+equipos+1+2.jpg" height="174" title="Competition scheme with 4 teams and 2 semifinals" width="320" /></a></div>
<br />
- But there’s another way to deal with them: eliminating one by one.<br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEV7jYFsDVKgllfqlLA1bCiiMhSSKuap567G-_DU1xO0VINz4VlqDlMkxCcsweNfRA8NjK0BlxsI3QKhEIoAyS_htwRhb5witGgKBfEKl_hTlOm2lcjeL0bfye02QdXOuL5IvtHPUp0eo/s1600/im21+4+equipos+1+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Competition scheme with 4 teams and no semifinals" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEV7jYFsDVKgllfqlLA1bCiiMhSSKuap567G-_DU1xO0VINz4VlqDlMkxCcsweNfRA8NjK0BlxsI3QKhEIoAyS_htwRhb5witGgKBfEKl_hTlOm2lcjeL0bfye02QdXOuL5IvtHPUp0eo/s1600/im21+4+equipos+1+2.jpg" height="238" title="Competition scheme with 4 teams and no semifinals" width="320" /></a></div>
<br />
<span style="text-align: justify;">- So with <b>4 teams</b>, we have <b>2 different trees</b>. Let's see with<b> 5</b> teams: there are <b>3 possible schemes</b>.</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhF_VaYkawhjUFQi6WPg2uMS0dH7XuW5sdyzTVCJxodvZIDwk5QaJZa2wmtq7-dIItcEa7aOjY9hn5dNeIPUNfoCLy9gmQWipFHrLHwdjvrlwNCzwmB7NdcutwOWCFzIJIqJFNbSmQPuZs/s1600/im22+5+equipos+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 5 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhF_VaYkawhjUFQi6WPg2uMS0dH7XuW5sdyzTVCJxodvZIDwk5QaJZa2wmtq7-dIItcEa7aOjY9hn5dNeIPUNfoCLy9gmQWipFHrLHwdjvrlwNCzwmB7NdcutwOWCFzIJIqJFNbSmQPuZs/s1600/im22+5+equipos+1.jpg" height="126" title="Possible competition scheme for a tournament with 5 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhChhwQXEKxyRtsyX9G5h8kpv5LDFVBCU9UDGsddFcGSbCM0Sxm7v_ln66zL_3Y9E31ZKghdPdfPsg300_Enud8uODGv4OZpjPnriSYDMRzmyfPFdFvBproAMD6eIdCompSTW5pmfibBxE/s1600/im23+5+equipos+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 5 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhChhwQXEKxyRtsyX9G5h8kpv5LDFVBCU9UDGsddFcGSbCM0Sxm7v_ln66zL_3Y9E31ZKghdPdfPsg300_Enud8uODGv4OZpjPnriSYDMRzmyfPFdFvBproAMD6eIdCompSTW5pmfibBxE/s1600/im23+5+equipos+2.jpg" height="141" title="Possible competition scheme for a tournament with 5 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidClEGE7c_Q10TbirfMesOwHIh6cVknfaKjbRAgGEJPXnLx_mFCZA08wPTXC9ZCmHOQbGJ0e4zIxpU5Puy2oBpDONhV3Jqnx0ugm85VBSMh5AbkLaWsuiUPR1wYdUkP1M_hxLXza-WyNI/s1600/im24+5+equipos+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 5 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidClEGE7c_Q10TbirfMesOwHIh6cVknfaKjbRAgGEJPXnLx_mFCZA08wPTXC9ZCmHOQbGJ0e4zIxpU5Puy2oBpDONhV3Jqnx0ugm85VBSMh5AbkLaWsuiUPR1wYdUkP1M_hxLXza-WyNI/s1600/im24+5+equipos+3.jpg" height="153" title="Possible competition scheme for a tournament with 5 teams" width="160" /></a></div>
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<br /></div>
- And with <b>6</b> teams, I can think of<b> 6</b> possible ways.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoorgf1YCiu5LpliRLmy6lY9qbrKmvDJsq1DV3K8oFTvNfnAL7EtIRmv8u02t7bW_VFL6OaKSKEJ41pgZNysCbnp6dJ308drP6yGJO97GzjidBa9hMVJ8FarhRaEnD7VEU9qADRQL4dzA/s1600/im25+6+equipos+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoorgf1YCiu5LpliRLmy6lY9qbrKmvDJsq1DV3K8oFTvNfnAL7EtIRmv8u02t7bW_VFL6OaKSKEJ41pgZNysCbnp6dJ308drP6yGJO97GzjidBa9hMVJ8FarhRaEnD7VEU9qADRQL4dzA/s1600/im25+6+equipos+1.jpg" height="171" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOURpJbZgXyr-PJXLnlq-nlg4WUtU_68BaV-5e7x7KQ4CyDNw7OcG6G5WKOhRwMH4SaiufQAnNq63FnKR3KwD8gP_mdj-h_TzPJqxA_DKFu3gDG2xcNb8Ytfd2nDVe1_giiK7JvLfrEV4/s1600/im26+6+equipos+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOURpJbZgXyr-PJXLnlq-nlg4WUtU_68BaV-5e7x7KQ4CyDNw7OcG6G5WKOhRwMH4SaiufQAnNq63FnKR3KwD8gP_mdj-h_TzPJqxA_DKFu3gDG2xcNb8Ytfd2nDVe1_giiK7JvLfrEV4/s1600/im26+6+equipos+2.jpg" height="153" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfGvbitasyIkJy2XNPwPmJas-bJYlEKuAx1ZaQRr3Ts36CE8FkZvJPIq6Re19lqoqfUOie7PptpIEQtZhvOXIRS8xbXVr5OqlSUEO6FVTvZxCGQNKkDu5S8GQPf-BL4x8dj0q32KxynKQ/s1600/im29+6+equipos+5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfGvbitasyIkJy2XNPwPmJas-bJYlEKuAx1ZaQRr3Ts36CE8FkZvJPIq6Re19lqoqfUOie7PptpIEQtZhvOXIRS8xbXVr5OqlSUEO6FVTvZxCGQNKkDu5S8GQPf-BL4x8dj0q32KxynKQ/s1600/im29+6+equipos+5.jpg" height="140" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjaGnfV4tWsb9xrw35vXCVpNtXiMu3yQNKocwom4v9kJK_pd75CTYvZnfbxK9sHchfbxuqFoVVJf1vs9zOyfzaZRvX9SQylDMaZ1Z-gZpw-1dOoEcZ_eFruwcd4fSURgnHlu-_BhlMKFgw/s1600/im27+6+equipos+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjaGnfV4tWsb9xrw35vXCVpNtXiMu3yQNKocwom4v9kJK_pd75CTYvZnfbxK9sHchfbxuqFoVVJf1vs9zOyfzaZRvX9SQylDMaZ1Z-gZpw-1dOoEcZ_eFruwcd4fSURgnHlu-_BhlMKFgw/s1600/im27+6+equipos+3.jpg" height="158" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtOYKv_1YNEwCp3m-QdxsQdR57QR1LsWhWcnQznTfivj3Nnp-rXyoxDycBcv9mwOio7Iy712SJOgAYrXXns3LsSsubMijmwNdfHTWaeKGPBcni-_UTrxTjrPCyDPzKeJXu8PyAN-cobUk/s1600/im28+6+equipos+4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtOYKv_1YNEwCp3m-QdxsQdR57QR1LsWhWcnQznTfivj3Nnp-rXyoxDycBcv9mwOio7Iy712SJOgAYrXXns3LsSsubMijmwNdfHTWaeKGPBcni-_UTrxTjrPCyDPzKeJXu8PyAN-cobUk/s1600/im28+6+equipos+4.jpg" height="160" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgYb1ELT-CLjS5oPr4Sj0np73gEs_DeXDgKMyoX3m2qjzT6WGjEI6ylKKSfbFh7LOZaDyMV4SJ-Gagdz4z0CijvFVZIJVxH-XEokfIGDy5mE4-L5wY5xgidAmKRzV4SIYRcWlSTqiaSPI/s1600/im30+6+equipos+6.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible competition scheme for a tournament with 6 teams" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgYb1ELT-CLjS5oPr4Sj0np73gEs_DeXDgKMyoX3m2qjzT6WGjEI6ylKKSfbFh7LOZaDyMV4SJ-Gagdz4z0CijvFVZIJVxH-XEokfIGDy5mE4-L5wY5xgidAmKRzV4SIYRcWlSTqiaSPI/s1600/im30+6+equipos+6.jpg" height="125" title="Possible competition scheme for a tournament with 6 teams" width="160" /></a></div>
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<br /></div>
<div style="text-align: justify;">
- With <b>7</b> teams we can form<b> 11</b> different trees. With <b>8</b> we build <b>23</b> trees, and with<b> 9</b> we can design <b>46</b> trees. But it's increasingly difficult to draw the different options. I think this way we won't get anywhere... </div>
<br />
<div style="text-align: justify;">
- Maybe we should find a formula to calculate how many distinct trees can be formed with 61 competitors. To do this, we'll study the <b>series </b>we have so far, to see if we can deduce how it follows.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCPb8KeR1R8KDzd5cNu2lmhJ1evVNgjOU9nHABxkaFvbrGBRGV2rZCxorNGWvY37kCoxOXbYMu3OvY1xk9TvKxyq1kT0YcRkovgVEiDMFFL2jRnFJuzPyyWIqFKc5RXGCx3lLk6fJQgCA/s1600/im31+serie+espa%25C3%25B1ol.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Possible different competition models, regarding on the number of participants" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCPb8KeR1R8KDzd5cNu2lmhJ1evVNgjOU9nHABxkaFvbrGBRGV2rZCxorNGWvY37kCoxOXbYMu3OvY1xk9TvKxyq1kT0YcRkovgVEiDMFFL2jRnFJuzPyyWIqFKc5RXGCx3lLk6fJQgCA/s1600/im31+serie+espa%25C3%25B1ol.jpg" title="Possible different competition models, regarding on the number of participants" /></a></div>
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<br /></div>
<div style="text-align: justify;">
- It doesn’t seem that the numbers we’ve obtained obey any <b>mathematical sequence</b>. Maybe we should search on internet for a solution to our problem.</div>
<br />
<div style="text-align: justify;">
- Yes, but the first thing we have to do is to define exactly what we're looking for.</div>
<br />
<div style="text-align: justify;">
<a href="http://es.wikipedia.org/wiki/%C3%81rbol_(teor%C3%ADa_de_grafos)" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" target="_blank"><img alt="Trees on Wikipedia" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNCQMZYLkh2J8yVfXZ8A1FXYD_NUzEDmW3IQqmM5K6ijyZOQRKkGSz9GrZWtmsgA3zl9Q4zt7IQ4IEIftMQSnMQnXQAXSkJV7FStxWEiI7xMCB_Fb8GpmPN_6sGThRy5nt3uAstiWruzM/s1600/im32+arbol.jpg" title="Trees on Wikipedia" /></a>- Well, these trees we're drawing are called '<b>graphs</b>' in mathematical terms. And as they have a hierarchical form, which simulates a tree (or a root), they’re also known by the name of ‘<b>trees</b>’ in Mathematics and Computer science.</div>
<br />
<div style="text-align: justify;">
- In addition, these trees are very special because from all of those little pitches (which we call <b>nodes</b>) representing the football matches, we start 2 lines (<b>branches </b>or children). There can’t be a game in which more than 2 teams meet, so we talk about ‘<b>binary trees</b>’. And no match can be played with only one team, so therefore, we say they’re '<b>strictly binary trees</b>’.</div>
<br />
<div style="text-align: justify;">
- On the other hand, we said before that it doesn’t matter the order of the different branches, as we're looking at the structure of the tree, and it doesn't care who plays the matches. So we're talking about ‘<b>strictly unordered binary trees</b>’.</div>
<br />
<div style="text-align: justify;">
- We just have to use the internet browser, write 'how many strictly unordered binary trees can be formed', and wait for the results...</div>
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisr0XhQckUduQF0imUlsTEACs4n86EtSHcJWWmca2c_hvn3rO1GbPUVPQr75F0_49N1bK8tFF8x71hYObAk6pBWStJUG_36dC6Rf-tqyGqZdmVXWR28hRs95tdg4sosNER_CfYE8E8LG4/s1600/im33+habitacionconpan.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="The mathematical experts and the Organising Committee are going to search on internet a solution for the problem" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisr0XhQckUduQF0imUlsTEACs4n86EtSHcJWWmca2c_hvn3rO1GbPUVPQr75F0_49N1bK8tFF8x71hYObAk6pBWStJUG_36dC6Rf-tqyGqZdmVXWR28hRs95tdg4sosNER_CfYE8E8LG4/s1600/im33+habitacionconpan.jpg" height="374" title="The mathematical experts and the Organising Committee are going to search on internet a solution for the problem" width="640" /></a><br />
<br />
<div style="text-align: justify;">
- Well, there aren’t many results. But I think none of them seems to fit what we're looking for. Perhaps we are not approaching the problem correctly.</div>
<br />
<div style="text-align: justify;">
- Maybe it's better to resume again the series of trees that we had previously calculated:</div>
<br />
<div style="text-align: center;">
<span style="font-size: x-large;">0 1 1 1 2 3 6 11 23 46 98</span></div>
<br />
<div style="text-align: justify;">
- And we introduce it on this very useful website, which recognizes all kinds of series from its terms:</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://oeis.org/" target="_blank"><img alt="OEIS. The On-Line Encyclopedia of Integer Sequences" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVkQWfEGk7JWhbRpEgXqlX_25NwhzeWLRUEbBEhm-WWaB9Q1387PFryWrOcCNWWYkIBV2FrPXtLTOjt5T_UvwyvnRmHIn-z3fy7-ma-LCKUl5MnitNq0FAMPcdhk0jBSHHeFRZqVkqQw8/s1600/im34+web+de+series.JPG" height="326" title="OEIS. The On-Line Encyclopedia of Integer Sequences" width="400" /></a></div>
<br />
<br />
<div style="text-align: justify;">
<a href="https://oeis.org/A001190/b001190.txt" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" target="_blank"><img alt="First 200 terms ot the series A001190 of Wedderburn-Etherington" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxUkDb4MQvkx5zP4VOxXSasGlj84sBeY_fXP0-Vmi6SYFvtVbjR-iYHIfHGV4OrLmoAIqdM_v1_TeycrStZ2eTSEkoqepeO3xVux4AYsfEV0hS-qb9htPd2E8PH_c-rKjbBJaFDSmjAXs/s1600/im35+serie+completa.JPG" height="200" title="First 200 terms ot the series A001190 of Wedderburn-Etherington" width="71" /></a>- Aha! Here we've got the solution: it’s <a href="https://oeis.org/A001190" target="_blank">series number A001190</a>, called the <a href="http://en.wikipedia.org/wiki/Wedderburn%E2%80%93Etherington_number" target="_blank">Wedderburn-Etherington</a> series. We can see that within the utilities described on the web, there’s the one concerned with our problem: the number of distinct ways of organizing a single-elimination tournament for n players (with the players name left blank), that's it, the amount of <b>possible different schedules</b> that can be set for a number n of participants.</div>
<br />
<div style="text-align: justify;">
- Now we just have to click the link where <b>T. D. Noe</b> calculates a table with <b>the first 200 terms of the series</b>, and pick the result for 61 teams:</div>
<br />
<div style="text-align: center;">
<span style="text-align: justify;"><span style="font-size: x-large;">844206159208807054529</span></span></div>
<br />
<div style="text-align: justify;">
- What a number! Let's put the commas, for to have an accurate idea of its magnitude.</div>
<br />
<div style="text-align: center;">
<span style="font-size: x-large;">844,206,159,208,807,054,529</span></div>
<br />
<div style="text-align: justify;">
- Over 844 trillion different trees! Even in the Amazon rainforest there aren’t so many trees!</div>
<br />
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDXfMszpbw2OKJgvy7xjEt8fj8nzIW_8MbJWkqjnheFH2XfiCzKdMLKf0GEECEvh-EohM_zgBf6eaDlDcSz4IM-ti7gOt2KEOcPz88jH45NKUVnU2VdCwad-LcV3s4PYuB1NTCJKCau5A/s1600/im36+habitantes+de+brasil.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Approximately Brazilian population in 2013" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDXfMszpbw2OKJgvy7xjEt8fj8nzIW_8MbJWkqjnheFH2XfiCzKdMLKf0GEECEvh-EohM_zgBf6eaDlDcSz4IM-ti7gOt2KEOcPz88jH45NKUVnU2VdCwad-LcV3s4PYuB1NTCJKCau5A/s1600/im36+habitantes+de+brasil.jpg" height="320" title="Approximately Brazilian population in 2013" width="320" /></a>- It looks like a pretty big number. And we said we wanted to study all the cases to see which system we adopt, right? I think we’ll have to work 24 hours a day.</div>
<br />
<div style="text-align: justify;">
- Well, let's think a bit. If one of us devotes 1 minute to each 'tree', and he/she doesn’t eat , sleep, nor rest, he/she needs a total of 160,617,610,995,447 years to finish the work.</div>
<br />
<div style="text-align: justify;">
- We should put to work on it all the Department.</div>
<br />
<div style="text-align: justify;">
- Nah, even if all Brazilians (over 201 million people), worked in this task it would take 7,989,625 years to complete it.</div>
<br />
<div style="text-align: justify;">
<span style="text-align: justify;">- Well, nowadays that almost everyone likes football, how about we ask the help of the entire <a href="http://www.worldometers.info/world-population/" target="_blank">world population</a>?</span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWb-T43VgMF5tFNVBKrBRAn6XKVu_3RIuBlteeMkA_6n92SZveZVFNF4-GPuDjcxNoWpigY7ZieYE0FRt7gJV2YiSRzt2_jw2EoMYIkbJjmVk1UkQJAR7dtQVXE4DaEjAXTLpvvH0gcvw/s1600/im37+mapamundi2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="World population" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWb-T43VgMF5tFNVBKrBRAn6XKVu_3RIuBlteeMkA_6n92SZveZVFNF4-GPuDjcxNoWpigY7ZieYE0FRt7gJV2YiSRzt2_jw2EoMYIkbJjmVk1UkQJAR7dtQVXE4DaEjAXTLpvvH0gcvw/s1600/im37+mapamundi2.jpg" height="292" title="World population" width="640" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<div style="text-align: justify;">
- In that case, it would take only <b>221,987 years</b> to study for 1 minute each of the possible ways of organizing our 61-teams tournament.<br />
<br />
- By the way, I assume you remember when I mentioned that we weren't interested in <b>differentiating </b>the various <b>teams</b>, and therefore we focused only on the structure of the championship.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9muj0vP4GkbtUv5YBaVnNPQrV6N2Vt3_aIraSmh1dd_LhdLyhUkVU_cg_URZk1Eu60yDZm-8FvYqBZ7hpDd2TUj35k3JHYd_7Xc73e6b81X9rihGaWTjb_MXhznMkv9Do_0wcKv_p83I/s1600/im19+3+equipos+1+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Two different ways of organising a 3-teams championship" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9muj0vP4GkbtUv5YBaVnNPQrV6N2Vt3_aIraSmh1dd_LhdLyhUkVU_cg_URZk1Eu60yDZm-8FvYqBZ7hpDd2TUj35k3JHYd_7Xc73e6b81X9rihGaWTjb_MXhznMkv9Do_0wcKv_p83I/s1600/im19+3+equipos+1+2.jpg" height="130" title="Two different ways of organising a 3-teams championship" width="400" /></a></div>
<br />
- Yes, but I didn’t quite understand why.<br />
<br />
- Let’s revisit the issue for one moment. We have 61 teams, so we want to know the number of different ways we can sort them. In Mathematics, we talk about the <b>permutations </b>that can occur in a set of 61 elements.<br />
<br />
- That’s right. And it’s easy to calculate. Any of the 61 national teams can be the first one to be chosen. For each one of them, we will pick another team among the remaining 60 teams. For each of these combinations of 2 teams, we have 59 teams to choose the third country. And so on.<br />
<br />
- Therefore, the total number of permutations of the 61 countries will be:<br />
<br />
<div style="text-align: center;">
<span style="font-size: x-large;">61 · 60 · 59 · 58 · ..... · 3 · 2 ·1 = 61!</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
- Indeed, for brevity we use the expression '<b>factor of</b> 61’, represented by the symbol of admiration. And if we calculate the product, we get the following result:</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<div style="text-align: center;">
<span style="font-size: x-large;">61! = 5.0758 · 10 <sup>83</sup></span></div>
</div>
</div>
<br />
<div style="text-align: justify;">
- That’s a number five followed by 83 more figures. If you thought that the number of trees we saw before looked huge (<span style="font-size: large;">8,4421 · 10 <sup>20</sup></span>), imagine how it will be this new number. And if we multiply the number of trees by the number of forms we can fit the 61 teams in them, we get a lovely number :<br />
<br />
<div style="text-align: center;">
<span style="font-size: x-large;">4.2850 · 10 <sup>104</sup> = 42,850 · 10 <sup>100</sup></span></div>
<br />
- Forty-two thousand eight hundred and fifty <b>googols </b>of different ways of organizing our championship! We must remember that a googol is greater than the number of hydrogen atoms in the <b>known universe</b>...<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuW0jcD3zVDBEp74QYZY4iDEG8vQpth5yAsjhxX_cbZHiS9gTG8TtCm42ykxHMey6r3WnJPk1u5_-tN5Tcih2LmyJkPHlVdZ2R-wDDrYvH5oerrkZ5SX4okdA8Db0Hb3jlQDHcZCW8S2Q/s1600/universo.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Microwave background radiation of the Universe" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuW0jcD3zVDBEp74QYZY4iDEG8vQpth5yAsjhxX_cbZHiS9gTG8TtCm42ykxHMey6r3WnJPk1u5_-tN5Tcih2LmyJkPHlVdZ2R-wDDrYvH5oerrkZ5SX4okdA8Db0Hb3jlQDHcZCW8S2Q/s1600/universo.jpg" height="369" title="Microwave background radiation of the Universe" width="640" /></a></div>
<br />
- Uff . Then, I think we’d better use the <b>traditional method</b>: 29 matches of round of 64, freeing three countries in this first round, and full direct eliminatory rounds since round of 32.</div>
<br />
- Brazil, and two others.<br />
<br />
- Of course!<br />
<br />
<div style="text-align: justify;">
- Come on then! Let’s decide now which new countries we’re going to invite for completing the schedule of 61 teams. And we should think of organising something special for the opening ceremony: <a href="http://www.matifutbol.com/en/balls.html" target="_blank">What about giving away some balls?</a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMoQuXhBcpWD9AGqdDH6Le6O2U9WlrrHdEIkezcabbRPVaj716yaDtsXhTywXmgIB8zTSYFjChEpSGkQVnrkFB-obGvILkB0786rZHG40gdC9QjK-wYJvGjuczmavTzNJhoitUEirpl68/s1600/maracanaconbalones.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMoQuXhBcpWD9AGqdDH6Le6O2U9WlrrHdEIkezcabbRPVaj716yaDtsXhTywXmgIB8zTSYFjChEpSGkQVnrkFB-obGvILkB0786rZHG40gdC9QjK-wYJvGjuczmavTzNJhoitUEirpl68/s1600/maracanaconbalones.jpg" /></a></div>
<br />
<div style="text-align: justify;">
- Right. But first of all, if anyone is interested in this topic of <a href="http://en.wikipedia.org/wiki/Tree_structure" target="_blank">tree structures</a>, here you have some links to deepen on it: <a href="http://en.wikipedia.org/wiki/Tree_(graph_theory)" target="_blank">Tree (graph theory)</a>, <a href="http://webhome.cs.uvic.ca/~ruskey/Theses/GangLiMScThesis.pdf" target="_blank">Generation of Rooted Trees and Free Trees</a>, <a href="http://en.wikipedia.org/wiki/Binary_tree" target="_blank">Binary Tree</a>.</div>
<br />
<div style="text-align: justify;">
- And here you have other links, if you liked the story and want to share it with your friends.</div>
<br />
- Goodbye. And may the best win!<br />
<div class="separator" style="clear: both; text-align: center;">
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<span style="color: #20124d; display: none; font-family: Verdana, sans-serif;">Si te gustó esta historia, puedes votar por ella en <a href="https://www.blogger.com/xxx%20poner%20enlace%20a%20men%C3%A9ame%20xxx%20http://www.meneame.net/story/sorprendente-caso-coche-extra" target="_blank">menéame</a> y <a href="https://www.blogger.com/xxx%20poner%20enlace%20a%20divoblogger%20xxx%20http://divoblogger.com/story?id=41970" target="_blank">divoblogger</a>. Muchas gracias.</span>WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0Río de Janeiro - Estado de Río de Janeiro, Brasil-22.9082998 -43.197077299999989-23.3788568 -43.845270799999987 -22.437742800000002 -42.548883799999992tag:blogger.com,1999:blog-8244131673148079224.post-10183596254515964072014-03-24T14:34:00.000+00:002014-03-24T14:34:50.086+00:00Probability and sunrise<!-- Page resume -->
<span style="display: none;">It's easy to choose between some options when you know all about them. But when not all data are available, it turns quite difficult to find a satisfactory way of making a decision. In this case, Michael Laudrup faces a decision among two offers, but he only knows the conditions of one of them. And when he tries to solve the problem through a mathematical procedure, he gets even more confused about the choice. Do you know anything about the exchange paradox?<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNX7fzH7napt3vUthjVPpbaR5fRuyyu13txDBk3AKI1NWuguCECrMswLBEqZ5djVuC3Af1DHrlJxGyLdwg5QFU28Q-qmSe86V_-iUDcwlLpyK3NLs5MxTU9tHsTHStSto2oh0ur8JHKG0/s400/laudruponthebeach.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNX7fzH7napt3vUthjVPpbaR5fRuyyu13txDBk3AKI1NWuguCECrMswLBEqZ5djVuC3Af1DHrlJxGyLdwg5QFU28Q-qmSe86V_-iUDcwlLpyK3NLs5MxTU9tHsTHStSto2oh0ur8JHKG0/s200/laudruponthebeach.jpg" height="150" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXIw6wyoOWOw2KdyDNZ2eHgghH5URBUAn0DXPcxrEBx2ZwQ6aTuruPzBc-J8cLz2bvRlUSssrKo3bjdsq0rlS031bptPKhvOFTurDWP-J9tCkX7qvVpaVjR6NuKYsv-M_p-lr-KPnVjiA/s400/satellite.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXIw6wyoOWOw2KdyDNZ2eHgghH5URBUAn0DXPcxrEBx2ZwQ6aTuruPzBc-J8cLz2bvRlUSssrKo3bjdsq0rlS031bptPKhvOFTurDWP-J9tCkX7qvVpaVjR6NuKYsv-M_p-lr-KPnVjiA/s200/satellite.jpg" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXbbgtkqoXJmNIYpv6K2Ka3HnOqwDMX6Ey6ebLXK3Szad9kRKkOnvTvPdavpjjHHb7sjnq_YaapvAq9TLB9VksJP9muyjFzb29xIAr92nPS5DhHYYRK0fc9g9mFSqZ83UQ4sAiB8O4Pmg/s400/sunrise2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXbbgtkqoXJmNIYpv6K2Ka3HnOqwDMX6Ey6ebLXK3Szad9kRKkOnvTvPdavpjjHHb7sjnq_YaapvAq9TLB9VksJP9muyjFzb29xIAr92nPS5DhHYYRK0fc9g9mFSqZ83UQ4sAiB8O4Pmg/s200/sunrise2.jpg" /></a></span>
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(This post participates on the <a href="http://www.tonysmaths.blogspot.co.uk/" target="_blank"><span class="negrita">109th edition</span></a> of the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank"><span class="negrita">Carnival of Mathematics</span></a>, hosted by <a href="http://www.tonysmaths.blogspot.co.uk/" target="_blank"><span class="negrita unidas">Tony's Maths blog</span></a>.)</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMxOxhCy7uIa626isp1Hir0QMf6Mzalh4w9Paxpvj4Q1lb7cLBGcA3Yeg3xBSxYqyMjirO6MirwQiQVBtiOuMArcAq8NHnrIhr4NWnSL2T63cg0eozdHlbxzliAtJ89Jox7D4zi689aF4/s1600/southpacificisland.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Coast landscape. Found on http://photoxperiences.blogspot.com.es/" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMxOxhCy7uIa626isp1Hir0QMf6Mzalh4w9Paxpvj4Q1lb7cLBGcA3Yeg3xBSxYqyMjirO6MirwQiQVBtiOuMArcAq8NHnrIhr4NWnSL2T63cg0eozdHlbxzliAtJ89Jox7D4zi689aF4/s1600/southpacificisland.jpg" height="200" title="Coast landscape. Found on http://photoxperiences.blogspot.com.es/" width="196" /></a>It’s an awesome sunrise at a beautiful <b>island in the South Pacific</b>. It’s a place so far away from any inhabited place, that internet signal hardly arrives to it.</div>
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Only every two hours, a <b>telecommunications </b>satellite passes over the island, and allows its inhabitants connect to the rest of the world just for a minute.</div>
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<div style="text-align: justify;">
<b>Michael Laudrup</b> has chosen this idyllic place for a relaxing and deserved vacation with his family.</div>
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Like every morning, he has come to the sea-shore to witness the <b>beautiful sunrise</b>. It's eight o'clock. Just at this time the satellite passes over, and an incoming message sounds in his mobile.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibwdmNm74EK-fHPNfNOX_8Tj22yIUTuZ66aQbdgqb46EXN1brISx__BXS5VQNQ63MOjkEqIND8o5xuGZR1D_e6s63H9W5Z_vhb9AAx1pA8tPI7QPem44CZuOT4qXgYg8MI5GyztIoiEC8/s1600/sunrise.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Awesome sunrise. Found on http://photoxperiences.blogspot.com.es/" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibwdmNm74EK-fHPNfNOX_8Tj22yIUTuZ66aQbdgqb46EXN1brISx__BXS5VQNQ63MOjkEqIND8o5xuGZR1D_e6s63H9W5Z_vhb9AAx1pA8tPI7QPem44CZuOT4qXgYg8MI5GyztIoiEC8/s1600/sunrise.jpg" height="224" title="Awesome sunrise. Found on http://photoxperiences.blogspot.com.es/" width="320" /></a>He doesn’t feel like picking up the phone, but he notices that it’s a message from his <b>agent</b>, so he finally decides to read it. The message says:</div>
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<div style="text-align: justify;">
'Michael, you have <b>2 offers to coach</b> next season. One is from Juventus, and the other is from Real Madrid. As you can see, one offer <b>doubles </b>the other.</div>
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<div style="text-align: justify;">
The problem is that they need an immediate response, so within 2 hours, when you have connection again, you should answer and tell me which of the two teams you have decided to join. Otherwise, both teams will seek another coach. Juventus gives you 2 million euros per season, and Real Madrid...'</div>
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Damn! Just then, the satellite had gone enough away so that the connection is cut off.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXIw6wyoOWOw2KdyDNZ2eHgghH5URBUAn0DXPcxrEBx2ZwQ6aTuruPzBc-J8cLz2bvRlUSssrKo3bjdsq0rlS031bptPKhvOFTurDWP-J9tCkX7qvVpaVjR6NuKYsv-M_p-lr-KPnVjiA/s1600/satellite.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="The communications satellite crosses over the island every 2 hours." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXIw6wyoOWOw2KdyDNZ2eHgghH5URBUAn0DXPcxrEBx2ZwQ6aTuruPzBc-J8cLz2bvRlUSssrKo3bjdsq0rlS031bptPKhvOFTurDWP-J9tCkX7qvVpaVjR6NuKYsv-M_p-lr-KPnVjiA/s1600/satellite.jpg" height="240" title="The communications satellite crosses over the island every 2 hours." width="320" /></a>Michael had two hours to decide the team he will sign for.<br />
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So he went down to the beach, and began to <b>think about the answer</b> he would give to his manager.<br />
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Both offers came from two teams he had played for, so he didn’t feel a special predilection for one of them.</div>
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<div style="text-align: justify;">
Juventus offer was very good, but… what if Real Madrid’s offer was better?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhINrHVdl4t7-YyMjw0vv3ZhRWxN6gC7RpjyMCjWe1ITvyEyY26bNY29Ahgc7g83duPrnGV5jHiwDVVTrpSVIsaglQBWrI3ak8HgoLt1tRX8egh5Iv-882YNDdSZvI9SofSgFta8aB6xcw/s1600/realmadridbadge.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Real Madrid badge" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhINrHVdl4t7-YyMjw0vv3ZhRWxN6gC7RpjyMCjWe1ITvyEyY26bNY29Ahgc7g83duPrnGV5jHiwDVVTrpSVIsaglQBWrI3ak8HgoLt1tRX8egh5Iv-882YNDdSZvI9SofSgFta8aB6xcw/s1600/realmadridbadge.gif" height="200" title="Real Madrid badge" width="142" /></a><b><span style="font-size: x-large;">vs.</span></b><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVKFXXW1nSmUl_96KIsGQAXY6_6MJuTvmyMokm-ucQCDXBPtZ10Lx3Ihfx3Yjlrxv4AsYNSDJSaSXc72nkFoT8QsfPUjwd0mqDmjWiJoJnEaIyjiuEuJJNBrpb7yOSlnvOKuQiBu-wXU8/s1600/juventusbadge.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Juventus badge" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVKFXXW1nSmUl_96KIsGQAXY6_6MJuTvmyMokm-ucQCDXBPtZ10Lx3Ihfx3Yjlrxv4AsYNSDJSaSXc72nkFoT8QsfPUjwd0mqDmjWiJoJnEaIyjiuEuJJNBrpb7yOSlnvOKuQiBu-wXU8/s1600/juventusbadge.gif" height="200" title="Juventus badge" width="120" /></a></div>
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It would be a difficult decision. He could draw lots, but he thought that for such an important choice it's worth looking for a logical way to make the decision. Perhaps the little notions of the <b>probability </b>theory he still remembered from his younger times could help him...</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNX7fzH7napt3vUthjVPpbaR5fRuyyu13txDBk3AKI1NWuguCECrMswLBEqZ5djVuC3Af1DHrlJxGyLdwg5QFU28Q-qmSe86V_-iUDcwlLpyK3NLs5MxTU9tHsTHStSto2oh0ur8JHKG0/s1600/laudruponthebeach.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Laudrup is thinking on which team he will sign for." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNX7fzH7napt3vUthjVPpbaR5fRuyyu13txDBk3AKI1NWuguCECrMswLBEqZ5djVuC3Af1DHrlJxGyLdwg5QFU28Q-qmSe86V_-iUDcwlLpyK3NLs5MxTU9tHsTHStSto2oh0ur8JHKG0/s1600/laudruponthebeach.jpg" height="480" title="Laudrup is thinking on which team he will sign for." width="640" /></a></div>
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Do you dare to help Michael to choose his new team for next season?</div>
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<!-- Homo mathematicus -->
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img alt="Think for a moment on the solution, and then follow with the story." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" height="252" title="Think for a moment on the solution, and then follow with the story." width="640" /></span></a></div>
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<div class="rojo negrita" style="float: left; margin-left: 2em;">
<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
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<div style="text-align: justify;">
<br />
At the beginning, Laudrup thinks that Juventus offer is quite tempting. And besides, there is a 50% chance for his election to be <b>the best</b> offer.<br />
<br /></div>
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhI4h8YGbMsjccJ_I-fzpA9xaWWtOjhXOV025L_Baq85YS09ugMezld7CgQRpdXPh9BLot79__hrnOd9fSN-TrFYJfa9VW43sfnbzTNWwZGjXS9NmpfGD1_35iWQLxHGULxHcAeZLmfeds/s1600/prueba7.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Laudrup played for Juventus. Will he be its next coach?" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhI4h8YGbMsjccJ_I-fzpA9xaWWtOjhXOV025L_Baq85YS09ugMezld7CgQRpdXPh9BLot79__hrnOd9fSN-TrFYJfa9VW43sfnbzTNWwZGjXS9NmpfGD1_35iWQLxHGULxHcAeZLmfeds/s1600/prueba7.jpg" title="Laudrup played for Juventus. Will he be its next coach?" /></a><span style="color: #134f5c;">I would definitely like to earn this amount of money every year... And if he chooses the offer from Real Madrid it’s possible that he gets less money, so after two hours I would answer: I sign for Juventus!</span></div>
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<span style="text-align: justify;">Yes, it seems a logical choice.</span><br />
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<div style="text-align: justify;">
<span style="color: #134f5c;">Definitely. If by changing his choice he can earn 2 million more, and he can only lose 1 million, all is clear, right?</span></div>
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<div style="text-align: justify;">
Well, here we would be facing an economics study on the marginal utility of these 2 million: Laudrup can be more concerned about losing 1 million than earning 2 more millions.<br />
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Nevertheless we'll assume that Michael cares the same an extra million he can win than a million he can lose, and we'll approach the problem from a mathematical point of view. Let's see: we have an offer of <span class="unidas">€ 2,000,000</span> from Juventus, right?</div>
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<span style="color: #134f5c;">Yes.</span><br />
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<div style="text-align: justify;">
And we know that one of the bids is <b>twice </b>the other. So Real Madrid may have offered 1 million or 4 million euros.</div>
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<span style="color: #134f5c;">That's correct.</span><br />
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<div style="text-align: justify;">
OK, now let's calculate what’s the <b>expected amount</b> of money Laudrup will get if he changes his mind and chooses joining Real Madrid.</div>
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<div style="text-align: justify;">
There is a 50% chance of winning 1 million, and a 50% chance of winning 4 millions. So the expected gain of this second option, that we’ll call E(Q<sub>2</sub>), will be the <b>average </b>of the two possible cases:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpt-sUXSuH2ZyuHz8ukfWhrlU7O2yxqxGPQ8FeDeh_o-NMJLoZbN0pKfClHukOz25EqytIekmhwHyUUYOolc8NF35s7HpYm-nxTJyMyo5NrKTuLfh8UNXkO3fnynnO5RqA7hR1yooWVEg/s1600/formula1.eng.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Calculating the expected gain E(Q2) = 1/2 (1,000,000) + 1/2 (4,000,000) = 2,500,000" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpt-sUXSuH2ZyuHz8ukfWhrlU7O2yxqxGPQ8FeDeh_o-NMJLoZbN0pKfClHukOz25EqytIekmhwHyUUYOolc8NF35s7HpYm-nxTJyMyo5NrKTuLfh8UNXkO3fnynnO5RqA7hR1yooWVEg/s1600/formula1.eng.jpg" title="Calculating the expected gain E(Q2) = 1/2 (1,000,000) + 1/2 (4,000,000) = 2,500,000" /></a></div>
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<div style="text-align: justify;">
<span style="color: #134f5c;">So, if he swaps his choice, his expected return on swapping will be 2 and a half million euros on average, half a million more than if he signs for Juventus, right?</span></div>
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<div style="text-align: justify;">
Indeed, if we apply the laws of probability, in this case Laudrup should choose the second option. But this fact doesn't only happen in this case, but in all cases, whenever the agent expresses the difference between the two offers in<b> relative terms</b>.</div>
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<span style="color: #134f5c;">I don’t quite understand it.</span><br />
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We have 2 deals, <span style="text-align: justify;">Q</span><sub style="text-align: justify;">a</sub> and <span style="text-align: justify;">Q</span><sub style="text-align: justify;">b</sub>, and we define r as the ratio between the two offers:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSiPhuk2luzLAUL7fzfhDB6kaGu8wWfPr2BMWX2KFeIuaV2ED-p5wxmWD_B3cMSTKBeGM0yv1dWr4oD_HFhvB-z5vP6Znu0qz57njn6-O22qetGh4iTxeY0wXxoOqpHSZz_SJz7YTWFZA/s1600/formula2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Ratio between the two offers r = Qa/Qb" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSiPhuk2luzLAUL7fzfhDB6kaGu8wWfPr2BMWX2KFeIuaV2ED-p5wxmWD_B3cMSTKBeGM0yv1dWr4oD_HFhvB-z5vP6Znu0qz57njn6-O22qetGh4iTxeY0wXxoOqpHSZz_SJz7YTWFZA/s1600/formula2.JPG" title="Ratio between the two offers r = Qa/Qb" /></a></div>
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<div style="text-align: justify;">
We don’t care which of the two values, Q<sub>a</sub> or Q<sub>b,</sub> is the largest. In any case, r will always be a <b>positive </b>number greater than zero (for the moment, Laudrup is not thinking on coaching for free, so Q<sub>a</sub> and Q<sub>b</sub> aren’t zero, nor is thinking in paying money for it, so Q<sub>a</sub> and Q<sub>b </sub>are two positive values greater than zero).</div>
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<div style="text-align: justify;">
If we call Q<sub>1</sub> to the offer we take first, and Q<sub>2</sub> to the offer we have rejected, we have that the value of Q<sub>2 </sub>can be (<span style="text-align: justify;">Q</span><sub style="text-align: justify;">1</sub> · r) or (<span style="text-align: justify;">Q</span><sub style="text-align: justify;">1</sub> / r).</div>
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<div style="text-align: justify;">
We calculate the expected value of Q<sub>2</sub>, which we call E(Q<sub>2</sub>), for the case that all the offers are <b>equally likely</b> of being the greatest one:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWUdmFKXA84dAiI_AcZJjjX7_LqKUZnLg7GWJDNYRIVqc1EPmxlzKE1ST44BJfUhNRy6pXo2KcAoyvKIKCQMjgdc59mqZbz2CPIxWYbc9WLbopFvpVGJDBuil5rR0GiSRm6Sc5TdXDCY8/s1600/formula3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Calculating the expected value E(Q2)= 1/2 (Q1/r) + 1/2 (r Q1) = 1/2 (r^2+1) (Q1/r)" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWUdmFKXA84dAiI_AcZJjjX7_LqKUZnLg7GWJDNYRIVqc1EPmxlzKE1ST44BJfUhNRy6pXo2KcAoyvKIKCQMjgdc59mqZbz2CPIxWYbc9WLbopFvpVGJDBuil5rR0GiSRm6Sc5TdXDCY8/s1600/formula3.JPG" title="Calculating the expected value E(Q2)= 1/2 (Q1/r) + 1/2 (r Q1) = 1/2 (r^2+1) (Q1/r)" /></a></div>
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We define k as<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYgXmeHCK4dW7zCsE1Lthx0bEPWhEux6RVN1eComCdW75rnQmBcpSY1nx9yMEBJfNVmJhO2RtqdSfDNjuCVPd_eug0Km4_tblchM5xVHvceCPK8yj8QXi0cX43NdCtjJJeEEmyYa7rCYA/s1600/formula4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="k = 1/2 (r^2+1) / r" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYgXmeHCK4dW7zCsE1Lthx0bEPWhEux6RVN1eComCdW75rnQmBcpSY1nx9yMEBJfNVmJhO2RtqdSfDNjuCVPd_eug0Km4_tblchM5xVHvceCPK8yj8QXi0cX43NdCtjJJeEEmyYa7rCYA/s1600/formula4.JPG" title="k = 1/2 (r^2+1) / r" /></a></div>
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We have that<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4dyIzY2ENboFra3MATULGPJdLnGWuz7oKQNurEhDGfgF-5NCbMCg9wuImEu_Y0ZDtZgX-H1spPzX3qHZFWDqLnbVKozLVr8j-awaRLkxfza2FrhjX2qUY7OBR4NSve-vYOwYzRmF7ArI/s1600/formula5.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="E(Q2) = k Q1" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4dyIzY2ENboFra3MATULGPJdLnGWuz7oKQNurEhDGfgF-5NCbMCg9wuImEu_Y0ZDtZgX-H1spPzX3qHZFWDqLnbVKozLVr8j-awaRLkxfza2FrhjX2qUY7OBR4NSve-vYOwYzRmF7ArI/s1600/formula5.JPG" title="E(Q2) = k Q1" /></a></div>
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<div style="text-align: justify;">
Let's see graphically how the value of k behaves according to the values of r, that is, depending on the difference between the two offers:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3QcrTswaYmqX59gkfqm9UsbrEJuPLZTKbm-2wedxI9AMSjvdOdIyUY24ueYFx1WXGLYjoCKEi5VnsddeqeZjzHwnF3I9gzhRIcJNEf_nC1ASi9a9J69zzdFd8v0CWo0cIuD2b6lGExBU/s1600/formula.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Graph of the function k = 1/2 (r^2 + 1) / r" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3QcrTswaYmqX59gkfqm9UsbrEJuPLZTKbm-2wedxI9AMSjvdOdIyUY24ueYFx1WXGLYjoCKEi5VnsddeqeZjzHwnF3I9gzhRIcJNEf_nC1ASi9a9J69zzdFd8v0CWo0cIuD2b6lGExBU/s1600/formula.jpg" height="426" title="Graph of the function k = 1/2 (r^2 + 1) / r" width="640" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRFtSjWLPb5yjxth6BW5_1i1RzXBV99oskGiRpH3JikPDdZInPhuMm3Mi8ImexdsulcaPUBORljPOvJSZv_72r1wLPBKzitgh5W7cQptYhEfzbh0hwGOIzsmjSGWMSyqBcLzZXFc1rT0A/s1600/formula11.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Second offer will always be better than the first one k >= 1 -> E(Q2) >= Q1" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRFtSjWLPb5yjxth6BW5_1i1RzXBV99oskGiRpH3JikPDdZInPhuMm3Mi8ImexdsulcaPUBORljPOvJSZv_72r1wLPBKzitgh5W7cQptYhEfzbh0hwGOIzsmjSGWMSyqBcLzZXFc1rT0A/s1600/formula11.JPG" title="Second offer will always be better than the first one k >= 1 -> E(Q2) >= Q1" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgM7HcZfDvQwa7jQkG7dGaBDCQ_-vwzrYeLyOwMJpN6yQHAh9sDfU_SELUdACTgy-5u_2DiU9pTgqGIiNyAGSQ4Jc_3-TBqq73iJ96JZNkn07bl_fuMZRzZ9LZ5g1z7dua4Idm4ppbqopA/s1600/prueba8.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Laudrup played for Real Madrid. Will he be also his coach?" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgM7HcZfDvQwa7jQkG7dGaBDCQ_-vwzrYeLyOwMJpN6yQHAh9sDfU_SELUdACTgy-5u_2DiU9pTgqGIiNyAGSQ4Jc_3-TBqq73iJ96JZNkn07bl_fuMZRzZ9LZ5g1z7dua4Idm4ppbqopA/s1600/prueba8.jpg" height="320" title="Laudrup played for Real Madrid. Will he be also his coach?" width="214" /></a><br />
<div style="text-align: justify;">
We can see that k is always greater than 1, except when the two offers are equal, so the expected value of the unknown offer <span style="text-align: start;">E(Q</span><sub style="text-align: start;">2</sub><span style="text-align: start;">) </span>is always greater than the one we know Q<sub>1</sub>. That way, we should always <b>choose the latter</b> offer.</div>
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<div style="text-align: justify;">
<span style="color: #134f5c;">Then, Laudrup will answer: I sign for Real Madrid!</span></div>
<br />
<div style="text-align: justify;">
Yes, but… can you imagine what would happen if his agent had mentioned in the first place the Real Madrid’s offer? Following the same reasoning, now Laudrup would be signing for Juventus…</div>
<br />
<div style="text-align: justify;">
However, we find this paradox only when the relationship between the two offers is expressed in relative terms. But imagine that his agent, instead of saying that an offer doubles the other, says that an offer is greater than the other in € 1,000,000 for example, that is, he quantifies the difference in absolute values. What happens then?</div>
<br />
<span style="color: #134f5c;">I don’t know.</span><br />
<br />
<div style="text-align: justify;">
We can recalculate the expected gain if Laudrup chooses Real Madrid’s offer:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0SMGtnSDpLNx3SEHrWE-SC6tIOdA422Eaaf7cScsYulTfq2XiSgy8kqm6EAF9c9DyfV8xjC1Qq68nS6cg-ryV3PV7FTY7dnGeA8iLI2QYWEUpd7Mf2cFwqKvFl9J6i4EoqohpVcAvfGc/s1600/formula6.eng.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Calculation of the expected return E(Q2) = 1/2 (1,000,000) + 1/2 (3,000,000) = 2,000,000" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0SMGtnSDpLNx3SEHrWE-SC6tIOdA422Eaaf7cScsYulTfq2XiSgy8kqm6EAF9c9DyfV8xjC1Qq68nS6cg-ryV3PV7FTY7dnGeA8iLI2QYWEUpd7Mf2cFwqKvFl9J6i4EoqohpVcAvfGc/s1600/formula6.eng.jpg" title="Calculation of the expected return E(Q2) = 1/2 (1,000,000) + 1/2 (3,000,000) = 2,000,000" /></a></div>
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<div style="text-align: justify;">
It turns out that he obtains an <b>expected gain </b>of € 2,000,000, just the same amount offered by Juventus!</div>
<br />
Let's see it in mathematical terms. If we now define r as:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPB2UIQMsbBECBqeKLiteg6N4GOCYK411BMGc2hsCA_sVPB4udPMpRoOnJPjDxyx5-6nG0r1Pf_K1V20wOFgVwOpnxRrdoVy7v76NSsZsI8LOdarXJNIsYnD08kC9IHNXuD3BbpZPS3oA/s1600/formula10.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="r = Qa - Qb" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPB2UIQMsbBECBqeKLiteg6N4GOCYK411BMGc2hsCA_sVPB4udPMpRoOnJPjDxyx5-6nG0r1Pf_K1V20wOFgVwOpnxRrdoVy7v76NSsZsI8LOdarXJNIsYnD08kC9IHNXuD3BbpZPS3oA/s1600/formula10.JPG" title="r = Qa - Qb" /></a></div>
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we get:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjG8tnrbmnZ6uwJRDpfvHw4LifVTu1ozpQ9A68mNCVD5ujEHN7yxhUkS9AFdWggCINFVk8SWxoie_FVaFeoOqpWd3ri77eMXHLVXlAtuNXQj219BHHBoqpVjQ4_Az2gYfHXvVAzotaPorQ/s1600/formula7.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="E(Q2) = 1/2 (Q1 + r) + 1/2 (Q1 - r)" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjG8tnrbmnZ6uwJRDpfvHw4LifVTu1ozpQ9A68mNCVD5ujEHN7yxhUkS9AFdWggCINFVk8SWxoie_FVaFeoOqpWd3ri77eMXHLVXlAtuNXQj219BHHBoqpVjQ4_Az2gYfHXvVAzotaPorQ/s1600/formula7.JPG" title="E(Q2) = 1/2 (Q1 + r) + 1/2 (Q1 - r)" /></a></div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhE0PP_YKZqbQ89BhyphenhyphenxF0agR88G4E_UKT3aJ9FY3skpXI654oRfdlqIuWzTb2lTbnmW9T4IzhvvDFuyNxQRDqeMe2fdBRv1W1L2gtT9jN6I83zOF5PIVy_63zhDA3LNhG2mE5Ende2CF0/s1600/formula8.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="E(Q2) = 1/2 (Q1 + Q2 - Q2) + 1/2 (Q1 - Q2 + Q1)" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhE0PP_YKZqbQ89BhyphenhyphenxF0agR88G4E_UKT3aJ9FY3skpXI654oRfdlqIuWzTb2lTbnmW9T4IzhvvDFuyNxQRDqeMe2fdBRv1W1L2gtT9jN6I83zOF5PIVy_63zhDA3LNhG2mE5Ende2CF0/s1600/formula8.JPG" title="E(Q2) = 1/2 (Q1 + Q2 - Q2) + 1/2 (Q1 - Q2 + Q1)" /></a></div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvzQPokNkIS2ipBOxckEmndyvxfhxAFveLej1GDyfrPpdC6D1AhrKb0OnFyQCOEOtU7rTA_6QGk48HScUErvg13GEpt1z9N9vNaoCaVed052f227WbJAdWfW7C-s1s8ZesoH97PLVwFic/s1600/formula9.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="E(Q2) = 1/2 (2 Q1) = Q1" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvzQPokNkIS2ipBOxckEmndyvxfhxAFveLej1GDyfrPpdC6D1AhrKb0OnFyQCOEOtU7rTA_6QGk48HScUErvg13GEpt1z9N9vNaoCaVed052f227WbJAdWfW7C-s1s8ZesoH97PLVwFic/s1600/formula9.JPG" title="E(Q2) = 1/2 (2 Q1) = Q1" /></a></div>
<br />
<div style="text-align: justify;">
<span style="color: #134f5c;">So if we express the difference in absolute terms, not relative, it happens that we get a more 'logical' <b>mathematical solution</b>...</span></div>
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<div style="text-align: justify;">
That’s true, in this case the mathematical result matches with what common sense dictates.</div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj1bSURFdFCkhK0kbHo1DblbW_whbCI-wIn0Bs487UYNwXgIfXXmiS4JmjPvWSrpuChnX_uhIYcpWWEy2LcdqGD7eq7S_RgIlle8S2Wug5pXbdxc7na2sZztqzOJI0U1m-vESJ2rdZPBk/s1600/prueba9.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Laudrup thinks about coaching his current team next season." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj1bSURFdFCkhK0kbHo1DblbW_whbCI-wIn0Bs487UYNwXgIfXXmiS4JmjPvWSrpuChnX_uhIYcpWWEy2LcdqGD7eq7S_RgIlle8S2Wug5pXbdxc7na2sZztqzOJI0U1m-vESJ2rdZPBk/s1600/prueba9.jpg" height="108" title="Laudrup thinks about coaching his current team next season." width="200" /></a><br />
<div style="text-align: justify;">
It's a hard time to decide on one of the 2 teams, because we estimate that both offers are equal, and because he has the same sympathy for both teams, so Michael will probably say: I stay in my current team!</div>
<br />
<div style="text-align: justify;">
<span style="color: #134f5c;">Well, such opportunities don’t come around every day. Can’t we help him in some way to decide on an offer?</span></div>
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<div style="text-align: justify;">
OK, after all these calculations, we're not going to leave him without a solution to his problem.</div>
<br />
<div style="text-align: justify;">
We’ve seen that according to how we interpret the data, the decision may be different. What should Laudrup do to take a right decision?</div>
<a href="http://www.matifutbol.com/en/decision.eng.html" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;" target="_blank"><img alt="Fernando Llorente and his hard decision on which team to sign for." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgU7bDSqTtKUY_ygvZpB0GDxaUH5C4cc2K9lGfXv9tgpxWIdJJA9pwxYEB-0IS-EI5tz0_zY1B4m6BpSNquVxHmJpF2NZAvFo6oWwxPOk0RPA3MitgUbsCTLqbNM1yfi1vgmsSVeXgEiRM/s1600/llorente1.jpg" title="Fernando Llorente and his hard decision on which team to sign for." /></a><br />
<div style="text-align: justify;">
Do you remember when Llorente didn’t know what to do, if signing for Tottenham Hotspur or Aston Villa? We saw it a while ago in our story ‘<a href="http://www.matifutbol.com/en/decision.eng.html" target="_blank">Llorente's hard decision</a>’.</div>
<br />
<div style="text-align: justify;">
<span style="color: #134f5c;">Ah, yes, I remember it. In that case what we had to do was applying the <b>theory of optimal stopping</b>, right?</span></div>
<br />
<div style="text-align: justify;">
That’s right. The first thing Laudrup has to do is thinking about how much money wants to earn by coaching next season, and when he picks an offer that equals or exceeds that amount, he must take it, without thinking that there may be better deals. And since the offer of Juventus seemed interesting enough and meets his expectations, he should definitely take it.</div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_T_WUK6prXQCsP-EGvWmxOGB4K03t7Roo0HY_-MwQXYvL8HfQauMTZSXLD5TQfROAP39aX0PFFhF4uKdvuIgxwCLAVlj9KUbjfo-if0F70CSrAHCYeW54uEPawmnPlIBg7zL0lVeBhnc/s1600/prueba7.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="It seems that finally Laudrup will coach Juventus team." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_T_WUK6prXQCsP-EGvWmxOGB4K03t7Roo0HY_-MwQXYvL8HfQauMTZSXLD5TQfROAP39aX0PFFhF4uKdvuIgxwCLAVlj9KUbjfo-if0F70CSrAHCYeW54uEPawmnPlIBg7zL0lVeBhnc/s1600/prueba7.jpg" height="200" title="It seems that finally Laudrup will coach Juventus team." width="167" /></a><span style="color: #134f5c;"></span><br />
<div style="text-align: justify;">
<span style="color: #134f5c;">So we can conclude that Laudrup’s answer will be: I sign for Juventus!</span></div>
<br />
Sorry, but it will not be so!<br />
<br />
<div style="text-align: justify;">
<span style="color: #134f5c;">How? Is there any other reasoning that we haven’t considered?</span></div>
<br />
<div style="text-align: justify;">
Surely. This issue has a direct relationship with theory of decision and Bayesian interpretation of probability, and it has been studied for a long time by experts in logic, philosophy, mathematics, etc. It’s called ‘<span style="color: red; font-size: large;"><b>the two envelopes paradox</b></span>’ or ‘<b>the paradox of change</b>’. On these links you can find more information about this paradox: <a href="http://wmbriggs.com/blog/?p=2645" target="_blank">The Two-Envelope Problem Solution: Part I</a>, <a href="http://wmbriggs.com/blog/?p=2649" target="_blank">The Two-Envelope Problem Solution: Part II</a>, <a href="http://www.gatsby.ucl.ac.uk/~turner/Notes/TwoEnvelopes/2envlps.pdf" target="_blank">The Two Envelopes Problem</a>.</div>
<br />
<div style="text-align: justify;">
<span style="color: #134f5c;">Yes, but at the end, what was Laudrup’s choice? You’ve mentioned that he rejected the offers of both teams.</span></div>
<br />
<div style="text-align: justify;">
Certainly, after 2 hours walking on the beach, he decided that neither in Madrid nor in Turin he could see such <b>gorgeous sunrises</b> as on this island. And he decided that he will no longer coach his team again. He’ll take a year off, resting in this magnificent place with his family.</div>
<br />
<div style="text-align: justify;">
And after sending the reply message to his agent, he did what he should have done when he came to the island: he turned off the phone.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXbbgtkqoXJmNIYpv6K2Ka3HnOqwDMX6Ey6ebLXK3Szad9kRKkOnvTvPdavpjjHHb7sjnq_YaapvAq9TLB9VksJP9muyjFzb29xIAr92nPS5DhHYYRK0fc9g9mFSqZ83UQ4sAiB8O4Pmg/s1600/sunrise2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Sunrise. Found on http://photoxperiences.blogspot.com.es/" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXbbgtkqoXJmNIYpv6K2Ka3HnOqwDMX6Ey6ebLXK3Szad9kRKkOnvTvPdavpjjHHb7sjnq_YaapvAq9TLB9VksJP9muyjFzb29xIAr92nPS5DhHYYRK0fc9g9mFSqZ83UQ4sAiB8O4Pmg/s1600/sunrise2.jpg" height="480" title="Sunrise. Found on http://photoxperiences.blogspot.com.es/" width="640" /></a></div>
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<span style="color: #20124d; display: none; font-family: Verdana, sans-serif;">Si te gustó esta historia, puedes votar por ella en <a href="http://www.meneame.net/story/sorprendente-caso-coche-extra" target="_blank">menéame</a> y <a href="http://divoblogger.com/story?id=41970" target="_blank">divoblogger</a>. Muchas gracias.</span>WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-3161526444397240332014-01-28T16:25:00.000+00:002014-01-31T22:28:03.805+00:00Fibonacci also plays football<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="display: none;">An interesting story that mixes football, mathematics, art, communications, sports bettings and nature, all around the Fibonacci sequence.<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUu1hNoVYNiv3cwJG1oDg24HhyRoNptl_oZJV544IFtAQ0lXuooKPaMD7GWkX9hx0al4gNo4mn5EsJYFbfNbSYrKy-jNegpuWFHpQ7uHBleaZZJPiLM4uc0p7MQaeaRjgFV3lUB1Zx7C4/s1600/leonardodepisa.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUu1hNoVYNiv3cwJG1oDg24HhyRoNptl_oZJV544IFtAQ0lXuooKPaMD7GWkX9hx0al4gNo4mn5EsJYFbfNbSYrKy-jNegpuWFHpQ7uHBleaZZJPiLM4uc0p7MQaeaRjgFV3lUB1Zx7C4/s1600/leonardodepisa.jpg" height="150" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhr8i5OvJ_VG2unZU2nF81msINqK20FHBGijRb_uXzHYF4q9OWL6k2knQWBrC7j6IqqhRnqLSf4TMp-s8880rYCKWi2WdnESHHpqe3ePiQaznjEZQU-g8XU9k05F-IU0cYITjkTjpyG_y4/s1600/vitrubio.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhr8i5OvJ_VG2unZU2nF81msINqK20FHBGijRb_uXzHYF4q9OWL6k2knQWBrC7j6IqqhRnqLSf4TMp-s8880rYCKWi2WdnESHHpqe3ePiQaznjEZQU-g8XU9k05F-IU0cYITjkTjpyG_y4/s1600/vitrubio.jpg" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaEP7q8wmMx2Zw0gPR5D_GbMJ74wv7VEgOY-1aEpb8RgKrCkWR00MDyN7RvV_jER8xHywTqmGA61KE0mOS-JYk_wn4xBjGoqez-eJ6u-SYz4yA7u3uH4rKNwP7vZufyZJiz_1POfNfAf0/s1600/espiraldorada2.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaEP7q8wmMx2Zw0gPR5D_GbMJ74wv7VEgOY-1aEpb8RgKrCkWR00MDyN7RvV_jER8xHywTqmGA61KE0mOS-JYk_wn4xBjGoqez-eJ6u-SYz4yA7u3uH4rKNwP7vZufyZJiz_1POfNfAf0/s1600/espiraldorada2.gif" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDNQWhppioV_rlyhRE7pEvLLVH_4i5uPWl4A_xtZqTmgjbTATdibuh_qcHjg2H3XI4pZ4mttVvgJazRmObz5t-w-icDza3drs3fE4WmNqIz57ODzaPK2i_PvuG3ssy4JDpDWwoXCYZ0rk/s1600/italiano3.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDNQWhppioV_rlyhRE7pEvLLVH_4i5uPWl4A_xtZqTmgjbTATdibuh_qcHjg2H3XI4pZ4mttVvgJazRmObz5t-w-icDza3drs3fE4WmNqIz57ODzaPK2i_PvuG3ssy4JDpDWwoXCYZ0rk/s1600/italiano3.gif" /></a></span>
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<span style="display: none;">(This post participes on the <a href="https://www.blogger.com/urldelaedici%C3%B3n" target="_blank"><span class="negrita">xxxth edition</span></a> of the <a href="https://www.blogger.com/urldelcarnaval" target="_blank"><span class="negrita">Carnival de Mathematics</span></a>, hosted by the blog <a href="http://cuentos-cuanticos.com/" target="_blank"><span class="negrita unidas">Cuentos cuánticos</span></a>.)</span></div>
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<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1tquPuhXMIVNW4S9M3Xb5zvLnPKsBT-eFqMmzZCBNSuvUZVKzZzQfYz6Eos5T7CYTKZj3Qjz0XsizRgupxTfLdjy1owRw8dklio8kGag5g7XP89bmDfsrvYDmCfLNIwQXlfcp1TMb2Vw/s1600/escudodelolympiacos.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Badge of the Athens soccer team Olympiacos F.C." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1tquPuhXMIVNW4S9M3Xb5zvLnPKsBT-eFqMmzZCBNSuvUZVKzZzQfYz6Eos5T7CYTKZj3Qjz0XsizRgupxTfLdjy1owRw8dklio8kGag5g7XP89bmDfsrvYDmCfLNIwQXlfcp1TMb2Vw/s1600/escudodelolympiacos.gif" height="284" title="Badge of the Athens soccer team Olympiacos F.C." width="320" /></a>A <b>new forward</b> is coming. The board of <a href="http://www.youtube.com/watch?v=V35zWZc2x2M" target="_blank">Olympiacos F.C.</a> (ΠΑΕ Ολυμπιακός) has decided to sign a new centre-forward, in order to strengthen the team for the next round of the Champions League.</div>
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<div style="text-align: justify;">
The team owner, Evangelos, calls his friend, the editor of Protathlitis newspaper, to give him the scoop:</div>
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<div style="text-align: justify;">
- Sotiris, we're signing a new striker. He’s a great <b>Italian </b> forward who plays in the Italian Serie B. Hardly anyone knows him, but he’s very good, plus we get him for little money. His name is 'Leo' (<b>Leonardo)</b>.</div>
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDNQWhppioV_rlyhRE7pEvLLVH_4i5uPWl4A_xtZqTmgjbTATdibuh_qcHjg2H3XI4pZ4mttVvgJazRmObz5t-w-icDza3drs3fE4WmNqIz57ODzaPK2i_PvuG3ssy4JDpDWwoXCYZ0rk/s1600/italiano3.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="New centre-forward of Olympiacos FC" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDNQWhppioV_rlyhRE7pEvLLVH_4i5uPWl4A_xtZqTmgjbTATdibuh_qcHjg2H3XI4pZ4mttVvgJazRmObz5t-w-icDza3drs3fE4WmNqIz57ODzaPK2i_PvuG3ssy4JDpDWwoXCYZ0rk/s1600/italiano3.gif" height="400" title="New centre-forward of Olympiacos FC" width="176" /></a>- Can I get ahead of the news?<br />
<br />
<div style="text-align: justify;">
- No, please. In <b>30 minutes</b> we’ll make a <b>press conference</b> to announce the new signing, and I don't want many people know the news in advance.</div>
<br />
<div style="text-align: justify;">
- So I can’t tell anyone? Let me inform my best friends by <b>SMS</b>. I promise you I won't spread it massively.</div>
<br />
<div style="text-align: justify;">
- Ok. I let you spread it, but with the following conditions: you may <b>not send</b> it <b>massively</b>. You can only send <b>one message at a time</b>. And you must tell this rule to the people you send the message, for them to do the same.</div>
<br />
<div style="text-align: justify;">
- Right, Evangelos, I’ll do it that way. But you haven’t told me the player's surname yet...</div>
<br />
<div style="text-align: justify;">
- You'll find out his surname if you look at the <b>rules </b>I've given you for <b>spreading </b>the message.</div>
<br />
<div style="text-align: justify;">
After hanging up, Sotiris, the <span style="text-align: justify;">Protathlitis</span> daily director, sends the first message to the editor in chief, to be ready for editing the news just when the conference takes place, in these terms:</div>
<div style="text-align: justify;">
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUFTakJ1D3hdnANZldaZ_ZOQY2nRtggEsak2iAFVbOeBCh9WutBS4JSs-L2q07Ssu9l6bzwVtaHj_07MyPX7HHIYjWQNgvJ6qeK2aR4c3fXtdbHbv5gblv8uCpQ5NYq7959xc_2NHVAeE/s1600/movil1eng.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="The message that Sotiris, the newspaper director, has sent to his friends" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUFTakJ1D3hdnANZldaZ_ZOQY2nRtggEsak2iAFVbOeBCh9WutBS4JSs-L2q07Ssu9l6bzwVtaHj_07MyPX7HHIYjWQNgvJ6qeK2aR4c3fXtdbHbv5gblv8uCpQ5NYq7959xc_2NHVAeE/s1600/movil1eng.jpg" height="200" title="The message that Sotiris, the newspaper director, has sent to his friends" width="121" /></a></div>
<b><i>'A player named Leonardo will be the new Olympiacos FC forward. I don’t know his last name, but apparently it has to do with how this message is send. You can resend this message to your contacts, but only one at a time, please.'</i></b><br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbeo0VxrU21F8WQZOEONDK1-ZWntjtnBWpx2ilL7rJaDUwFufEJljpaP_Sdxea3WnGbY-oeWGbC5FjkJn2TFtZkvXbBa9EGMAIOxILofMYbKtnqte9I-vtiMpN-ZeOuEES6NEOTKdxrWA/s1600/periodico.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Cover of the newspaper Protathlitis where the new signing is announced" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbeo0VxrU21F8WQZOEONDK1-ZWntjtnBWpx2ilL7rJaDUwFufEJljpaP_Sdxea3WnGbY-oeWGbC5FjkJn2TFtZkvXbBa9EGMAIOxILofMYbKtnqte9I-vtiMpN-ZeOuEES6NEOTKdxrWA/s1600/periodico.jpg" height="320" title="Cover of the newspaper Protathlitis where the new signing is announced" width="259" /></a></div>
<div style="text-align: justify;">
After sending the message to the editor, the director sends a new SMS, this time to a friend, a fan of Olympiacos. And so he continues sending messages for <b>30 minutes</b> before the conference starts.</div>
<br />
<br />
<div style="text-align: justify;">
Likewise, the editor, after reading the message, <b>resends</b> it to the layout designer, in order to prepare the front page. And within the next minutes, he goes on resending the message to other people, always <b>one at a time</b>.</div>
<br />
And that’s what happens to the rest of recipients of the message.<br />
<br />
<div style="text-align: justify;">
We know that SMS <b>sending </b>is <b>instantaneous</b>, that everyone takes aproximately <b class="unidas">1 minute</b> to <b>read</b> the message, and <b>1 minute</b> more to decide the person to whom will <b>forward</b> it. And we know that nobody has received the message from two different issuers.</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDXf1BOfyiP7vZpXcA5tkFFaQjBmQ2KSnaTeIFeonY4kxJBQCb52ereHt7i4tfsQPu8AaZX-oQ23LlG8AAX5HOQse1wut-RnBMcf-um4zolYzKenGcUj1nnoPMqkUoRaS10sUNzYbpfqM/s1600/difusi%C3%B3n.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="So the news spreads" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDXf1BOfyiP7vZpXcA5tkFFaQjBmQ2KSnaTeIFeonY4kxJBQCb52ereHt7i4tfsQPu8AaZX-oQ23LlG8AAX5HOQse1wut-RnBMcf-um4zolYzKenGcUj1nnoPMqkUoRaS10sUNzYbpfqM/s1600/difusi%C3%B3n.gif" height="144" title="So the news spreads" width="640" /></a></div>
<br />
<div style="text-align: justify;">
With these data, and 30 minutes later, can you calculate how many people will know the news before the press conference starts? 100 people? 500? 1,000? And, most importantly, can you imagine <b>what’s the second name</b> of the Italian footballer?</div>
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<!-- End of main article -->
<!-- Homo mathematicus -->
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img alt="Think for a moment the answer, you'll feel better even if you do not guess right" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" height="252" title="Think for a moment the answer, you'll feel better even if you do not guess right" width="640" /></span></a></div>
<div class="separator" style="clear: both; text-align: justify;">
<br /></div>
<!-- Second half -->
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<div class="rojo negrita" style="float: left; margin-left: 2em;">
<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
<!-- Main article -->
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<br />
<div style="text-align: justify;">
<span style="color: #073763;">I don’t know how to start with this problem...</span></div>
<br />
So we best start from the beginning :-)<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: justify;">
We will call ‘minute 0’ to the moment the owner of Olympiacos calls the newspaper director <span class="unidas">(person 1 = P1)</span>. There's nobody who already knows the news, that is, <b><span style="color: red; font-size: 120%;">0</span></b> people know the scoop (excluding the members of the board of the club).<br />
<br />
In the <b>first minute</b>, they talk about the new player. So, at the end of the first minute, there's <b><span style="color: #cc0000; font-size: 120%;">1</span></b> person who knows the news.<br />
<br />
During the <b>second minute</b>, the newspaper director decides that the first person who should know the news is the editor in chief (person 2 = P2). He writes the SMS message and sends it to him. So when 2 minutes have passed, there's still only <span style="color: #cc0000; font-size: 120%;"><b>1</b></span> one person who knows the name of the footballer.</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="text-align: justify;">
Now let's see what happens in the <b>next minute</b>. The director decides to send a new message, this time to his friend, the fan of Olympiacos (P3). Meanwhile, the editor in chief has read the message the director sent him. Now, there are <b><span style="color: #cc0000; font-size: 120%;">2</span></b> people who know about the signing, and there's one more person that has received the message, but has not read it.</div>
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<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="text-align: justify;">
In the <b>fourth minute</b>, the director sends a new message, this time to other friend (P4). The editor, once he has read the message, has resent it to the layout designer (P5). And the fan of Olympiacos has received the message from the director and has already read it. Now there're <span style="color: #cc0000; font-size: 120%; text-align: justify;"><b>3</b></span> people aware of the deal, and 2 more to which the message has reached them, but haven’t read it yet.</div>
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<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhc0rskEXqWGqYbGmwmgSUBM_fHFEyMcMhytWvXW82Mppv5DTc2PFesybMVeJSH9gqRGSdArvqEBcm-bebpHfnyx8boAqreuJ0b8miHOLj_3V5uCo9oa4AV_z5HNkwUT9Lc8F-UUv3SqI/s1600/tab3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="What's happening during the first four minutes?" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhc0rskEXqWGqYbGmwmgSUBM_fHFEyMcMhytWvXW82Mppv5DTc2PFesybMVeJSH9gqRGSdArvqEBcm-bebpHfnyx8boAqreuJ0b8miHOLj_3V5uCo9oa4AV_z5HNkwUT9Lc8F-UUv3SqI/s1600/tab3.JPG" height="194" title="What's happening during the first four minutes?" width="640" /></a></div>
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<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="text-align: justify;">
In the <b>fifth minute</b>, the director resends the message to another friend (P6). The editor does the same with a friend (P7). The fan of Olympiacos sends his first message, to P8. Meanwhile, P4 and P5 have read the message. Thus, we have <span style="color: #cc0000; font-size: 120%; text-align: justify;"><b>5</b></span> people who know the news.</div>
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<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="text-align: justify;">
In the <b>sixth minute</b>, that’s what happens: the director, the editor and the fan of Olympiacos send another new message (to P9, P10 and P11), P4 and P5 send their first messages (to P12 and P13), and P6, P7 and P8 read the message. Now there are <span style="color: #cc0000; font-size: 120%; text-align: justify;"><b>8</b></span> people in the know of the signingl.</div>
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<br />
<div style="text-align: justify;">
In <b>minute 7</b>, the first five people send a new message. The next 3 make their first delivery. And the remaining five have just read the incoming message. Now there are <b><span style="color: #cc0000; font-size: 120%;">13</span></b> people who know the subject. And eight more will receive the message shortly.</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyQn221_4ZYU6uR47AvVJSTa4xbVcX4J87TgiL32qa1x1jyJJIq3tw0yFvfwAPnzLiSjLnvob_H_-gHd3cT2p9HTeVz-Us7hRGfUzV_UoKYxRgx5BwosLrzEEws01zYVd6VKo3KIDB6PI/s1600/tab4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="what's happening during the first seven minutes?" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyQn221_4ZYU6uR47AvVJSTa4xbVcX4J87TgiL32qa1x1jyJJIq3tw0yFvfwAPnzLiSjLnvob_H_-gHd3cT2p9HTeVz-Us7hRGfUzV_UoKYxRgx5BwosLrzEEws01zYVd6VKo3KIDB6PI/s1600/tab4.JPG" height="306" title="what's happening during the first seven minutes?" width="640" /></a></div>
<br />
<div style="text-align: justify;">
We could go on until the <b>30th minute</b>, but now we’ve got <b>enough clues</b> to figure out what's happening.</div>
<br />
<div style="text-align: justify;">
<span style="color: #073763;">Yeah? Well, I just don’t see what happens... And with this rythm I don't think that many people will know the notice before the press conference starts</span></div>
<br />
<div style="text-align: justify;">
Wait a minute. Let's look at the number of people who know the name of the new player at the end of each minute:</div>
<br />
<div style="text-align: center;">
<span style="color: #cc0000; font-size: 120%;"><b>0, 1, 1, 2, 3, 5, 8, 13...</b></span></div>
<br />
<span style="color: #073763;">I can't understand how these numbers are <b>related </b>to each other...</span><br />
<br />
<div style="text-align: justify;">
Try as follows: pick the first <b>2</b> numbers, <b>add them up</b> and tell me the result.</div>
<br />
<div style="text-align: center;">
<span style="color: #073763;">0 + 1 = 1</span></div>
<br />
<div style="text-align: justify;">
We’ve obtained the third number in the sequence. Now take the second and the third number, and add them up again.</div>
<br />
<div style="text-align: center;">
<span style="color: #073763;">1 + 1 = 2</span></div>
<br />
<span style="color: #073763;">Are we going to do the same with the third and fourth?</span><br />
<br />
Yes, please.<br />
<br />
<div style="text-align: center;">
<span style="color: #073763;">1 + 2 = 3</span></div>
<br />
<div style="text-align: justify;">
<span style="color: #073763;">I see how it works. Each number is obtained by adding the previous two, right?</span></div>
<br />
That's right. In <b>mathematical terms</b> we’ll define it as follows:<br />
<br />
<div style="text-align: center;">
F(n) = F(n-1) + F(n-2); F(0) = 0; F(1) = 1</div>
<br />
<div style="text-align: justify;">
This way, we’ll generate the following numbers until we reach number 30 in the series, which will be the one that indicates the number of people who knew the news before the press conference started.</div>
<br />
<div style="text-align: justify;">
Let's write down the numbers of the sequence, that mean the <b>amount of people who know the news</b>, and in brackets we'll write the corresponding <b>minute</b>:</div>
<br />
0 (0), 1 (1), 1 (2), 2 (3), 3 (4), 5 (5), 8 (6), 13 (7), 21 (8), 34 (9), 55 (10), 89 (11), 144 (12), 233 (13), 377 (14), 610 (15), 987 (16), 1.597 (17), 2,584 (18), 4,181 (19), 6,765 (20), 10,946 (21), 17,711 (22), 28,657 (23), 46,368 (24), 75,025 (25), 121,393 (26), 196,418 (27), 317,811 (28), 514,229 (29), <b><span style="color: #cc0000;">832,040 (30)</span></b><br />
<br />
<div style="text-align: justify;">
<span style="color: #073763;">Nearly <b>a million</b> people, in just<b> 30 minutes</b>, without performing mass messages!</span></div>
<br />
<div style="text-align: justify;">
Indeed, this sequence seems that progresses very slowly (in the first 10 minutes only 55 people know the news), but then it has an <b>exponential</b> behaviour.</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXoL1Tx-1qxp9FJr9TA5tFCOEM7aYOK__zGZDOlY6e8C5QOQBEaWuumZgSPtSu8WMtgeP9r9zvH1ZKNnIU2cFLD5AWvKD3lI-7rIepX2vzKRjGkKBLy7czpLbaEzQpijEFgPclmvDQ6Dc/s1600/grafico.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="People who know the story as time progresses" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXoL1Tx-1qxp9FJr9TA5tFCOEM7aYOK__zGZDOlY6e8C5QOQBEaWuumZgSPtSu8WMtgeP9r9zvH1ZKNnIU2cFLD5AWvKD3lI-7rIepX2vzKRjGkKBLy7czpLbaEzQpijEFgPclmvDQ6Dc/s1600/grafico.JPG" height="388" title="People who know the story as time progresses" width="640" /></a></div>
<br />
<div style="text-align: justify;">
In fact, if we continue with the series, and assuming that all inhabitants of the planet have mobile, and network coverage, the news will be known <b>worldwide</b> in just <b class="unidas">49 minutes</b>.</div>
<br />
<div style="text-align: justify;">
<span style="color: #073763;">But if there are several people who <b>send</b> the message <b>to the same person</b>, this no longer fulfills...</span></div>
<br />
<div style="text-align: justify;">
That’s why, at the beginning of the story, we said that we knew that nobody has received the message from two different issuers. So we avoid that multiple people send the message to the same recipient. This doesn’t happen in real life, so the figures would be somewhat different from those calculated in theory.</div>
<br />
<div style="text-align: justify;">
In addition, we must consider another detail. Not all the people are interconnected. There may be '<b>islands</b>' of people who will never receive the message. Imagine, for example, that all the Greek people have solely contacts of Greek people in their phones, and none from another country. The message would not leave Greece! Or think that people living in Oslo are all interconnected, but don’t have contacts of other persons outside the city. The message would spread throughout the world, except by Oslo.</div>
<br />
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDhARW1lZNkJVqd6n1PMhVkQrBwOXsNCvQh9uOpUjj68sZ4q-15Pvaj_dEN8ACY_xKOW4p2L5v9G9KO6991vxoClmdFZU51cDtV7P6wH4tCyVZ72Kf-XL8ZyeqMm8MFXBLl4nhAwt8ITQ/s1600/Retratodefibonacci2.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Leonardo Fibonacci, Leonardo Pisano or Leonardo of Pisa" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDhARW1lZNkJVqd6n1PMhVkQrBwOXsNCvQh9uOpUjj68sZ4q-15Pvaj_dEN8ACY_xKOW4p2L5v9G9KO6991vxoClmdFZU51cDtV7P6wH4tCyVZ72Kf-XL8ZyeqMm8MFXBLl4nhAwt8ITQ/s1600/Retratodefibonacci2.jpg" height="400" title="Leonardo Fibonacci, Leonardo Pisano or Leonardo of Pisa" width="281" /></a>Despite these drawbacks that can occur, it’s likely that, by the end of the press conference, almost everyone will already know the name of the new striker. However, the people who <b>have guessed the surname</b> of the player will be just a <b>few</b>.<br />
<br /></div>
<span style="color: #073763;">And, what can be his surname?</span><br />
<br />
<div style="text-align: justify;">
You see, this sequence we've seen, in which each number is obtained by adding the two previous, is called <b><span style="font-size: 120%;">Fibonacci sequence</span></b>. Fibonacci was a mathematician of the 12th-13th centuries, significant because he introduced the use of Arabic numerals into Europe, and because he discovered, among other works, this succession when studying a problem about breeding of rabbits. However, this sequence was already noted by some Indian mathematicians some centuries before.</div>
<br />
<span style="color: #073763;">Then, the player is called Leonardo Fibonacci, right?</span><br />
<br />
<div style="text-align: justify;">
Actually his name was not Fibonacci. Fi-Bonacci means "son of Bonacci", which was how his father was known, an Italian tradesman. His real name was Leonardo Pisano (Leonardo of <a href="http://www.youtube.com/watch?v=5WDvg-Rsw3s" target="_blank">Pisa</a>).</div>
<br />
And by chance there’s an Italian striker whose name's: <b><span style="color: #cc0000;">Leonardo Pisano</span></b>. So, we have the <b>full name</b> of the forward!<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUu1hNoVYNiv3cwJG1oDg24HhyRoNptl_oZJV544IFtAQ0lXuooKPaMD7GWkX9hx0al4gNo4mn5EsJYFbfNbSYrKy-jNegpuWFHpQ7uHBleaZZJPiLM4uc0p7MQaeaRjgFV3lUB1Zx7C4/s1600/leonardodepisa.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="The full name of the player: Leonardo Pisano" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUu1hNoVYNiv3cwJG1oDg24HhyRoNptl_oZJV544IFtAQ0lXuooKPaMD7GWkX9hx0al4gNo4mn5EsJYFbfNbSYrKy-jNegpuWFHpQ7uHBleaZZJPiLM4uc0p7MQaeaRjgFV3lUB1Zx7C4/s1600/leonardodepisa.jpg" height="464" title="The full name of the player: Leonardo Pisano" width="640" /></a></div>
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<span style="color: #073763;">Well, I had never heard about Fibonacci nor his series of numbers.</span></div>
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It's astounding, because it’s a <b>sequence </b>that appears in the most unexpected places, which leads to many mathematicians and laymen to engage in researching its <b>characteristics and applications</b>.</div>
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Among its properties, which would fill an entire encyclopedia, there are some really amazing.</div>
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For example, the <b>square </b>of each number in the series is equal to the <b>product </b>of the two <b>adjacent</b> numbers, adding or subtracting 1 (the difference alternates positive-negative-positive-negative -...)</div>
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0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, <span style="color: #cc0000;">144</span>, <span style="color: #cc0000;">233</span>, <span style="color: #cc0000;">377</span>, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711...<br />
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144<sup>2</sup> = 20,736 89 x 233 = 20,737 = 144<sup>2</sup> <span style="color: #cc0000;"><b>+ 1</b></span><br />
233<sup>2</sup> = 54,289 144 x 377 = 54,288 = 233<sup>2</sup><span style="color: #073763;"> <b>- 1</b></span><br />
377<sup>2</sup> = 142,129 233 x 610 = 142,130 = 377<sup>2</sup> <span style="color: #cc0000;"><b>+ 1</b></span><br />
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Moreover, if we pick 4 successive Fibonacci numbers, we get that the <b>difference</b> of the <b>squares</b> of the two <b>central</b> numbers equals the <b>product</b> of the two<b> extremes </b><br />
<div class="unidas">
<span style="text-align: left;">C</span><sup style="text-align: left;">2</sup><span style="text-align: left;">-B</span><sup style="text-align: left;">2</sup><span style="text-align: left;">=AxD.</span></div>
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0, 1, 1, 2, 3, 5, 8, 13, <span style="color: #cc0000;">21</span>, <span style="color: #cc0000;">34</span>, <span style="color: #cc0000;">55</span>, <span style="color: #cc0000;">89</span>, 144, 233, 377, 610, <span style="color: #073763;">987</span>, <span style="color: #073763;">1597</span>, <span style="color: #073763;">2584</span>, <span style="color: #073763;">4181</span>, 6765, 10946, 17711...<br />
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55<sup>2</sup>-34<sup>2</sup> = 1,869 = 21 x 89<br />
2,584<sup>2</sup>-1,597<sup>2</sup> = 4,126,647 = 987 x 4,181<br />
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If instead of 4 consecutive numbers, we take <b>10 numbers </b>and we <b>add</b> them together, we see that the sum is equal to the <b>seventh</b> number <b>multiplied</b> by 11.</div>
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0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, <span style="color: #cc0000;">89</span>, <span style="color: red;">144</span>, <span style="color: #cc0000;">233</span>, <span style="color: #cc0000;">377</span>, <span style="color: #cc0000;">610</span>, <span style="color: #cc0000;">987</span>, <span style="background-color: yellow; color: red;">1597</span>, <span style="color: red;">2584</span>, <span style="color: red;">4181</span>, <span style="color: red;">6765</span>, 10946, 17711...<br />
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89+144+233+377+610+987+1,597+2,584+4,181+6,765 = 17,567 = 1,597 x 11<br />
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If you look at the <b>latest figures</b> from the Fibonacci numbers, we can see that every 60 numbers we find the <b>same figures</b>. If you look at the last 2 figures, they repeat every 300 numbers. The last 3 figures are repeated in 1500 numbers. And so on.</div>
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<span style="text-align: justify;">If we forget about the first 0, the third number is 2, and we find a <b>multiple of 2</b> every 3 numbers. The fourth number is 3, and every 4 numbers we get a <b>multiple of 3</b>. The fifth number is 5, and every 5 numbers we find a <b>multiple of 5</b>. The sixth number is 8, and we have a <b>multiple of 8</b> every 6 numbers. And so on.</span><br />
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1, 1, <span style="color: #274e13;">2</span>, 3, 5, <span style="color: #274e13;">8</span>, 13, 21, <span style="color: #274e13;">34</span>, 55, 89, <span style="color: #274e13;">144</span>, 233, 377, <span style="color: #274e13;">610</span>, 987, 1597, <span style="color: #274e13;">2584</span>, 4181, 6765, <span style="color: #274e13;">10946</span>, 17711...<br />
1, 1, 2, <span style="color: red;">3</span>, 5, 8, 13, <span style="color: red;">21</span>, 34, 55, 89, <span style="color: red;">144</span>, 233, 377, 610, <span style="color: red;">987</span>, 1597, 2584, 4181, <span style="color: red;">6765</span>, 10946, 17711...<br />
1, 1, 2, 3, <span style="color: #073763;">5</span>, 8, 13, 21, 34, <span style="color: #073763;">55</span>, 89, 144, 233, 377, <span style="color: #073763;">610</span>, 987, 1597, 2584, 4181, <span style="color: #073763;">6765</span>, 10946, 17711,<br />
1, 1, 2, 3, 5, <span style="color: magenta;">8</span>, 13, 21, 34, 55, 89, <span style="color: magenta;">144</span>, 233, 377, 610, 987, 1597, <span style="color: magenta;">2584</span>, 4181, 6765, 10946, 17711,<br />
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<b>The sum of all numbers</b> up to a given one equals the number two positions farther, subtracting 1</div>
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<span style="color: #cc0000;">0</span>,<span style="color: #cc0000;"> 1</span>, <span style="color: red;">1</span>, <span style="color: #cc0000;">2</span>, <span style="color: #cc0000;">3</span>, <span style="color: red;">5</span>, <span style="color: red;">8</span>, <span style="color: red;">13</span>, <span style="color: #cc0000;">21</span>, <span style="color: red;">34</span>, <span style="color: red;">55</span>, <span style="color: red;">89</span>, 144, <span style="color: blue;">233</span>, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711..<br />
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0+1+1+2+3+5+8+13+21+34+55+89 = 232 = 233-1<br />
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And if we <b>divide </b>each number by the immediately preceding, we see that, as we move forward in succession, the result of this division is getting closer and closer to the value of the <b>golden ratio</b>, also known as <b>golden section</b>, <b>golden mean</b>, <b>golden number</b> or <b>Phi</b> (<b>φ</b>).</div>
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1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711...<br />
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1/1=<span style="color: red;">1</span> 2/1=<span style="color: red;">2</span> 3/2=<span style="color: red;">1.5</span> 5/3=<span style="color: red;">1.666...</span> 8/5=<span style="color: red;">1.6</span> 13/8=<span style="color: red;">1.625</span> 21/13=<span style="color: red;">1.615...</span> 34/21=<span style="color: red;">1.619</span> 55/34=<span style="color: red;">1.617...</span><br />
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<span style="text-align: justify;">φ = 1.61803398874989484820458683...</span></div>
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This ‘golden ratio’ sounds me a lot. It's widely used in art, right?</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSH6VvQrUHFVFL2IpTV53PGnCjzjtgimHjRMdxCQEG3hjGznkYhu_rBHjIEaws9MR8q5T3WN3JKPsyBFu8dt6KtIEX_pJ0jzoP-how_MNPHltag7jbbTf2oQZiyf67SLKzuAhDQ6SzFfE/s1600/hombredevitrubio.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Vitruvian Man of Leonardo da Vinci" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSH6VvQrUHFVFL2IpTV53PGnCjzjtgimHjRMdxCQEG3hjGznkYhu_rBHjIEaws9MR8q5T3WN3JKPsyBFu8dt6KtIEX_pJ0jzoP-how_MNPHltag7jbbTf2oQZiyf67SLKzuAhDQ6SzFfE/s1600/hombredevitrubio.jpg" height="200" title="Vitruvian Man of Leonardo da Vinci" width="189" /></a><br />
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Yes. Many artists used this ratio in their art works. Thus we have the <b><a href="http://www.youtube.com/watch?v=tBdEr2-C-U4" target="_blank">Vitruvian Man</a></b>, in which another <b>Leonardo</b>, <v>da Vinci </v><span style="text-align: left;">this time, draws a man starting from rectangles formed with the golden ratio. We observe the same on the Mona Lisa, or on the picture of the Last Supper. And other painters such as </span><b style="text-align: left;">Michelangelo</b><span style="text-align: left;"> and </span><b style="text-align: left;">Dürer</b><span style="text-align: left;"> also applied it widely in their works.</span></div>
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<span style="color: #073763;">So this proportion has been used from the Renaissance...</span></div>
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No. It was known long before. Thus, we find it in the <b>Athenian Parthenon</b> (s. V BC), when analyzing the relationship between the parts, the roof and columns. And the Babylonian and Assyrian civilizations also used it.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqsvz_Z4BpUMRRirkudnJEPGOVwySGM8_puOThqgQt80MaAOpAPk5C6UPHlKRiU4taHvgqjISl_Bl_0tHQPQ1_4EiNVKPYxnaBt1Swbo96w-DNhQByNikIHb6Yj2hcG5GluqeBs8WqCpg/s1600/partenon2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Proportions of the Parthenon temple, and the golden ratio" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqsvz_Z4BpUMRRirkudnJEPGOVwySGM8_puOThqgQt80MaAOpAPk5C6UPHlKRiU4taHvgqjISl_Bl_0tHQPQ1_4EiNVKPYxnaBt1Swbo96w-DNhQByNikIHb6Yj2hcG5GluqeBs8WqCpg/s1600/partenon2.jpg" height="396" title="Proportions of the Parthenon temple, and the golden ratio" width="640" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-Y64OfKnbByT5UZG6Zg1i90LrYOllN28BKI4yY0F9shff4UEPgW0Ad23N6HULvWA8dVjUIurThZNvcVIDpVrWSOTUr4iP7CD2OuDWGbJXUMcnMwdz95m2umZDTU5DPzXfR9sVoT3GNIg/s1600/violin.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="The golden ratio in the design of violins" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-Y64OfKnbByT5UZG6Zg1i90LrYOllN28BKI4yY0F9shff4UEPgW0Ad23N6HULvWA8dVjUIurThZNvcVIDpVrWSOTUr4iP7CD2OuDWGbJXUMcnMwdz95m2umZDTU5DPzXfR9sVoT3GNIg/s1600/violin.jpg" height="320" title="The golden ratio in the design of violins" width="174" /></a><span style="color: #073763;"></span><br />
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<span style="color: #073763;">So, it’s also <b>applied</b> to other various arts apart from of painting.</span></div>
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Yes, in addition to <b>painting</b> and <b>sculpture</b>, we can observe the golden ratio in the formal structure of many works of <b>classical music</b> of Mozart, Beethoven, Schubert, Béla Bartók and Debussy. And even in the arrangement of the shape of violins.</div>
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<span style="color: #073763;">I see that this sequence has multiple mathematical and artistic applications.</span></div>
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And there’re more applications. We can find apps in other areas, truly amazing.</div>
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In nature, we can see how this series appears in the spiral arrangement of <b>sunflower seeds</b>. Thus there’re sunflowers containing 21 <b>spirals</b> in one direction and 34 in the other. And others have 55 and 89 spirals, or even 89 and 144, but always about <b>consecutive Fibonacci numbers</b>.</div>
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<span style="color: #073763;">What a strange layout!</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFAxjPZO1E5YvTBsMxCCT6lOOF9Pia3dOk2vNVl4IGrGY0BqmLE0lcHgiw7_0zd82cEcPSigdZFSgJSq4Z-d9kOW4hv1JH5MemZB9HN53zUuGFZ-lzYyuNKQhNFvUc5GUNLsffuyNBB9E/s1600/girasol2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Spiral arrangement of sunflower seeds" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFAxjPZO1E5YvTBsMxCCT6lOOF9Pia3dOk2vNVl4IGrGY0BqmLE0lcHgiw7_0zd82cEcPSigdZFSgJSq4Z-d9kOW4hv1JH5MemZB9HN53zUuGFZ-lzYyuNKQhNFvUc5GUNLsffuyNBB9E/s1600/girasol2.jpg" height="480" title="Spiral arrangement of sunflower seeds" width="640" /></a></div>
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Well, it turns out that this is <b>the best way</b> on a circular flower <b>to fit</b> as many seeds as possible. We also find this layout in the <b>pinecones</b> with their double set of intersecting spirals: the total of right-turning and left-turning spirals correspond to the Fibonacci sequence: 5 and 8, or 8 and 13. <b>Daisies</b> group their seeds in 21 spirals in one direction and 34 in the other, and usually have 13, 21, 34, 55 or 89 petals.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSJCIt5bg4JIrI-yq0tLS6quwIUkJ8ZgqY8O34KkgVxcQ2v7td3lPTkzb95Bo1CY9Nf-K64KbTHQziuMTcGir80au3HTbKxjyS3TUJlKVwJhfum6gKue7skqP_YeRXURgA-EGsQ5uGSxw/s1600/hojas.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="How the leaves are distributed in the branches" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSJCIt5bg4JIrI-yq0tLS6quwIUkJ8ZgqY8O34KkgVxcQ2v7td3lPTkzb95Bo1CY9Nf-K64KbTHQziuMTcGir80au3HTbKxjyS3TUJlKVwJhfum6gKue7skqP_YeRXURgA-EGsQ5uGSxw/s1600/hojas.gif" height="320" title="How the leaves are distributed in the branches" width="164" /></a>If you look at the <b>branches</b> or <b>leaves</b> of plants, we see that they are always distributed so that they can receive the <b>maximum sunlight</b> and <b>rainwater </b>as possible. For this aim, its position around the stem or branch is determined by the Fibonacci sequence. And also occurs with <b>roots</b>, in order to cover the maximum possible terrain and therefore food, and so as to interfere as little as possible between them.</div>
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<span style="color: #073763;">So the <a href="http://www.youtube.com/watch?v=pMnc45ojXQc" target="_blank">plant world</a> is full of sequences of Fibonacci...</span></div>
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And also the animal world. For example, in <b>human body</b>, proportions between the <b>distance </b>from shoulder to fingers and the distance from elbow to the fingers, the height of the hip and knee, the joints of the hands and feet, or the height of being human and height of the belly button, among others, are determined by the <b>golden ratio</b>, so intimately linked to the <b>Fibonacci</b> numbers.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhr8i5OvJ_VG2unZU2nF81msINqK20FHBGijRb_uXzHYF4q9OWL6k2knQWBrC7j6IqqhRnqLSf4TMp-s8880rYCKWi2WdnESHHpqe3ePiQaznjEZQU-g8XU9k05F-IU0cYITjkTjpyG_y4/s1600/vitrubio.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="The golden ratio in human proportions" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhr8i5OvJ_VG2unZU2nF81msINqK20FHBGijRb_uXzHYF4q9OWL6k2knQWBrC7j6IqqhRnqLSf4TMp-s8880rYCKWi2WdnESHHpqe3ePiQaznjEZQU-g8XU9k05F-IU0cYITjkTjpyG_y4/s1600/vitrubio.jpg" height="488" title="The golden ratio in human proportions" width="640" /></a></div>
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In the <b>bees</b>’ world, we find these numbers when we count the number of possible <b>routes</b> a bee can take by the hexagonal cells of a <b>honeycomb</b>.</div>
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We also find this pattern in how hares or rabbits <b>multiply </b>(that was the original problem in which Fibonacci stated this series) as well as other animals. And the DNA molecule measures 34 armstrongs long by 21 armstrongs wide!</div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMWbGRr6fw0RpJhdr812K5BdIfs4YGl0qa7_TiGFJBKDXZyda_4FPTvTGje0c3FzAikXvFvJyRvmd8LlYvQQ2yF-o9u1Eilqcbut6ootyajq5ARM3IS9FWX35bW-c1pNtu4Jod6VFmJ-Y/s1600/nautilus3.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Spiral shell of the nautilus" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMWbGRr6fw0RpJhdr812K5BdIfs4YGl0qa7_TiGFJBKDXZyda_4FPTvTGje0c3FzAikXvFvJyRvmd8LlYvQQ2yF-o9u1Eilqcbut6ootyajq5ARM3IS9FWX35bW-c1pNtu4Jod6VFmJ-Y/s1600/nautilus3.jpg" height="312" title="Spiral shell of the nautilus" width="320" /></a><br />
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<span style="color: #073763;">There’s only left to find these numbers in the <b>marine world</b>...</span></div>
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Indeed. And for that, nothing better than looking at the shell of the <b>Nautilus</b>.</div>
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In it, each full convolution is at a distance from the center 1.618 (<b>φ</b>) times of the previous round. This spiral, called <b>golden spiral</b> (or Dürer spiral), can be roughly drawn by using the numbers of the Fibonacci sequence. And can be found in many animals as in the shells of snails or in the horns of ruminants, and even in the form of some galaxies.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9kkpMVr0yyXSjboN5aq2bhniVf1NPBrG6LdEjpQXYPbvf0Oik-lQ-QdRxXKVYL-89vs0mfzWPim6fOKfhTgix2yi771cwhA6bjhRjDQxb7BPCkbVUUouLCKQMrBm8RKzOjwPH7604HEQ/s1600/espiraldorada3.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Spiral based on Fibonacci numbers" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9kkpMVr0yyXSjboN5aq2bhniVf1NPBrG6LdEjpQXYPbvf0Oik-lQ-QdRxXKVYL-89vs0mfzWPim6fOKfhTgix2yi771cwhA6bjhRjDQxb7BPCkbVUUouLCKQMrBm8RKzOjwPH7604HEQ/s1600/espiraldorada3.gif" height="404" title="Spiral based on Fibonacci numbers" width="640" /></a></div>
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In <b>stock market</b>, when a certain value has been going up or down for long periods, and changes its trend, the limit of the estimated variation or correction tends to correspond to the inverse of the golden number 1/<b>φ= </b>61,8%. And Fibonacci numbers are widely used to identify <b>changes in market trends</b>, by setting time periods of 5, 8, 13 and 21 years in the graphic indices or values.</div>
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<span style="color: #073763;">And can we find any application of these numbers in the world of <b>football?</b></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTb6ozBdWEXDW0SOM6KqYV_5pPw1t-W3eHv92qjWnacYgoDyZ4NAJOOJTJuSWfYzgIqtL-Ovmj_xIcTmkEgs_dLg5q4aSfIAD9CncsjekgfgMBqbHFZz6oQKrMBJEQfXGS8Mc6bKoT3ig/s1600/apuestas.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="The Fibonacci system on sports betting" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTb6ozBdWEXDW0SOM6KqYV_5pPw1t-W3eHv92qjWnacYgoDyZ4NAJOOJTJuSWfYzgIqtL-Ovmj_xIcTmkEgs_dLg5q4aSfIAD9CncsjekgfgMBqbHFZz6oQKrMBJEQfXGS8Mc6bKoT3ig/s1600/apuestas.jpg" height="243" title="The Fibonacci system on sports betting" width="320" /></a>Not so much in football itself, but in something closely related to it: <b>sports betting</b>. For them, the <b>Fibonacci system</b> has been determined as one of the safest methods, especially when you bet on favorite..</div>
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<div style="text-align: justify;">
It works as follows: the player will be making bets whose amount is determined by the numbers of the Fibonacci sequence. If you <b>lose</b> a bet, you should <b>keep betting</b> using the <b>following number</b> and amount of the sequence. And if you <b>win</b>, you must <b>go back 2 numbers</b> in the sequence, and bet that amount.</div>
<br />
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269...<br />
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<div style="text-align: justify;">
This is a very good system for <b>betting on favorite</b>, as it’s very difficult to miss more than 5 consecutive times. The main drawback is that the <b>growth rate</b> of potential <b>gains</b> is a little <b>slow</b>, and for having big results you have to invest more time, but instead losses are minimized.</div>
<br />
<span style="color: #073763;">And these Fibonacci numbers have something to do with <b><a href="http://www.youtube.com/watch?v=BURjKRfOA9g" target="_blank">fractals</a></b>, right?</span><br />
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<div style="text-align: justify;">
Yes, they do. Do you remember the tables we have been drawing to see the evolution of the people who knew the news as the minutes passed? Well, we could have studied the <b>evolution</b> of the <b>spreading </b>of the news in a more graphically way.</div>
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<div style="text-align: justify;">
We will assign a <b>small circle</b> to each person involved in the communication. We'll leave them <b>blank</b> when they <b>don’t yet known</b> the news because they haven’t yet read them, and we’ll paint them <b>green</b> one minute after receiving the news, because we assume that they <b>have already read</b> them.</div>
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<div style="text-align: justify;">
We draw a <b>blue arrow</b> between two circles to indicate that a person is <b>sending a message</b> to another. We’ll paint a <b>yellow</b> line to show that the person is <b>reading the message</b>, and a <b>red</b> line to indicate that the person is <b>thinking </b>on the next receiver of the message.</div>
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And now look at our new scheme:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZHF2XJ9neSwK9XyCzXx36ijOQdhbmxbMP8h6jcEOYNzJiZSZgK3HhQbpe-8_Dd0wQsf_Xh7-AEbDGpo8OctCbSIBw-TGrs76a_V_r1w7OYCQujWinJj2I1lzd_XITjDTS_p_qhmx86DY/s1600/cuadricula.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Diagram of how the news spread" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZHF2XJ9neSwK9XyCzXx36ijOQdhbmxbMP8h6jcEOYNzJiZSZgK3HhQbpe-8_Dd0wQsf_Xh7-AEbDGpo8OctCbSIBw-TGrs76a_V_r1w7OYCQujWinJj2I1lzd_XITjDTS_p_qhmx86DY/s1600/cuadricula.jpg" height="248" title="Diagram of how the news spread" width="640" /></a></div>
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It’s clear that the layout of the scheme is a <b>fractal</b>: the main 'tree' (root in this case) repeats its form for any person you choose</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgygcJZIH3SNud81OTscp31JfRm5ZBlgyApVh6bC6ahL5W3-KvTnlKbBwc3EQZtB4TDfWEHPGCh_jE6EWd05FVxAmbQkxPAZuGRQfXO-dKe5ZFaVJG7hxCF7uZCpujAt9wDxIID33jHzw8/s1600/cuadricula2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Fractal formed with the spreading of the message through individual SMS" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgygcJZIH3SNud81OTscp31JfRm5ZBlgyApVh6bC6ahL5W3-KvTnlKbBwc3EQZtB4TDfWEHPGCh_jE6EWd05FVxAmbQkxPAZuGRQfXO-dKe5ZFaVJG7hxCF7uZCpujAt9wDxIID33jHzw8/s1600/cuadricula2.jpg" height="248" title="Fractal formed with the spreading of the message through individual SMS" width="640" /></a></div>
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<div style="text-align: justify;">
<span style="color: #073763;">It's true. Just one more question. To go getting the terms of the Fibonacci sequence, we have been adding the previous two. But would be there a <b>formula </b>to <b>calculate a specific term</b> of the sequence, i.e. the 49th term, without having to make all the previous sums?</span></div>
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Yes, of course. And guess who we find in the formula!</div>
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<span style="color: #073763;">I’ve got no idea.</span><br />
<br />
Our beloved ‘<b>golden number</b>’ o Phi (φ):<br />
<br />
This is the formula :<br />
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<div style="text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilefQw7kGQ37jPmPEwvgfI-GkJ90jA8EG7L8sZvQ-ir9Xg3zRqZyGGpfLxPLe67kta7jc7g0yCqxI4wnlo9G0B94ySIGdDAOPJLXuVJXIQysmxkPIo7G8QV0xymDrb2RDjX8SMrQuYmbk/s1600/movil2eng.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="I like this story" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilefQw7kGQ37jPmPEwvgfI-GkJ90jA8EG7L8sZvQ-ir9Xg3zRqZyGGpfLxPLe67kta7jc7g0yCqxI4wnlo9G0B94ySIGdDAOPJLXuVJXIQysmxkPIo7G8QV0xymDrb2RDjX8SMrQuYmbk/s1600/movil2eng.jpg" height="320" title="I like this story" width="193" /></a>ƒ<sub>n</sub> = [(φ<sup>n</sup>-(1-φ)<sup>n</sup>]/√5
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<div style="text-align: justify;">
And if we apply this formula (attributed to Binet) to number 49, we get that, at <b>minute 49</b> there will be <b>7,778,742,049</b> persons who would know the scoop.</div>
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<span style="color: #073763;">It's clear that secrets are only secrets if you don’t tell them to anyone...</span></div>
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That 's true. In any case, this story is no secret, so you <b>may spread it</b> so widely as you want, even <b>massively</b> <span class="unidas" style="color: #cc0000; font-size: 120%;">;-)</span> </div>
<div style="text-align: justify;">
<br /></div>
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And if you're interested in learning more about Fibonacci, you can visit any of these wonderful links: <a href="http://www.etereaestudios.com/docs_html/nbyn_htm/about_index.htm" target="_blank">Nature by numbers: the theory behind this movie</a>, <a href="http://www.inspirationgreen.com/fibonacci-sequence-in-nature.html" target="_blank">The Fibonacci sequence in nature</a>, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html" target="_blank">The Fibonacci numbers and golden section in nature</a>, <a href="http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html" target="_blank">Nature, the golden ratio, and Fibonacci too...</a>, <a href="http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm" target="_blank">Fibonacci in nature</a>, <a href="http://www.intelligentbettingtips.com/blog/sports/soccer/soccer-betting-system-fibonacci-betting-strategy/" rel="nofollow" target="_blank">Soccer betting system Fibonacci strategy</a>, <a href="http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm" target="_blank">Fibonacci Numbers in Nature</a>.</div>
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WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-9920069091908472902013-12-22T16:25:00.000+00:002013-12-22T17:09:16.821+00:00Magic 2014<!-- Resumen de la página -->
<span style="display: none;">The turn of the year gives us an excellent excuse to go over various mathematical properties of natural numbers, making a comparison between the numbers 13 and 14. With this story, Matifutbol wishes you a Merry Christmas and hopes that 2014 will be a magical year for you. <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWu_GNgdL3EIJAQJ02sXdMJI9hvkZflJ1YCMXid7MU1No2ymp7IXvL00_ju8leCJNu8YBoT8IJ2P8sgzTv6C7HpfDNJNbBfVATecw9jN5VosMNOVBrUuKeVFil13Hc87u0g_iNsPmgQ0U/s640/happy2014.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWu_GNgdL3EIJAQJ02sXdMJI9hvkZflJ1YCMXid7MU1No2ymp7IXvL00_ju8leCJNu8YBoT8IJ2P8sgzTv6C7HpfDNJNbBfVATecw9jN5VosMNOVBrUuKeVFil13Hc87u0g_iNsPmgQ0U/s640/happy2014.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1LXw_l9yukPYzYPnwPV3xloVZODp0bfRQ1umKyCZX20KJpZ3pfXbDOez7XtvhTLq0FTe4wmnw1V3CE18Ki2i0zLKW8RhVNMK3qlx7zX8_vwwr1YNn02YVpm4KxaYBozSEFnjGJcTMMuM/s1600/piramide.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1LXw_l9yukPYzYPnwPV3xloVZODp0bfRQ1umKyCZX20KJpZ3pfXbDOez7XtvhTLq0FTe4wmnw1V3CE18Ki2i0zLKW8RhVNMK3qlx7zX8_vwwr1YNn02YVpm4KxaYBozSEFnjGJcTMMuM/s1600/piramide.gif" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_HEkaJee9X8SmPb7ns_9x7SLqaLOtP26MEacJ0SiHt0WiJUhT-d82gB5jo8sw8-b2b8yFwBCThZUXNDMEveX-q7QctKNkkIUEvSu8oYx-rYH_yMPfXjud6u4ChLSO4xsTGC9LS9tIynk/s200/13fantasmagorico.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_HEkaJee9X8SmPb7ns_9x7SLqaLOtP26MEacJ0SiHt0WiJUhT-d82gB5jo8sw8-b2b8yFwBCThZUXNDMEveX-q7QctKNkkIUEvSu8oYx-rYH_yMPfXjud6u4ChLSO4xsTGC9LS9tIynk/s200/13fantasmagorico.png" /></a></span>
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<span style="display: none;"> </span>
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<span style="display: none;">(This post participates on the <a href="http://ztfnews.wordpress.com/2013/11/08/edicion-4-12310562-del-carnaval-de-matematicas-20-27-de-noviembre-de-2013/" target="_blank"><span class="negrita">Edición 4.12310562</span></a> del <a href="http://carnavaldematematicas.bligoo.es/" target="_blank"><span class="negrita">Carnaval de Matemáticas</span></a>, cuyo anfitrión es el blog <a href="http://ztfnews.wordpress.com/" target="_blank"><span class="negrita unidas">::ZTFNews.org</span></a>.)</span></div>
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<span style="font-family: Verdana, sans-serif;">FIRST HALF</span></div>
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<span style="font-family: inherit;">This time,
<span style="color: blue; font-size: 130%;">Matifutbol </span>has brought together some of the <b>best football players</b> to wish you a <span style="color: red; font-size: 130%;">Happy 2014!</span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp-JcDi62X4z29HYOh_d4A-ror24cuIRfBF5X6dbiFMZUhceeglYQZfyNSXv4MS_XbtD6_M_rWX6aUYayfPuZ_EC3p12LyY4bd5KDG2hvsI8RuzlEh0OTtwEkxlf-pS9DxfKbso86jsYc/s1600/happy2014.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="520" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp-JcDi62X4z29HYOh_d4A-ror24cuIRfBF5X6dbiFMZUhceeglYQZfyNSXv4MS_XbtD6_M_rWX6aUYayfPuZ_EC3p12LyY4bd5KDG2hvsI8RuzlEh0OTtwEkxlf-pS9DxfKbso86jsYc/s640/happy2014.jpg" width="640" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjeouUKIOnKDq1TJtE5URbMsunP5zz4RExXRzU7V3jYxP-Ez-PKDBvmkRkwaWb-I2ABd8UyLrfOLIN2WKjrw0pr-Ha8oSqYsJyg9BcznrKma2awonQpJOKN55ha2bmdLFWbZMkdbx1SMCQ/s1600/Feliz+2014+(13).jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: inherit;"></span></a></div>
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<span style="font-family: inherit;">I hope you'll have a great new year, and that all your math operations will be good for you!</span><br />
<br />
<div style="text-align: justify;">
<span style="color: #073763; font-family: inherit;">Thank you very much, but.... have you noticed that there are <b>13 </b>players on your card? What a <b>bad luck</b>! Also, as year 2014 is coming, I think you should have invited 14 players for the photo.</span><br />
<span style="color: #073763; font-family: inherit;"><br /></span></div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDj_mpPkzTUZ5vXUvLkP5XB76RxIzE_y8ia2sM0BLAdWxv2_Xwi6RHYPSgIdSdDa1nlpicm21IlUK42SWB-8KbVnphkigPIUmDxxW-Mn39K6_p-9Bm1eCp770B89dfb8MosxI69iRGkyc/s1600/de13a14.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: inherit;"><img border="0" height="105" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDj_mpPkzTUZ5vXUvLkP5XB76RxIzE_y8ia2sM0BLAdWxv2_Xwi6RHYPSgIdSdDa1nlpicm21IlUK42SWB-8KbVnphkigPIUmDxxW-Mn39K6_p-9Bm1eCp770B89dfb8MosxI69iRGkyc/s320/de13a14.gif" width="320" /></span></a><br />
<div style="text-align: justify;">
<span style="font-family: inherit;">Well, I can't see why number 13 has to mean bad luck. In any case, we can easily modify the card.</span></div>
<span style="font-family: inherit;"><br /></span>
<span style="color: #073763; font-family: inherit; text-align: justify;">Yes? How? It will be difficult to gather them back to take a new picture with one more player...</span><br />
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">There's a way to do it without putting them together again, using a <b>translational</b> move.</span></div>
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Can you imagine how?</span><br />
<span style="font-family: inherit;"><br /></span>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkZNjtAol92vhyphenhyphenzF8zPYQBJoWZPhgKjAjLQWtah0o5An2tD_vbR4BFGOi9RuMj21ey8lkQZWBzOQZhzGHdtdTPG4bs74FQqjHb743a1Xp2TrlYl2sLfPQfZKvWlmc7Z1MCgwRXo1zfMcI/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: inherit;"><img border="0" height="250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkZNjtAol92vhyphenhyphenzF8zPYQBJoWZPhgKjAjLQWtah0o5An2tD_vbR4BFGOi9RuMj21ey8lkQZWBzOQZhzGHdtdTPG4bs74FQqjHb743a1Xp2TrlYl2sLfPQfZKvWlmc7Z1MCgwRXo1zfMcI/s640/homomathematicus.jpg" width="640" /></span></a></div>
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<span style="font-family: inherit;">SECOND HALF</span></div>
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<span style="font-family: inherit;"><br /></span>
<span style="color: #073763; font-family: inherit;">Well, I can't imagine how to solve it...</span><br />
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<div style="text-align: justify;">
<span style="font-family: inherit;">In fact, there's nothing to solve. Number 13 is just a normal number, like any other.</span></div>
<span style="font-family: inherit;"><br /></span>
<span style="color: #073763; font-family: inherit;">Yes, but it's said to bring back luck. There must be something bad in it...</span><br />
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<div style="text-align: justify;">
<span style="font-family: inherit;">The truth is that this is something that comes from old times. Actually, it's a superstition that comes from the times of our ancestors, who worshiped the forces of nature.</span></div>
<span style="font-family: inherit;"><br /></span>
<span style="color: #073763; font-family: inherit;">Has nature anything to do with number 13?</span><br />
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<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaakPK12KRrV7Qo0et6m-3wPtOHQwYSQL3VLgh0L6Fi2gZIcnW8_vnikT6D-i3q0ScCnfkpZdLSoOKScwjYmIfQM6QbDmkcVrnhpIiCxmSoozmuOG06ZMDU-Z3ucVZpZMaICNucwDAYEc/s1600/lunaroja.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: inherit;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaakPK12KRrV7Qo0et6m-3wPtOHQwYSQL3VLgh0L6Fi2gZIcnW8_vnikT6D-i3q0ScCnfkpZdLSoOKScwjYmIfQM6QbDmkcVrnhpIiCxmSoozmuOG06ZMDU-Z3ucVZpZMaICNucwDAYEc/s1600/lunaroja.png" /></span></a><span style="font-family: inherit;">Yes, a lot, especially the <b>Moon</b>. Our ancestors discovered that there were years with 12 new moons, and years with 13 moons. Actually, the average is <b><span style="font-size: 130%;">12.41</span></b> new moons per year. Before we set calendars with 12 months, we used calendar with 13 months, one of which was shorter than the others.</span></div>
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<div style="text-align: justify;">
<span style="font-family: inherit;">This incomplete thirteenth moon, which caused an atypical month, with fewer days than the rest, gave more headaches than benefits to the shamans who had to handle the calendar to set their rituals and festivals, so they finally ended up by 'hating' this number.</span></div>
<span style="font-family: inherit;"><br /></span>
<span style="color: #073763; font-family: inherit;">That's how all started?</span><br />
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">This explanation is very likely, because almost all ancient cultures and civilizations throughout the world share some background on this issue.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">Thus, <b>13 </b>is considered a damned number by the ancient <b>Persians</b>, where number 13 was identified with chaos. <b>Zoroaster </b>also identifies it with the devil, and in Norse mythology the god Loki, Lord of the lies, appears as the 13<sup>th</sup> god assistant to a fateful dinner (dinners and number 13 tend to be closely related).</span></div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhk1GTdm_fHSB1GKlDPuDujmYz9oyp7-uZ-CTgD019yrtNJLW58VNEyj-eMbMO50FOSA9L039XOyOEAo-desvsAPwAEKzN5fXV3l9aFDa2ULK2zfylzr89b6txgUWHKudkJYkbUhiXAIcM/s1600/120.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><span style="font-family: inherit;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhk1GTdm_fHSB1GKlDPuDujmYz9oyp7-uZ-CTgD019yrtNJLW58VNEyj-eMbMO50FOSA9L039XOyOEAo-desvsAPwAEKzN5fXV3l9aFDa2ULK2zfylzr89b6txgUWHKudkJYkbUhiXAIcM/s200/120.jpg" width="150" /></span></a><br />
<div style="text-align: justify;">
<span style="font-family: inherit;"><b>Egyptians </b>also reserved number 13 for the last stage of life, the death. This is also taken by <b>Tarot</b>, that also assigns the death card to this number.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">And in Christianity we find numerous references. From the thirteenth guest at the Last Supper of Jesus Christ (Judas Iscariot), the man who betrayed him, to the thirteenth chapter of Apocalypsis, in which the coming of Antichrist is predicted.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">In any case, we could find an endless number of negative references regarding any number you want. But clearly there's something special about this number 13 that makes us to fix more attention on it than on others.</span></div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_HEkaJee9X8SmPb7ns_9x7SLqaLOtP26MEacJ0SiHt0WiJUhT-d82gB5jo8sw8-b2b8yFwBCThZUXNDMEveX-q7QctKNkkIUEvSu8oYx-rYH_yMPfXjud6u4ChLSO4xsTGC9LS9tIynk/s1600/13fantasmagorico.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: inherit;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_HEkaJee9X8SmPb7ns_9x7SLqaLOtP26MEacJ0SiHt0WiJUhT-d82gB5jo8sw8-b2b8yFwBCThZUXNDMEveX-q7QctKNkkIUEvSu8oYx-rYH_yMPfXjud6u4ChLSO4xsTGC9LS9tIynk/s200/13fantasmagorico.png" width="168" /></span></a><br />
<div style="text-align: justify;">
<span style="color: #073763; font-family: inherit;">Yes, it's true that many people has some superstitious misgivings about this number.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">And some people even take it to the extreme. We talk about <b>triskaidekaphobia</b>, which consists of the extreme and irrational fear to the number 13. Thus, due to this phobia, we can find countries where there's no 13<sup>th</sup> floor in buildings, hotels with 13<sup>th</sup> floor used for service tasks, streets without number 13, airplanes without seats row number 13, or athletes who avoid wearing this number on their backs. In other cases, 13<sup>th</sup> floor is replaced by 12 bis, for example, and there are also computer programs that hop from version 12 to 14.</span></div>
<br />
<div style="text-align: justify;">
<span style="color: #073763; font-family: inherit;">But all these issues are completely contrary to <b>Mathematics</b>, aren't they?</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">Yes, they are, but not at all. <b>Numerology </b>(which is the set of beliefs or traditions that aims to stablish a mystical relationship between numbers and living things, physical and spiritual forces) has always evolved in parallel with the development of Mathematics.</span></div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTgJ2J_fbqVRUEfjFx-BnFNtANtTSmwYCaX8SD-YTdvS8JnPXkqspnsM6G4sqEqfG80zRCuftd6N2CjntuaZx2YxU6IDQZVht9vaigph0Mmvfn5JtRnWQcFWuz_NWxjOEn9-uQ36SQ7hs/s1600/pitagoras.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><span style="font-family: inherit;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTgJ2J_fbqVRUEfjFx-BnFNtANtTSmwYCaX8SD-YTdvS8JnPXkqspnsM6G4sqEqfG80zRCuftd6N2CjntuaZx2YxU6IDQZVht9vaigph0Mmvfn5JtRnWQcFWuz_NWxjOEn9-uQ36SQ7hs/s200/pitagoras.gif" width="170" /></span></a><br />
<div style="text-align: justify;">
<span style="font-family: inherit;">Actually, a famous mathematician as <b><span style="font-size: 130%;">Pythagoras</span></b>, developed in his time (V century BC) a whole study on the relationship between numbers and the laws of nature. And also scholars of <b>Kabbalah</b> studied such relations, or <b>Eastern cultures</b>, that assign to numbers certain properties and meanings.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">But regardless of this, number 13, only for mathematical purposes, it's also a very interesting number.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">13 is a <b><span style="color: #660000; font-size: 130%;">prime</span></b> number, that is, it's only divisible by one and itself. It's also the first <span style="color: #660000; font-size: 130%;">emirp</span> number, ie, if we write it backwards, 31, we have another prime number.</span><br />
<span style="font-family: inherit;"><br /></span></div>
<span style="font-family: inherit;">Moreover, it's one of the terms of the <b><span style="color: #274e13; font-size: 130%;">Fibonacci</span></b> sequence, in which each number is obtained by adding the previous two:</span><br />
<br />
<div style="text-align: center;">
<span style="font-family: inherit; font-size: 130%;">1, 1, 2, 3, 5, 8, 13, 21, 34, 55...</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">It belongs to a <b><span style="color: #20124d; font-size: 130%;">Pythagorean triple</span></b>: 5, 12, 13. This means that they are 3 positive integers that satisfy the condition that a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup>, that is, that can be associated with the lenghts of the two legs/catheti and the hypotenuse of a right triangle. Moreover, it's a <span style="color: #20124d; font-size: 130%;">primitive</span> Pythagorean triple, ie, the greatest common divisor of the 3 numbers is 1.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">Also, there are only 13 <span style="font-size: 130%;">Archimedean solids</span> (excluding isomorphic forms). And 13 were the number of books <span style="font-size: 130%;">Euclid</span> wrote about Mathematics: 'The Elements'.</span><br />
<span style="font-family: inherit;"><br /></span>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfRhVtkVBC3aAgOdG9nTPsvUtjpiBQyrt-_gPlZCXGSC5sLdFFauTzIHXPkptgSmxMe8L0n7DBaQ5PLKcCaPe1yLUcarSilfG6g7Rod4Mcmx4j4Ji-rLJZ8cNzcw_JhZ9cxTb_glIMfxA/s1600/13feliz.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: inherit;"><img border="0" height="169" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfRhVtkVBC3aAgOdG9nTPsvUtjpiBQyrt-_gPlZCXGSC5sLdFFauTzIHXPkptgSmxMe8L0n7DBaQ5PLKcCaPe1yLUcarSilfG6g7Rod4Mcmx4j4Ji-rLJZ8cNzcw_JhZ9cxTb_glIMfxA/s200/13feliz.gif" width="200" /></span></a><br />
<div style="text-align: justify;">
<span style="font-family: inherit;">With numbers 5 and 563, 13 is one of the 3 known <b><span style="color: #073763; font-size: 130%;">Wilson primes</span></b>, ie, that satisfies the property that 13<sup>2</sup> divides (13-1)! + 1.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">13 is a <b><span style="color: #783f04; font-size: 130%;">happy</span></b> number, as you can see on this <a href="http://mathsball.blogspot.com.es/2013/07/football-and-happiness.html" target="_blank">link</a>. It's the third <b>ordered Bell number</b> or <b><span style="color: #134f5c; font-size: 130%;">Fubini </span>number</b> (related to combinatorial). And we could add more properties.</span></div>
<div style="text-align: justify;">
<span style="font-family: inherit;"><br /></span></div>
<div style="text-align: justify;">
<span style="color: #073763; font-family: inherit;">And is this important? What good is it to be a prime number or to belong to the Fibonacci sequence, for example?</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;"><span style="font-size: 130%;"><b>Prime numbers</b></span> have recently become important in the issue of documental encryption. Plus they're considered as the foundation over which all other natural numbers are constructed.</span></div>
<br />
<div style="text-align: justify;">
<a href="http://www.neoteo.com/la-sucesion-de-fibonacci-en-la-naturaleza/" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" target="_blank"><span style="font-family: inherit;"><img border="0" height="135" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmfJT2OVHfVAwquxH17-Bes_G1UsMOM6Y7PneWoENjip586cMry5ZHH6ia-CG-kR4B6UejuK00TUwzmqEfNjL0TKYdrs9ap79AQfeIWBcryggJUVGB12dYLSllML36Rwq7Wm2jv5t3_Qw/s200/hojas.jpg" width="200" /></span></a><span style="font-family: inherit;">And <b><span style="font-size: 130%;">Fibonacci</span> </b>sequence, in which each term is the sum of the previous two, is found in many places in nature: in the reproductive rates of animals or viruses, in the distribution of leaves around the stem of the plants for better use of the sun light, in pinapples, in sunflowers seeds, in daisies seeds, in arms of spiral galaxies and hurricanes, in equinodermos bodies, or in the family tree of bees .... It has also many applications in computing and games theory. And in addition, any natural number can be written as the sum of a limited number of terms in the Fibonacci sequence, each one different from the others.</span></div>
<span style="font-family: inherit;"><br /></span>
<span style="color: #073763; font-family: inherit;">But this will also happen with any number, such as number 14.</span><br />
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">Don't believe so. Unlike number 13, which seems to be a rather unique number in mattematical terms, number <b>14 </b>has some peculiarities hardly worth mentioning.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">It's a <b><span style="color: #20124d; font-size: 130%;">square pyramidal</span></b> number, that is, it's the result of the sum of the first 3 square numbers:</span><br />
<span style="font-family: inherit;"><br /></span></div>
<div style="text-align: center;">
<span style="font-family: inherit;"><span style="font-size: 130%;">14 = 1<sup>2</sup>+2<sup>2</sup>+3<sup>2</sup></span>
</span></div>
<div class="MsoNormal">
<span style="font-family: inherit;"><br /></span></div>
<div style="text-align: justify;">
<span style="font-family: inherit;">If we visualize these squares as layeres of spheres placed one above the other, this number represents a pyramid with a square base.</span><br />
<div class="MsoNormal">
<span lang="EN-US" style="font-family: inherit;"><o:p></o:p></span></div>
</div>
<div style="text-align: justify;">
<span style="font-family: inherit;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1LXw_l9yukPYzYPnwPV3xloVZODp0bfRQ1umKyCZX20KJpZ3pfXbDOez7XtvhTLq0FTe4wmnw1V3CE18Ki2i0zLKW8RhVNMK3qlx7zX8_vwwr1YNn02YVpm4KxaYBozSEFnjGJcTMMuM/s1600/piramide.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: inherit;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1LXw_l9yukPYzYPnwPV3xloVZODp0bfRQ1umKyCZX20KJpZ3pfXbDOez7XtvhTLq0FTe4wmnw1V3CE18Ki2i0zLKW8RhVNMK3qlx7zX8_vwwr1YNn02YVpm4KxaYBozSEFnjGJcTMMuM/s1600/piramide.gif" /></span></a></div>
<div style="text-align: justify;">
<span style="font-family: inherit;">Number 14 is a discrete <b><span style="color: #7f6000; font-size: 130%;">semiprime </span></b>or biprime number<span style="font-family: inherit;">, <span style="font-size: 12pt;">because it’s a natural
number which is</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">the result of the product of two prime
numbers 14=2*7.</span></span></span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">It's a <b><span style="color: #274e13; font-size: 130%;">Catalan </span></b>number (related to combinatorial problems), it's a <b>open meandric</b> number (related to intersections of curves and lines), it's a <b>Pell-Lucas</b> numb<span style="font-family: inherit;">er (</span><span lang="EN-US" style="font-size: 12pt;"><span style="font-family: inherit;">belongs to a sequence of
numbers related to the approximation by rational numbers to the square root of
2),</span> and it’s the maximum number of electrons that can fit in the </span><b>f atomic sublevel</b>.</span></div>
<br />
<div style="text-align: justify;">
<div class="MsoNormal">
<span lang="EN-US" style="font-family: inherit;">All these
are important features, but less relevant and practical than number 13’s properties.<o:p></o:p></span></div>
</div>
<br />
<div style="text-align: justify;">
<div class="MsoNormal">
<span lang="EN-US" style="font-family: inherit;"><span style="color: #073763;">So it’s seems
that calling 13 players, not 14, has been a good idea. But, what can you tell
me numbers 2013 and 2014?</span><o:p></o:p></span></div>
</div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiduH65A8s96-0UI4sJld2B9cf7eWbZFJLMqiY0VU-83V2Ac4a8Gc-DbkoeGZtLqZRmjk5hhnPTf59bT-1OZRbw-UIcNjTYj5rUsq2xA3tjD2M2QXf-uyNe3_XmNqT73lV9LNiMr3UQMXE/s1600/2013vs2014.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><span style="font-family: inherit;"><img border="0" height="75" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiduH65A8s96-0UI4sJld2B9cf7eWbZFJLMqiY0VU-83V2Ac4a8Gc-DbkoeGZtLqZRmjk5hhnPTf59bT-1OZRbw-UIcNjTYj5rUsq2xA3tjD2M2QXf-uyNe3_XmNqT73lV9LNiMr3UQMXE/s320/2013vs2014.gif" width="320" /></span></a></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">They're both sphenic numbers, ie, they are the product of 3 distinct prime factors</span><br />
<br />
<div style="text-align: center;">
<span style="font-family: inherit; font-size: 130%;">2013=3*11*61 and 2014=2*19*53</span></div>
</div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: inherit;"><br /></span></div>
<span style="font-family: inherit;">But 2013 is also the sum of three consecutive <b><span style="color: #783f04; font-size: 130%;">sphenic </span></b>numbers:</span><br />
<br />
<div style="text-align: center;">
<span style="font-family: inherit; font-size: 130%;">2013 = 665 (5*7*19) + 670 (2*5*67) + 678 (2*3*113)</span></div>
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Both have 8 <b>divisors</b>: <span style="color: blue;">1, 3, 11, 33, 61, 183, 671, 2013</span> and <span style="color: red;">1, 2, 19, 38, 53, 106, 1007, 2014</span>.</span><br />
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">And none of them is neither a <b>Fibonacci</b> number, nor a <b>Bell</b>, <b>Catalan</b>, factorial, regular prime, or perfect number.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: inherit;">Regarding to happiness, we can say that number <b>2013 </b>ís a <b>happy</b> number, while <b>2014 </b>is not.</span></div>
<br />
<div style="text-align: justify;">
<div class="MsoNormal">
<span lang="EN-US" style="font-family: inherit;"><span style="color: #073763;">So there’s
not much difference between the 2 numbers... However, I still think I don’t like
the card with only 13 players.</span><o:p></o:p></span></div>
</div>
<br />
<div style="text-align: justify;">
<div class="MsoNormal">
<span lang="EN-US" style="font-family: inherit;">For they
have all gone to their homes, and it will be hard to bring them all together again, I propose
an exercise of translation.<o:p></o:p></span></div>
</div>
<span style="font-family: inherit;"><br /></span>
<span style="color: #073763; font-family: inherit;">What's that of <b><span style="font-size: 130%;">translation</span></b>?</span><br />
<br />
<div style="text-align: justify;">
<div class="MsoNormal">
<span lang="EN-US" style="font-family: inherit;">It’s a process by which each point of the plane is shifted to another point at a distance.
It’s a direct move with no change in direction, ie, all moved objects maintain
their shape, orientation and size, so that the translated images are identical
to the initial figures.<o:p></o:p></span></div>
</div>
<br />
<div class="MsoNormal">
<span lang="EN-US" style="font-family: inherit;"><span style="color: #073763;">Okay, but I
don’t know what are you going to achieve with a simple movement of the players.</span><o:p></o:p></span></div>
<br />
<div style="text-align: justify;">
<div class="MsoNormal">
<span lang="EN-US" style="font-family: inherit;">I’m not
changing only their places, but I 'm going to do something else. Do you want to
see it?<o:p></o:p></span></div>
</div>
<span style="font-family: inherit;"><br /></span>
<span style="color: #073763; font-family: inherit;">Yes, I want
to see what you do with the players. So I click the video.</span><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: inherit;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='640' height='532' src='https://www.youtube.com/embed/ECiC2l5wa9Y?feature=player_embedded' frameborder='0'></iframe></span></div>
<br />
<div style="text-align: justify;">
<div class="MsoNormal">
<span lang="EN-US" style="font-family: inherit;"><span style="color: #073763;">That’s a
really funny way to convert the 13 players in 14, but now some of them are not
recognizable...</span><o:p></o:p></span></div>
</div>
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">Yes, but the important thing is that the message itself has remained the same: <span style="color: blue; font-size: 130%;">Matifutbol </span>wishes you a <span style="color: red; font-size: 130%;">Happy new year 2014!</span></span><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: inherit; margin-left: 1em; margin-right: 1em;"></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWu_GNgdL3EIJAQJ02sXdMJI9hvkZflJ1YCMXid7MU1No2ymp7IXvL00_ju8leCJNu8YBoT8IJ2P8sgzTv6C7HpfDNJNbBfVATecw9jN5VosMNOVBrUuKeVFil13Hc87u0g_iNsPmgQ0U/s1600/happy2014.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="520" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWu_GNgdL3EIJAQJ02sXdMJI9hvkZflJ1YCMXid7MU1No2ymp7IXvL00_ju8leCJNu8YBoT8IJ2P8sgzTv6C7HpfDNJNbBfVATecw9jN5VosMNOVBrUuKeVFil13Hc87u0g_iNsPmgQ0U/s640/happy2014.jpg" width="640" /></a></div>
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<span style="color: #20124d; display: none; font-family: Verdana, sans-serif;">Si te gustó esta historia, puedes votar por ella en <a href="http://www.meneame.net/story/sorprendente-caso-coche-extra" target="_blank">menéame</a> y <a href="http://divoblogger.com/story?id=41970" target="_blank">divoblogger</a>. Muchas gracias.</span></div>
WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-47594398540073358612013-11-24T22:15:00.001+00:002013-12-15T16:38:23.063+00:00The amazing case of the extra car.<!-- Resumen de la página -->
<span style="display: none;">The sponsor of the Football World Championship 2015 in Canada is going to give away some gifts to the best scorers. But a last minute change in the number of gifts creates a curious change in their apportionment.<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbETZBTxF0l04SiNAXQJwUmCiQrsz1bYKmnKCAQu9uN6u-AkZmpdjAC6NiEDERoju0Onn6CYJwhMrLdPmfPAH73qHeNHEzUJcjIG-_WK6VRuspckuZQHXcXM9C9lpjPTKRyq7CW0WgxXk/s1600/Yukon+cars.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbETZBTxF0l04SiNAXQJwUmCiQrsz1bYKmnKCAQu9uN6u-AkZmpdjAC6NiEDERoju0Onn6CYJwhMrLdPmfPAH73qHeNHEzUJcjIG-_WK6VRuspckuZQHXcXM9C9lpjPTKRyq7CW0WgxXk/s200/Yukon+cars.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmO4fS1YvUi31errtDw-_iToks7TW9ExFqVQi_HVcIGk0SIXP0bpjk47vz145qOdwWeZpq6sVO0YD1rbzzJ-ykla7RQSAw8mUfmPUO-Wm_P0Rh3R7qoNixlQInnEzLbDnuVOz-yTzLGzU/s1600/logo+canada+2015+2.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmO4fS1YvUi31errtDw-_iToks7TW9ExFqVQi_HVcIGk0SIXP0bpjk47vz145qOdwWeZpq6sVO0YD1rbzzJ-ykla7RQSAw8mUfmPUO-Wm_P0Rh3R7qoNixlQInnEzLbDnuVOz-yTzLGzU/s1600/logo+canada+2015+2.gif" /></a></span>
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<a href="http://matematicasyfutbol.blogspot.com/2013/11/el-sorprendente-caso-del-coche-extra.html" style="clear: left; float: right; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq2wq5rjLaANT4l-sjeQg9UTchvqOSy36d_j8_VYlCaL-ZkzhY0ta22s-gCo946Ss-8f_rkY9hK-ZExg36uhE2Ajp7lX3OTLKNnvSaTVgKiu3C0Sm4zRwAFLwytg5TaAyhdjurO7FQngc/s1600/versionespanyola.gif" width="60" /></a>
<span style="display: none;"><a href="http://matesifutbol.blogspot.com/2013/10/esmorzar-amb-matematiques.html" style="float: left; margin-bottom: 1em; margin-right: 1em;"><img 60="" border="0" height="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYxdbwexXkGFKRd5fqYMvnJMNLgIRhh0uM66729vdeZ-Abndi7cdXiv5TEkLcy_Twtq1EPBrZls7dwg-RMfCUTkMCeWhXgOfLVvWxIziVzGJUsKuv8wK0fBWurrnUfMl4HCLhWoAhNAlo/s1600/versiocatalana.gif width=" /></a></span>
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<span style="font-family: Verdana, sans-serif;">FIRST HALF</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmO4fS1YvUi31errtDw-_iToks7TW9ExFqVQi_HVcIGk0SIXP0bpjk47vz145qOdwWeZpq6sVO0YD1rbzzJ-ykla7RQSAw8mUfmPUO-Wm_P0Rh3R7qoNixlQInnEzLbDnuVOz-yTzLGzU/s1600/logo+canada+2015+2.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmO4fS1YvUi31errtDw-_iToks7TW9ExFqVQi_HVcIGk0SIXP0bpjk47vz145qOdwWeZpq6sVO0YD1rbzzJ-ykla7RQSAw8mUfmPUO-Wm_P0Rh3R7qoNixlQInnEzLbDnuVOz-yTzLGzU/s1600/logo+canada+2015+2.gif" /></span></a><span style="font-family: Verdana, sans-serif;">The <b>Football World Championship</b> 2015 in
<b><span style="color: red;">Canada</span> has finished</b>.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;">It's the <b>closing ceremony</b> and some awards are going to be delivered between the winners of the various categories.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;">In the section of best <span style="font-size: 130%;"><b>scorers</b></span>, first <b>3</b> qualified players are:<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQma3Qm_9QykDblGHXdwUVVL696MFMvOdIunZKOj5vSFYML1hwXb5YrKwQT_-1b48a4ahzvlfAvYTi-Tppgrrk4hFUd2YK8LRu0IDcvdhlqOwRBMhIEsICkO5kCnZchFOIjogTetVo_hw/s1600/podium.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQma3Qm_9QykDblGHXdwUVVL696MFMvOdIunZKOj5vSFYML1hwXb5YrKwQT_-1b48a4ahzvlfAvYTi-Tppgrrk4hFUd2YK8LRu0IDcvdhlqOwRBMhIEsICkO5kCnZchFOIjogTetVo_hw/s320/podium.gif" width="320" /></span></a></div>
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<span style="font-family: Verdana, sans-serif;">1.- <b><span style="color: blue;">Yuki Ogimi</span></b>, Japanese footballer, with <span class="unidas">12 goals</span>.</span></div>
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<span style="font-family: Verdana, sans-serif;"><span style="text-align: left;">2.- </span><span style="color: #b45f06; text-align: left;"><b>Lotta Schelin</b></span><span style="text-align: left;">, Swedish player, also with 12 goals.</span></span><br />
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3.- <span style="color: red;"><b>Christine Sinclair</b></span>, forward and captain of the Canadian team, with 4 goals.<span style="color: navy;"><o:p></o:p></span></span></div>
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<span style="font-family: Verdana, sans-serif;">There's a <b>draw to 12 goals</b> in the first place. Tha Japanese <b>Ogimi </b>is the winner because she has made more <b>goal assists</b>.</span><br />
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Second place, also with 12 goals, but less assits than Ogimi, is for Swedish player <b>Schelin</b>.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;">And the third scorer is <b>Sinclair</b>, the Canadian player, with 4 goals.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;">The Championship <b>sponsor</b>, Yukon Cars has decided to <b><span style="font-size: 130%;">give away 40 cars</span></b>. <o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbETZBTxF0l04SiNAXQJwUmCiQrsz1bYKmnKCAQu9uN6u-AkZmpdjAC6NiEDERoju0Onn6CYJwhMrLdPmfPAH73qHeNHEzUJcjIG-_WK6VRuspckuZQHXcXM9C9lpjPTKRyq7CW0WgxXk/s1600/Yukon+cars.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbETZBTxF0l04SiNAXQJwUmCiQrsz1bYKmnKCAQu9uN6u-AkZmpdjAC6NiEDERoju0Onn6CYJwhMrLdPmfPAH73qHeNHEzUJcjIG-_WK6VRuspckuZQHXcXM9C9lpjPTKRyq7CW0WgxXk/s200/Yukon+cars.jpg" width="200" /></span></a><o:p><span style="font-family: Verdana, sans-serif;"> </span></o:p></div>
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<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">They'll
give a car to the president of the Organizing Committee, another one to
President of</span><span style="line-height: 115%;"> the </span><span style="line-height: 115%;">Referees’ Committee, and
the</span> <b class="unidas">38 remaining cars</b> will be apportioned among the 3 top <b>scorers </b>in proportion to the goals scored by each one.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: Verdana, sans-serif;">At the last
minute, President of the Organizing Committee decides to reject his gift
because he has just been appointed CEO of Manitoba Motors, a direct competitor
of Yukon Cars.<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmG3d6t2reL37bt3X0ZHqYs6zenu8eqPA1hrGKKepsyzqXbpdYafIJ1wurvlph-qLW11m22BisyOC6IjBSHQR40WWiTRGNt3oj0LI8FzuAnYczWX-WXaNIrviZVmJ8LsKanFKTNm5sR6U/s1600/manitoba+motors.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><span style="font-family: Verdana, sans-serif;"><img border="0" height="130" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmG3d6t2reL37bt3X0ZHqYs6zenu8eqPA1hrGKKepsyzqXbpdYafIJ1wurvlph-qLW11m22BisyOC6IjBSHQR40WWiTRGNt3oj0LI8FzuAnYczWX-WXaNIrviZVmJ8LsKanFKTNm5sR6U/s200/manitoba+motors.jpg" width="200" /></span></a></div>
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<span lang="EN-US" style="font-family: Verdana, sans-serif;">Thus, his
car will join the other 38 to be distributed among the 3 best scorers.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;">These news are greeted with joy by all players. Well, all except for <b>Christine Sinclair</b>, who is, in addition to good footballer, a <b>fond of Mathematics</b>.</span></div>
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<span style="font-family: Verdana, sans-serif;">Why do you think <b>Christine </b>didn't like this late change?<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;"><o:p> </o:p> </span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiivqk-9p_nwc-LqWuIgMBA0sbDLkK83ZI7bSMmz3LRqfvkxlWKVMxGruY0O7TIvi_bcxEAbSDs5v0sovc74nUeHiqEJDpuwLkobvsNc_gi_RJ9CSW6yiuUBk4pMNzwwbhfmrZp3-SF0co/s640/homomathematicus.jpg" width="640" /></span></a></div>
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<span style="display: none;"><span style="font-family: Verdana, sans-serif;">In a few days we will post the solution. Meanwhile, you can think about what's happening. We encourage you to give us your solution by writing below a comment, or just entering its event page <a href="https://www.facebook.com/events/439583686164501/" target="_blank">on facebook</a> or <a href="https://plus.google.com/events/c282p0oo71dvd5avs28kqg630tg" target="_blank">on google+</a>.</span></span></div>
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<span style="display: none;"><span style="font-family: Verdana, sans-serif;"><b>Thank you</b> very much.</span></span></div>
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<span style="font-family: Verdana, sans-serif;">SECOND HALF</span></div>
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<span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">Apparently it's no sense she's sad because there's one more car to apportion, right?</span><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;">First of all, it seems that if the sponsor is giving away <b>one more car</b>, this shouldn't harm anyone, on the contrary, <b>someone will benefit</b> because of it.<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifof8HSVabBZT6ghy7QDzoxFqrituH3twtLCSHFeSHFoU9fCx9miXRRLROaFErj4Z8yehiGKT4BK_E6c9423IGgAaET2I3c5HX1-4OaPcDo1ZxOYbKaKnWakZCDvRV-4YjFnilwgsGu1U/s1600/39+yukon+cars.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" height="155" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifof8HSVabBZT6ghy7QDzoxFqrituH3twtLCSHFeSHFoU9fCx9miXRRLROaFErj4Z8yehiGKT4BK_E6c9423IGgAaET2I3c5HX1-4OaPcDo1ZxOYbKaKnWakZCDvRV-4YjFnilwgsGu1U/s400/39+yukon+cars.jpg" width="400" /></span></a></div>
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<span style="font-family: Verdana, sans-serif;">Moreover, as that the top 2 scorers are tied in goals, and the extra car can't be distributed between both players in a fair way, it seems logical that Christine will be who gets the extra car.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;">Nevertheless, <b>Christine Sinclair</b> is right to be <b>worried</b>.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">Why? Hasn't she got enough parking space for so many cars?</span><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: Verdana, sans-serif;">No, that's
not the problem. Actually, she's going to offer all the cars she gets among her
teammates, who have helped her to score these goals.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">Perhaps, like the president of the Organizing Committee, she has either a family meber or a friend who works in Manitoba Motors, the rival brand?</span><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: Verdana, sans-serif;">No, not
really. In addition, she loves cars manufactured by Yukon Cars.<o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">Then I don't understand...</span><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif;">Let's see the <b>first gifts allocation </b>to the three players, when the sponsor was going to apportion <b><span style="font-size: 130%;">38 cars</span></b>.<o:p></o:p></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhua8wjdLSFyb-6LYNYQmydfpkdT_3WiUGfDl6ZXDmR44zxsMFuXJ0cXnCAViv0Iex11ap7UY_YWcSmPicKnweMx2LKZjRlFLHUsRShYqpbZ70K2MtIZnpRWYFOqYIY5C6A4f0YiqUC9A4/s1600/tabla1+eng.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" height="188" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhua8wjdLSFyb-6LYNYQmydfpkdT_3WiUGfDl6ZXDmR44zxsMFuXJ0cXnCAViv0Iex11ap7UY_YWcSmPicKnweMx2LKZjRlFLHUsRShYqpbZ70K2MtIZnpRWYFOqYIY5C6A4f0YiqUC9A4/s640/tabla1+eng.JPG" width="640" /></span></a></div>
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<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">We
see that Ogimi and Schelin should have received 16.2857 cars (mathematicians
would refer to a 'quota' of 16.2857) and Sinclair should have got 5.4286 cars.
As far as it's not possible to</span> <b>divide a car</b>, <span lang="EN-US" style="line-height: 115%; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">we
determine that the Japanese and the Swedish will get 16 cars, and Christine
will get 5 cars (which corresponds to the 'integer part' of their quotas). So
far we've distributed 16+16+5 = 37 cars. And the </span><b>remainig car</b> belongs to Christine, because she's the player whose <b>fractional part</b> is <b>the largest </b>and closest to next unit.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: Verdana, sans-serif;"><span style="color: #134f5c;">I didn't
understand very well this last deal...</span><o:p></o:p></span></div>
</div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">Well,
Ogimi and Schelin, in addition to their 16 cars, should have got 0.2857
additional cars, while Christine should have received 0.4286 more cars. So
Christine is more entitled to take the remaining car than the other two players.
This way of apportionment is called in mathematical terms as the</span> ‘<b>largest remainder method</b>’ or ‘<b>Hamilton's method</b>’<span style="font-size: 10.0pt;">.</span><o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="MsoBodyText">
<div class="MsoNormal">
<span lang="EN-US" style="font-family: Verdana, sans-serif;"><span style="color: #134f5c;">Now I see
it more clearly.</span><o:p></o:p></span></div>
</div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="">
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">Let's
see what happens when the sponsor decides</span> <b><span style="font-size: 130%;">to give away one more car</span></b>.</span></div>
</div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br />
</span><br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDgh-guOwSJDhB1BkunYKAV8z4OusuWKh8KAtxEMNm2XKAsM6DDZmKLjl80GfUlxUv4MCftss0keAmxb3YgIfETZw162JGfQY-f5JBAto7YWaIeE_XQYhLF2ls7iibom2IJQrkVGmr048/s1600/tabla2+eng.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Verdana, sans-serif;"><img border="0" height="188" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDgh-guOwSJDhB1BkunYKAV8z4OusuWKh8KAtxEMNm2XKAsM6DDZmKLjl80GfUlxUv4MCftss0keAmxb3YgIfETZw162JGfQY-f5JBAto7YWaIeE_XQYhLF2ls7iibom2IJQrkVGmr048/s640/tabla2+eng.JPG" width="640" /></span></a></div>
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">Now
we see that the first 2 players are given 16 cars, and Christine receives 5
cars. With </span><b>the two remaing cars</b>, we'll use
the same method as before.</span><br />
<div class="MsoNormal">
<span lang="EN-US" style="font-family: Verdana, sans-serif;"><o:p></o:p></span></div>
<span style="font-family: Verdana, sans-serif;"><br />
<span style="line-height: 115%;">Ogimi
and Schelin should get 0.7143 more cars, while Sinclair should get only 0.5714
more cars. Therefore, the 2 remaining cars are now assigned to each of the
first 2 scorers, because their </span><b>fractional parts are larger</b>.<o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="MsoBodyText" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">So now we give 17 cars to Ogimi, another 17 to Schelin, and only 5 cars to Christine, don't we?</span><o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">That's
right. It turns out that increasing one gift, not only</span> <b>Christine </b><span lang="EN-US" style="line-height: 115%; mso-ansi-language: EN-US; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">doesn't
get one more car, but she</span> <b>loses one</b> of hers.<o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="MsoBodyText">
<span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">But this question will be very unusual, right?</span><o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">You
shouldn't think so. Actually, this issue has received special attention from
mathematicians, who call it by the name of</span> <b><span style="color: red; font-size: 130%;">Alabama Paradox</span></b>. <o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="MsoBodyText">
<span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">Why this name?</span><o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjN4Hb-Wz829wanT1hNN6P6laZQjb3-qwzJ4FQFq1hdfD_eCSKCPlHYglNpCG10LcD-H_Pl-7zvkGfi2UQnayRX_yZTk_8SD6pjytgLLr3bLcDoXBHOeTeOmDcAZhuSvgewfawXAxI8YT4/s1600/alabama.gif" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><span style="font-family: Verdana, sans-serif;"><img border="0" height="174" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjN4Hb-Wz829wanT1hNN6P6laZQjb3-qwzJ4FQFq1hdfD_eCSKCPlHYglNpCG10LcD-H_Pl-7zvkGfi2UQnayRX_yZTk_8SD6pjytgLLr3bLcDoXBHOeTeOmDcAZhuSvgewfawXAxI8YT4/s320/alabama.gif" width="320" /></span></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-family: Verdana, sans-serif;">Map from the Nations Online Project</span></td></tr>
</tbody></table>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;">This mathematical paradox was detected for the first time in the <b>United States House of Representatives</b>. In it, the number of each state's <b>seats </b>are redistributed every 10 years, regarding on their <b>population </b>growths.<o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">After the 1880 census, a study was made on a possible extension of the number
of seats in the House. In this study it was found that the state of</span> <b>Alabama </b><span style="line-height: 115%;">would
have got 8 seats in a chamber of 299 representatives, but only 7 with a House
size of 300.</span></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="MsoBodyText" style="text-align: justify;">
<div class="MsoNormal">
<span lang="EN-US" style="font-family: Verdana, sans-serif;"><span style="color: #134f5c;">And is
there any method of proportional apportionment in which these paradoxes do not
occur?</span><o:p></o:p></span></div>
</div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">Lots
of mathematicians have tried to give a solution as fair as possible to the
issue of</span> <b>allocation of seats</b> <o:p></o:p><span style="line-height: 115%;">after
a vote, but they haven't achieved a perfect method.</span></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">Thus,
numerous ways have been created to assign seats based on the votes cast, like
the</span> <b>Hamilton's method</b> we've seen here. <o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;">We've got some <b><i><span style="color: blue;">divisor methods</span></i></b>, such al <b>Jefferson's </b>method (also known as<b> d’Hondt </b>method), <b>Webster's </b>method (method of<b> odd numbers or S</b><b>ainte-Laguë's </b>method), <b>Huntington-Hill's </b>method, <b>Dean's </b>method, <b>Adams' </b>method, or the <b>Danish </b>method, among others.</span><br />
<span style="font-family: Verdana, sans-serif;"><o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="">
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;"><span style="line-height: 115%;">In
these systems we can find several <b><i><span style="color: #cc0000;">paradoxes</span></i></b>: </span><b>Alabama </b>paradox, the <b>population </b>paradox, or the <b>new states </b>paradox<b>. </b>That means that every method generates some bias or favoritism to some population vs. other.<o:p></o:p></span></div>
</div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;">And we also have <span style="color: blue;"><i><b>preferential voting systems</b></i></span>, among which we have the <b>simple or relative majority</b>, the<b> second round</b>, the <b>Borda </b>count, the <b>Condorcet </b>method, the <b>single transferable vote, </b>or the <b>'approval voting'</b>.<o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="">
<a href="http://www.matifutbol.com/es/guante.html" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;" target="_blank"><span style="font-family: Verdana, sans-serif;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-fiJik_ATbCi4Uwy2665lMGHxi5esGIZtqBJcHQfa7sC-2XADAynOwklvyNFDNtnJVFYvSlLQz-o3T6vwqEPNoPcf8rlnLC3BxfbU6pEuagGbo5CFJv_1A1KHmSLe2I2wcO806Wri7Z0/s1600/iker1.jpg" /></span></a></div>
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;">But they are also imperfect. We can remember our story about <span style="color: blue; font-size: 130%;"><a href="http://www.matifutbol.com/en/glove.html" target="_blank">the Arrow paradox</a></span>, for example.</span></div>
<span style="font-family: Verdana, sans-serif;"><o:p></o:p><br />
</span><br />
<div class="">
<span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">But some methods are </span><b style="color: #0c343d;">better </b><span style="color: #0c343d;">than others, right?</span></span></div>
<div class="MsoBodyText">
<span style="font-family: Verdana, sans-serif;"><o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;">Yes, <b>Webster's </b>method (also called <b>Sainte-Laguë`s </b>method <b>or</b> <span style="font-size: 130%;"><b>method of odd divisors</b>)</span> seems to be the one which produces <b>minor injustices</b>, but it's not the most used. That's because there uses to be a <b>political will</b> to prioritize the <b>good governance </b>instead of the strictly proportional representativeness of the Parliaments, and to benefit the <b>major parties</b>, or such sort of things.</span><br />
<span style="font-family: Verdana, sans-serif;"><o:p></o:p></span></div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="" style="text-align: justify;">
<div class="MsoNormal">
<span lang="EN-US" style="font-family: Verdana, sans-serif;">Now that
we've seen all the problems of the distribution, as in our case, <b>Christine </b>did
have reason to worry about when she heard they would deliver a car more...<o:p></o:p></span></div>
</div>
<div class="">
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
<div class="">
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;">No doubt, she would have previously read any of these <b><span style="font-size: 130%;">great links</span></b>: <i><a href="http://www.cut-the-knot.org/ctk/Democracy.shtml" target="_blank">'The Constitution and Paradoxes'</a>, <a href="http://mathispower4u.wordpress.com/2013/09/03/apportionment-the-alabama-paradox/" target="_blank">'Apportionment: The Alabama Paradox'</a> o <a href="http://rangevoting.org/Apportion.html" target="_blank">'Apportionment and rounding schemes'</a>.</i> </span><br />
<span style="font-family: Verdana, sans-serif;"><br /></span></div>
</div>
<div class="">
<div style="text-align: justify;">
<span style="font-family: Verdana, sans-serif;">And regarding on these issues of <b>fair divisions</b>, <b>voting </b>and <b>paradoxes</b>, surely Christine have also read some of our stories: <a href="http://www.matifutbol.com/en/penalty.html" target="_blank"><i>And now, who should kick the penalty?</i></a>, <a href="http://www.matifutbol.com/en/fairdivision.html" target="_blank"><i>The problem of the fair division</i></a>, or <a href="http://www.matifutbol.com/en/glove.html" target="_blank"><i>The golden goalkeeper</i></a>.</span><br />
<span style="font-family: Verdana, sans-serif;"><br /></span>
<br />
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">So, should
Christine ask for an apportionment based on Sainte-Laguë method?</span><o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Verdana, sans-serif;">Well,
despite being the method that produces less distortions, in this particular
case, its application would cause an <b>additional problem</b>, as <i>Tom (C.M.) Thomson</i>
discovered (many thanks for the comment). While it's also true that Christine
may benefit from the use of that method.<o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">Why?</span><o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Verdana, sans-serif;">Well, this
method is used as follows :<o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Verdana, sans-serif;">We add up
the goals scored by each player, and we calculate the quotients resulting from
dividing the number of goals by the successive odd numbers. And we'll give the
cars according to the highest resulting quotients.<o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Verdana, sans-serif;">Let's see
how the 38th and 39th cars would be assigned, according to the following table:<o:p></o:p></span></span></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Verdana, sans-serif;"><br /></span></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVl7nuXc3Fg2gu-492yuhfCG7F5mpKCXyMSdNJBO1lBwihZJVDrRpoTwLBv6IfeXzo48yprs5uN_R4mRY1HciSW3s0JJDNsg1DxYQaeUoUIDd8TDFjVZARHswvASChkSJ1UkzS8WAZOTw/s1600/table+saint+lague.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVl7nuXc3Fg2gu-492yuhfCG7F5mpKCXyMSdNJBO1lBwihZJVDrRpoTwLBv6IfeXzo48yprs5uN_R4mRY1HciSW3s0JJDNsg1DxYQaeUoUIDd8TDFjVZARHswvASChkSJ1UkzS8WAZOTw/s1600/table+saint+lague.JPG" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Verdana, sans-serif;">The cars
would be assigned to the players depending on the <b>highest quotients</b>: 12 - 12 - 4 - 4 - 4 - 2.4 - 2.4 - 1.714 - 1.714 - 1.333 - 1.333 - 1.333 - 1.091 - 1.091 - etc...<o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpibnmEIpoSeywS0g79fJDOVrpick9eb6dygpfKodW246u_d6jvqreV2DisWQrcMVWz1RTRAZ-YhQB3a3rjYpR5tfpCLE8aUutRwv25AfOSBj4mnWwQdOOqMrVcNkX1ktx1wRi-e-d4zU/s1600/christine+sinclair.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpibnmEIpoSeywS0g79fJDOVrpick9eb6dygpfKodW246u_d6jvqreV2DisWQrcMVWz1RTRAZ-YhQB3a3rjYpR5tfpCLE8aUutRwv25AfOSBj4mnWwQdOOqMrVcNkX1ktx1wRi-e-d4zU/s1600/christine+sinclair.gif" /></a><span lang="EN-US"><span style="font-family: Verdana, sans-serif;">As you can
see, the first 37 are easily given out (yellow cells). But with cars 38th and
39th we have a serious problem, since the three players are equally entitled to
the cars, as they've got the same next quotient: 0.364.<o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Verdana, sans-serif;">Obviously,
there should be some kind of rule to break ties. But if there's no fixed rule, the remaining 2 cars should be delivered by lot, so Christine would have a <b>2/3</b> chance of
getting the 6th car for her.<o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<br />
<div class="MsoNormal">
<span lang="EN-US"><span style="font-family: Verdana, sans-serif;"><span style="color: #0c343d;">Then, no doubt, Christine should ask for this method to be used! </span><o:p></o:p></span></span></div>
</div>
<br /></div>
WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-21749107353117682862013-10-24T21:50:00.002+01:002013-11-12T16:21:25.433+00:00Breakfast at Les Deux Magots<a href="http://matematicasyfutbol.blogspot.com/2013/10/desayuno-con-matematicas_24.html" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq2wq5rjLaANT4l-sjeQg9UTchvqOSy36d_j8_VYlCaL-ZkzhY0ta22s-gCo946Ss-8f_rkY9hK-ZExg36uhE2Ajp7lX3OTLKNnvSaTVgKiu3C0Sm4zRwAFLwytg5TaAyhdjurO7FQngc/s1600/versionespanyola.gif" width="60" /></a>
<a href="http://matesifutbol.blogspot.com/2013/10/esmorzar-amb-matematiques.html" style="float: left; margin-bottom: 1em; margin-right: 1em;"><img 60="" border="0" height="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYxdbwexXkGFKRd5fqYMvnJMNLgIRhh0uM66729vdeZ-Abndi7cdXiv5TEkLcy_Twtq1EPBrZls7dwg-RMfCUTkMCeWhXgOfLVvWxIziVzGJUsKuv8wK0fBWurrnUfMl4HCLhWoAhNAlo/s1600/versiocatalana.gif width=" /></a>
<br />
<div class="rojo negrita" style="float: left; margin-left: 2em;">
FIRST HALF</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQVsW54jrp8jFv7HbDdUv5u_9Qh6ngYD6QzyycVU5r4nkjc5b7S2fTstLywAmdkP2TgyyI5YkncggWL-YV9_ndR0WLCTmkSjCoxzKHQvEmuP3XMPbHsg2zPAP_VnQPEcIVAOVla3rm6Bc/s1600/ibratriste.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Zlatan Ibrahimovic intrigued by the friendship paradox" border="0" height="230" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQVsW54jrp8jFv7HbDdUv5u_9Qh6ngYD6QzyycVU5r4nkjc5b7S2fTstLywAmdkP2TgyyI5YkncggWL-YV9_ndR0WLCTmkSjCoxzKHQvEmuP3XMPbHsg2zPAP_VnQPEcIVAOVla3rm6Bc/s320/ibratriste.jpg" title="Zlatan Ibrahimovic intrigued by the friendship paradox" width="320" /></a></div>
<div style="text-align: justify;">
<b>(This entry participates on the <a href="http://math-frolic.blogspot.co.uk/2013/11/come-frolic-at-104th-carnival-of-math.html" target="_blank">104th edition</a> of the <a href="http://aperiodical.com/carnival-of-mathematics/" target="_blank">Carnival of Mathematics</a> hosted by <a href="http://math-frolic.blogspot.co.uk/" target="_blank">Math-Frolic!</a>)</b><br />
<b><br /></b>
<b>Zlatan Ibrahimovic </b>is sad, very sad. This is because he thinks he has few <b>friends </b>among his teammates. </div>
<br />
<div style="text-align: justify;">
When he takes a look at his companions, he realizes that most of his friends have more friends than him. And not only in the pitch or in the locker room, but they also have more followers and friends in social networks (Facebook , Twitter, Linkedin, Google+...). </div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguuP_uaol-LnFVhv7hHmg84VD1_jO4WV1ZNtG9FRIK49pwYabDc9J7WPHt7GEq1c7tPzy5oIuwFWdFxSLNkJAsd6x10Wlxm5oNti-XPb_Nfi4mOmNLdxwOLeeZbw1AvkgSzla3yPmeg88/s1600/desayunofrances2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img alt="Awesome continental breakfast" border="0" height="134" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguuP_uaol-LnFVhv7hHmg84VD1_jO4WV1ZNtG9FRIK49pwYabDc9J7WPHt7GEq1c7tPzy5oIuwFWdFxSLNkJAsd6x10Wlxm5oNti-XPb_Nfi4mOmNLdxwOLeeZbw1AvkgSzla3yPmeg88/s200/desayunofrances2.jpg" title="Awesome continental breakfast" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGh2zDQKc5_bkCSNRobywaMnKCI6TPqP9fJ7DF_1BVjNpF9T3vAUQeI841ntO-iawl8njZ-0NroOcV-Fwq9PiE15xCh6pRb3v-mKLZBft9T8nWC94HQOJd5JQRDRq-Nvb0sGewsB5siZU/s1600/redessociales.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 0em;"></a>He has proposed himself the firm intention to change this situation, and he's looking for someone to give him advice on how to make more friends. So he meets with <b>Thiago Silva</b> at <i>Les Deux Magots</i> to have breakfast and talk about it.<br />
<br /></div>
<div style="text-align: justify;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidGypjcpdRU6TgDY0EEvwJjpqycbyF1D_PseGORzbQBkXCRFTdjzZwb7FztKa70Q6Xs6ab5pdtPdoQ3TeKxA7rKlsres7p8ZdInzTEEhTBHUrKM7f7FvvbePMZWhVdTsbOYGUIjzTMTHQ/s1600/leseeuxmagots.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Restaurant Les Deux Magots, Paris." border="0" height="132" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidGypjcpdRU6TgDY0EEvwJjpqycbyF1D_PseGORzbQBkXCRFTdjzZwb7FztKa70Q6Xs6ab5pdtPdoQ3TeKxA7rKlsres7p8ZdInzTEEhTBHUrKM7f7FvvbePMZWhVdTsbOYGUIjzTMTHQ/s200/leseeuxmagots.jpg" title="Restaurant Les Deux Magots, Paris." width="200" /></a>It 's a lovely day in Paris, so Zlatan and Thiago sit outside to have a nice <b>breakfast</b>. Zlatan <b>exposes the problem</b> to Thiago:</div>
<br />
<div style="text-align: justify;">
<span style="color: #660000;">My friends seem so much more popular than me. I've noticed that they've got more friends, and I'd like to do anything to have so many friends as them.</span><span style="color: #990000;"> </span></div>
<div>
<br /></div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguuP_uaol-LnFVhv7hHmg84VD1_jO4WV1ZNtG9FRIK49pwYabDc9J7WPHt7GEq1c7tPzy5oIuwFWdFxSLNkJAsd6x10Wlxm5oNti-XPb_Nfi4mOmNLdxwOLeeZbw1AvkgSzla3yPmeg88/s1600/desayunofrances2.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><br /></a><span style="text-align: justify;">Thiago Silva talks as follows:</span><br />
<br />
<div style="text-align: justify;">
<span style="color: #073763;">You shouldn’t worry about what you’re saying. It's very common, it's not your fault, it's just <b>Maths</b>. </span></div>
<br />
<div style="text-align: justify;">
Do you think that it’s also the fault of Mathematics? Think a moment, and pass to see the solution in the second half.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkZNjtAol92vhyphenhyphenzF8zPYQBJoWZPhgKjAjLQWtah0o5An2tD_vbR4BFGOi9RuMj21ey8lkQZWBzOQZhzGHdtdTPG4bs74FQqjHb743a1Xp2TrlYl2sLfPQfZKvWlmc7Z1MCgwRXo1zfMcI/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkZNjtAol92vhyphenhyphenzF8zPYQBJoWZPhgKjAjLQWtah0o5An2tD_vbR4BFGOi9RuMj21ey8lkQZWBzOQZhzGHdtdTPG4bs74FQqjHb743a1Xp2TrlYl2sLfPQfZKvWlmc7Z1MCgwRXo1zfMcI/s640/homomathematicus.jpg" width="640" /></a></div>
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<div class="rojo negrita">
SECOND HALF</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyxlyv7m3F59_FZIuTR6ZuL5AxJ7VW6bIRvt6g5BnDkURvR2j8gp3lIRVE320nFiEqYO9w_1GqT_jM8_RUbf1X7TJ7jl2MsytEvRt-DGkAETITlXmy1SJrZZyjcgoR0uPzYgoX_oK-XUM/s1600/thiago2.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Thiago Silva about to explain to Ibra the friendship paradox." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyxlyv7m3F59_FZIuTR6ZuL5AxJ7VW6bIRvt6g5BnDkURvR2j8gp3lIRVE320nFiEqYO9w_1GqT_jM8_RUbf1X7TJ7jl2MsytEvRt-DGkAETITlXmy1SJrZZyjcgoR0uPzYgoX_oK-XUM/s1600/thiago2.jpg" title="Thiago Silva about to explain to Ibra the friendship paradox." /></a></div>
<div style="text-align: justify;">
Well, as Thiago Silva says, it’s a simple <b>mathematical problem</b>, and not otherwise. Let's see how Thiago explains it to Ibra:</div>
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<div style="text-align: justify;">
<span style="color: #073763;">Do you know the <b>friendship paradox</b>? According to it, your friends have more chances of having more friends than you do...</span></div>
<br />
<div style="text-align: justify;">
<span style="color: #660000;">Well, once I heard that people usually are more likely to be friends with those people who have many friends than with those with few friends ...</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<span style="color: #073763;">Yes, that’s what Satoshi Kanazawa psychologist said: people use to establish friendship more often with people who have many friends that with those which have few. Although it doesn’t match exactly with this problem; we should rather look at the studies of Feld sociologist.</span></div>
<br />
<span style="color: #660000;">Is this Feld the one who discovered the friendship paradox?</span><br />
<br />
<div style="text-align: justify;">
<span style="color: #073763;">Yes. Indeed, Feld’s frienshid paradox is a property of Graph Theory, by which it’s established that it’s likely that our friends have more friends than we have...</span></div>
<div style="text-align: justify;">
<br /></div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3LRZtSYe_sZpDMEsa7Ni9Wst9UyFQQ8qjAnx-KSETypX2RwbXiYKMRsDYYAGO68aF33Eo5kl6suyRAQSzrjb-JLs32pkWK1TcfjCudopTwRTIDBeumlvr_wuX0Yea6r5LFMhfWKmmB-g/s1600/papeljugadores.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="List of the 11 lineup players of the Paris Saint Germain team." border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3LRZtSYe_sZpDMEsa7Ni9Wst9UyFQQ8qjAnx-KSETypX2RwbXiYKMRsDYYAGO68aF33Eo5kl6suyRAQSzrjb-JLs32pkWK1TcfjCudopTwRTIDBeumlvr_wuX0Yea6r5LFMhfWKmmB-g/s320/papeljugadores.jpg" title="List of the 11 lineup players of the Paris Saint Germain team." width="256" /></a><span style="color: #660000;">How is it possible?</span><br />
<br />
<div style="text-align: justify;">
Thiago picks up a notebook and begins to put on paper all the team’s friendship relations.</div>
<br />
<div style="text-align: justify;">
<span style="color: #073763;">Let's see it graphically. We’ll write in this paper the 11 lineup players (list on the right).</span><br />
<span style="color: #073763;"></span>
<span style="color: #073763;">Now, we'll identify the <b>players </b>with a <b>blue circle</b> (vertices), and we'll write inside them their numbers. We'll connect with a <b>red line</b> (edges) the circles of those who are <b>friends</b>.</span></div>
<br />
<div style="text-align: justify;">
<span style="color: #073763;">Beside each player we'll write in <b>lilac</b> a figure representing the <b>number of friends</b> he has (just counting the number of red lines from his circle). That way we can see graphically all existing friendship relations on the team.</span><br />
<span style="color: #073763;"><br /></span>
<span style="color: #073763;">As you can see, you are the blue circle with number 10 inside it. There’re 3 red lines connecting your circle with your 3 friends: Pastore (27), Lucas (29) and Lavezzi (22). That’s why I wrote a lilac color number <b>3</b> beside your circle, because you’ve got 3 friends.</span></div>
<div style="text-align: justify;">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3xUIbI2-F-renvnJYpYf9WbeS1hjVndC7Vv3vGvZrzPAmgRdBhnU4xf_6m9kV4R7avXm251BZWmffi53a1jlXosLrFaPEMSY1B1hYSxFyKJWv45MdyepHMeZ6rP_gv7BIVXkRsxu2uVw/s1600/grafodeamistades.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Graph of the friendship relations among the players." border="0" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3xUIbI2-F-renvnJYpYf9WbeS1hjVndC7Vv3vGvZrzPAmgRdBhnU4xf_6m9kV4R7avXm251BZWmffi53a1jlXosLrFaPEMSY1B1hYSxFyKJWv45MdyepHMeZ6rP_gv7BIVXkRsxu2uVw/s400/grafodeamistades.gif" title="Graph of the friendship relations among the players." width="400" /></a></div>
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<div style="color: #073763; text-align: justify;">
Now we’re going to build a table from this graph. In the line corresponding to each player we’ll write the total of friends of each player’s friends in the cells where his row crosses with his friends' columns. In the table beside, in the first column we’ll put the amount of friends each player has. In the second one we’ll add up all his friends’ friends. And in the third one we’ll calculate the friends average of his friends, by dividing the sum of the friends of the friends by the number of friends.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCQv8lj8D3epPFm7c6NrdMb5Xezz_j4NxvLl5mIu62NRaSLV6LOUX1ZDChdvzEHYHbNXkoXq-AH7g1OZ659niJ5ObOEQOPdaNZ_0HqYY2NZDzlU2iP9FdBHu7DsvpLqFvZGXo3NdRggJw/s1600/11players.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Table of the friendship relations among the players." border="0" height="328" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCQv8lj8D3epPFm7c6NrdMb5Xezz_j4NxvLl5mIu62NRaSLV6LOUX1ZDChdvzEHYHbNXkoXq-AH7g1OZ659niJ5ObOEQOPdaNZ_0HqYY2NZDzlU2iP9FdBHu7DsvpLqFvZGXo3NdRggJw/s640/11players.JPG" title="Table of the friendship relations among the players." width="640" /></a></div>
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<div style="text-align: justify;">
<span style="color: #073763;">Let's go first with you, <b>Ibra</b>. You're <b>number 10</b>. Look at row 10: you've got <span class="unidas">3 friends</span>: Lavezzi (22), Lucas (29) and Pastore (27). Lavezzi (22) has <span class="unidas">3 friends</span>, so we've written a 3 in the cell where your row (10) meets his column (22). Lucas (29) has got <span class="unidas">7 friends</span>, so we write a 7 in the cell where your row (10) crosses with his column (29). And Pastore (27) has got <span class="unidas">2 friends</span>, so we've written a 2 in the correspondent cell.</span></div>
<span style="color: #073763;"><br /></span>
<span style="color: #073763;">We’ll do it that way with all the players, and now we fix our attention to the table on the right.</span><br />
<span style="color: #073763;"><br /></span>
<span style="color: #073763;"></span>
<br />
<div style="text-align: justify;">
<span style="color: #073763;">If we pay our attention to how many players have more friends than the average of their friends, we get, surprisingly, that <b>only 3 of them </b> (23, 2 y 29, <b>in red color</b>) have <b>more friends than the average of their friends</b> (number on third column is greater than number on first column), and the <b>remaining 8</b> have <b>fewer friends</b>. And as we can see, the total average of the friends </span><br />
<span style="color: #073763;"><br /></span>
<span style="color: #073763;"></span>
<br />
<div style="text-align: center;">
<span style="color: #073763;"><span style="font-size: large;">42 / 11 = 3.82 </span></span></div>
<span style="color: #073763;"><br /></span>
<span style="color: #073763;">is much lower than the total average of the friends of the friends.</span>
<br />
<div style="text-align: center;">
<span style="color: #073763; font-size: large;"><br /></span>
<span style="color: #073763; font-size: large;">182 / 42 = 4.33</span></div>
</div>
<br />
<span style="color: #660000;">Why does this happen?</span><br />
<br />
<div style="text-align: justify;">
<span style="color: #073763;">This has to do with the properties of the <b>geometric mean</b> and the <b>statistical variance</b>. Let's take another paper, and we’ll focus only on the friendship relations among the 4 defenders.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9mN10XjAqj8RGRxbPHrderaeok-fr16Bb50JPaFNc2yvceiRhc_s4hVtdvv_PVT_JDkarX8Yygx1GzfmUEOQCIIFSZiDBEpIX-CZ8aQE92T7PFwfSfsx8fD0Ue-5WyQVmZkAbgbXONmE/s1600/grafodefensas.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Graph of the friendship relations among the defenders." border="0" height="177" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9mN10XjAqj8RGRxbPHrderaeok-fr16Bb50JPaFNc2yvceiRhc_s4hVtdvv_PVT_JDkarX8Yygx1GzfmUEOQCIIFSZiDBEpIX-CZ8aQE92T7PFwfSfsx8fD0Ue-5WyQVmZkAbgbXONmE/s320/grafodefensas.gif" title="Graph of the friendship relations among the defenders." width="320" /></a></div>
<br />
<div style="text-align: justify;">
<span style="color: #073763;">We notice that Van der Wiel (23) has got 2 friends, Jallet (26) has 2 friends, Maxwell (17) has got only 1 friend, and I, Thiago (2), have got 3 friends.</span><br />
<span style="color: #073763;"><br /></span></div>
<div style="text-align: justify;">
<span style="color: #073763;">We calculate the friends average of the defenders: <span style="font-size: large;">(3+2+2+1)/4 = 2</span>. We can see that defenders have <b>2 friends</b> each one <b>by average</b>.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhn7wZ-Pxai556sHBtGNJa6dJ2DHg8Hp7jyTIhnPcmGF1m76DEbCVX2pYMS62C4VosImFzYpaGqOZDKh764lVH3tsWxZ2xpCCBZj9fB1w7-6-k2D-qJnVxf3vqLld2so0to4mclKG1-myY/s1600/4players.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Table of the friendship relations among the defenders." border="0" height="183" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhn7wZ-Pxai556sHBtGNJa6dJ2DHg8Hp7jyTIhnPcmGF1m76DEbCVX2pYMS62C4VosImFzYpaGqOZDKh764lVH3tsWxZ2xpCCBZj9fB1w7-6-k2D-qJnVxf3vqLld2so0to4mclKG1-myY/s400/4players.JPG" title="Table of the friendship relations among the defenders." width="400" /></a></div>
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<div style="text-align: justify;">
<span style="color: #073763;">Now we’ll calculate the friends mean of the defenders’ friends. Van der Wiel (23) has 2 friends, who have <span style="font-size: large;">2</span> and <span style="font-size: large;">3</span> friends, respectively. Jallet (26) has got 2 friends, who have <span style="font-size: large;">2</span> and <span style="font-size: large;">3</span> friends. Maxwell (17) has one friend, who has <span style="font-size: large;">3</span> friends. And I (2) have got three friends, with <span style="font-size: large;">2</span>, <span style="font-size: large;">2</span> and <span style="font-size: large;">1</span> friend, respectively.</span></div>
<span style="color: #073763;"><br /></span>
<span style="color: #073763;">Thus, we’ve got the following data:</span><br />
<br />
<div style="text-align: justify;">
<div style="text-align: center;">
<span style="font-size: x-large;">2, 3, 2, 3, 2, 2, 1, 3. </span></div>
</div>
<br />
<span style="color: #073763;">We sort them like this:</span><br />
<br />
<div style="text-align: center;">
<span style="font-size: x-large;">3, 3, 3, 2, 2, 2, 2, 1</span></div>
<br />
<span style="color: #073763;">And we calculate their average:</span><br />
<br />
<div style="text-align: center;">
<span style="font-size: large;">(3+3+3 + 2+2 + 2+2 + 1) / 8 = </span></div>
<div style="text-align: center;">
<span style="font-size: large;">(3·3 + 2·2 + 2·2 + 1·1) / 8 =</span></div>
<div style="text-align: center;">
<span style="font-size: large;">(3<sup>2</sup> + 2<sup>2</sup> + 2<sup>2</sup> + 1<sup>2</sup>) / 8 = </span></div>
<div style="text-align: center;">
<span style="font-size: large;">2.25</span></div>
<br />
<div style="text-align: justify;">
<span style="color: #073763;">We see how, in a seemingly amazing way, <b>the friends average of defenders’ friends </b> rises up to <b><span style="font-size: large;">2.25</span></b>, higher than <b>the friends average of defenders</b> we calculated before, which was <b><span style="font-size: large;">2</span></b>.</span></div>
<br />
<span style="color: #660000;">That’s true. And always happens the same?</span><br />
<br />
<div style="text-align: justify;">
<span style="color: #073763;">Yes, always. Unless everyone has the same number of friends. In that case the two means are identical.</span></div>
<br />
<span style="color: #660000;">And why is this happening?</span><br />
<br />
<div style="text-align: justify;">
<span style="color: #073763;">I suppose you've noticed that, when I calculated the mean of friends of friends, <span class="unidas">I rearranged</span> the data we had in a special order.</span></div>
<br />
<div style="text-align: justify;">
<div style="text-align: center;">
<span style="font-size: x-large;">2, 3, 2, 3, 2, 2, 1, 3</span><span style="font-size: x-large;"> → </span><span style="font-size: x-large;">3, 3, 3, 2, 2, 2, 2, 1</span></div>
</div>
<br />
<span style="color: #660000;">Yes, I've noticed it. Why did you do that?</span><br />
<br />
<div style="text-align: justify;">
<span style="color: #073763;">I did it this way so you realized the following question: people who have many friends are incorporated many times within the sum, in fact, so many times as friends they have (3 times number 3), while people who have few friends are included less times (1 time number 1). This means that those players who have more friends have also more weigh in the average </span> <span style="font-size: large;">(</span><span style="font-size: x-large;">3+3+3</span><span style="font-size: large;">+2+2+2+2</span><span style="font-size: x-small;">+</span>1<span style="font-size: large;">)</span><span style="color: #073763;">, so that the overall mean tends up.</span></div>
<span style="text-align: justify;"><span style="color: #073763;"><br /></span></span>
<span style="text-align: justify;"><span style="color: #073763;">However, in the friends mean, we add only once each number </span> </span><span style="font-size: large; text-align: justify;">(3+2+2+1)</span><span style="text-align: justify;"><span style="color: #073763;">, that is, all people have the same weight, both those who have many friends as those who have few.</span></span><br />
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<span style="color: #660000;">I think now I understand it, but I’m not sure that one average has to be larger than the other one for that reason.</span></div>
<br />
<span style="color: #073763;">Well, then let's go to the <b>mathematical proof</b>.</span><br />
<div class="separator" style="clear: both; text-align: center;">
<span style="color: #073763;"><br /></span></div>
<div style="text-align: justify;">
<span style="color: #073763;">Let's call <b><span style="font-size: large;">x<sub>i</sub></span> </b>the number of friends player <i><b><span style="font-size: large;">i</span></b></i> has, and <b><span style="font-size: large;"><i>μ</i></span> </b>the friends mean of a <span class="unidas"><b><span style="font-size: large;">n</span></b>-players</span> team, which will be equal to:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMG8XKO1bVHbv_GO4VdnVoCHJG5H13D8sgQ1nONcyOaUEx1cm820kizJw4Iu26g-tTIETiA_-0wBPHm7EctOKCqrIYoQf24CaE3alixlmYRmrafsikOdHjVxdtn_VubVwNLml3SkqIXkU/s1600/formula105.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Arithmetic mean formula." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMG8XKO1bVHbv_GO4VdnVoCHJG5H13D8sgQ1nONcyOaUEx1cm820kizJw4Iu26g-tTIETiA_-0wBPHm7EctOKCqrIYoQf24CaE3alixlmYRmrafsikOdHjVxdtn_VubVwNLml3SkqIXkU/s1600/formula105.gif" title="Arithmetic mean formula." /></a></div>
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<span style="color: #073763;">Now let's see how we calculate the friends average of the friends. We’ve seen on the example about the defenders that the sum of the friends of friends corresponds to <span style="font-size: large;">∑(x<sub>i</sub><sup>2</sup>)</span>, and we know that the amount of friendship relations is <span style="font-size: large;">∑x<sub>i</sub></span>. So the friends mean of the friends, which we name <i style="font-size: x-large;"><b>μ'</b></i> will be:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj55k8tTOqkaNW2O_l_wHMWNDaIeNhRqXbHFXT1rADCs2sLtRrSlY8HpnscM5qcqA7AspmQ-21H6sJdmA-cfvmm4o495hh5sh5yVEywDsodtC2dTkL90fE0lLWGBh8raFxmPG-8_zcPAX4/s1600/formula106.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Formula of the arithmetic mean of the friends of the friends." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj55k8tTOqkaNW2O_l_wHMWNDaIeNhRqXbHFXT1rADCs2sLtRrSlY8HpnscM5qcqA7AspmQ-21H6sJdmA-cfvmm4o495hh5sh5yVEywDsodtC2dTkL90fE0lLWGBh8raFxmPG-8_zcPAX4/s1600/formula106.gif" title="Formula of the arithmetic mean of the friends of the friends." /></a></div>
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<span style="color: #073763;">Now we need to check if <b><i style="font-size: x-large;">μ</i> </b>is greater than, lesser than, or equal to <i style="font-size: x-large;"><b>μ'</b></i>.</span><br />
<span style="color: #073763;"><br /></span>
<span style="color: #073763;">For proving this, we’ll use the formula of the statistical variance (<span style="font-size: large;"><i>σ<sup>2</sup></i></span>) of a discrete random variable. We know that:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiIGPwCNbiyIVTLkqdj4sRh-gf_lW9VYXcCz5ycmE_lf6VmyZzXssOo_TnAOu_JYGz1V4gR0K_yjcJpGGkNMEpHEASM12pr0c59nCh31z1P35xlwQfCmTt4_8yMGOdIz4iNnlqZCHWJQY/s1600/formula107.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Formula of the statistical variance." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiIGPwCNbiyIVTLkqdj4sRh-gf_lW9VYXcCz5ycmE_lf6VmyZzXssOo_TnAOu_JYGz1V4gR0K_yjcJpGGkNMEpHEASM12pr0c59nCh31z1P35xlwQfCmTt4_8yMGOdIz4iNnlqZCHWJQY/s1600/formula107.gif" title="Formula of the statistical variance." /></a></div>
<span style="color: #073763;">It follows like:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWbY8O4t8nVqZOuUEZHu9SJ_m9Pw0BRR4wU9REnO2Qyr0ulqTTB97Z7smOjl9Ti3sXf4guMJ_U80pYdHcw6dzfj31WMxIxUyJL4ppwfqVfRDwukKz40Bj9HoS4O7g94rm2g-HnxfBuJaM/s1600/formula102.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWbY8O4t8nVqZOuUEZHu9SJ_m9Pw0BRR4wU9REnO2Qyr0ulqTTB97Z7smOjl9Ti3sXf4guMJ_U80pYdHcw6dzfj31WMxIxUyJL4ppwfqVfRDwukKz40Bj9HoS4O7g94rm2g-HnxfBuJaM/s1600/formula102.gif" /></a></div>
<span style="color: #073763;">We divide both terms by <span style="font-size: large;">∑(x<sub>i</sub>)</span>, and we get:</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh43VbnLT-gy8KaNjYuqQedI9lgyg-D5zFnRXyxuGEVwED0Ut_0MTZCpHpLVk2EoAlnJqU7qV5rjpU1sEBsSqdwtQYkFiF6_tK-MdhoXNtJ6S-9EuJN0hcj4kNkGWQcXGKJW-Qse-2JRfQ/s1600/formula101.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img alt="Comparison between the friends average and the friends average of the friends." border="0" height="91" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh43VbnLT-gy8KaNjYuqQedI9lgyg-D5zFnRXyxuGEVwED0Ut_0MTZCpHpLVk2EoAlnJqU7qV5rjpU1sEBsSqdwtQYkFiF6_tK-MdhoXNtJ6S-9EuJN0hcj4kNkGWQcXGKJW-Qse-2JRfQ/s640/formula101.gif" title="Comparison between the friends average and the friends average of the friends." width="640" /></a><br />
<span style="color: #073763;">So:</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBoyenLdOsqQEK4_4qEu5RwWOPEVkhqfgSdUVoVh5IGlOQHZpRZvrCuiKcvVSAKmznWCol9kgQvID4FdVhQdVsDLtLPrNK3shjp0K9iIcJVSoptcn0B-wJyg-fH0sKSt8rKJn0UIViSxk/s1600/formula103.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Comparison between the friends average and the friends average of the friends." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBoyenLdOsqQEK4_4qEu5RwWOPEVkhqfgSdUVoVh5IGlOQHZpRZvrCuiKcvVSAKmznWCol9kgQvID4FdVhQdVsDLtLPrNK3shjp0K9iIcJVSoptcn0B-wJyg-fH0sKSt8rKJn0UIViSxk/s1600/formula103.gif" title="Comparison between the friends average and the friends average of the friends." /></a></div>
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<span style="color: #073763;">In other words: <span style="font-size: large;"><b><i>μ'</i> = <i>μ</i></b></span> plus a value greater or equal to zero, because both the statistical variance and the friends average are always non-negative.</span></div>
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<span style="color: #073763;">Therefore, in a group, <b>the friends average of friends will always be greater than or equal to the average of friends</b>.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn26G6ptE4NWvp6grUwmmkPIQvb0eiAX32PYB1w38zlWiK00tVRwwVV9im1V1qJqvj_UN3XneaQ1zM8-T2woZ1DqAzcaPSoCzkqVeTS2JomB5h7qRBF7nejaLpEhz4cqPt80f5-sLog8U/s1600/formula108.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Comparison between the friends average and the friends average of the friends." border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn26G6ptE4NWvp6grUwmmkPIQvb0eiAX32PYB1w38zlWiK00tVRwwVV9im1V1qJqvj_UN3XneaQ1zM8-T2woZ1DqAzcaPSoCzkqVeTS2JomB5h7qRBF7nejaLpEhz4cqPt80f5-sLog8U/s1600/formula108.gif" title="Comparison between the friends average and the friends average of the friends." /></a></div>
<div style="text-align: justify;">
<span style="color: #660000;">Well, with this <b>mathematical demonstration</b>, you definitely convinced me. And beyond the fact that people to not be depressed by having fewer friends than their friends, have all these studies other applications?</span></div>
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<div style="text-align: justify;">
<span style="color: #073763;">Well, this paradox has many applications in many other areas of life. </span><span style="color: #073763;">For example, it's used for the </span><b style="color: #073763;">control of diseases</b><span style="color: #073763;">. Thereby, our club does blood tests to some players to detect any contagious illness and prevent it. Every month they pick 5 players randomly, and does some clinical tests to them. Can you imagine a better way of improving this monitoring task?</span></div>
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<span style="color: #660000;">I think that...</span><br />
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<div style="color: #073763;">
No doubt they should pick 5 players randomly, and then ask them each choose a friend. We know these friends of friends have a friends average higher than the initially chosen players, so they’ll also have a much higher chance of being infected, because as they have more friends they are more exposed to infection. This way we get a sample of players more significant than randomly selected.</div>
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<span style="color: #660000;">That's what I was thinking. </span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJpLfrp0l5q5Hljzdo7cj_X7dCVj-zz7tiFnTINLzcxnjCECY6C7U8hJBf2eEdhCDei-Xl4C-1QS1_H-I3OaHmkD4smNWgZFXIfpM8HRZlu6EumgPZX-XajOEWNob2_WPAnuBS3Bmry8s/s1600/gimnasio1.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;">t<img alt="Gym." border="0" height="132" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJpLfrp0l5q5Hljzdo7cj_X7dCVj-zz7tiFnTINLzcxnjCECY6C7U8hJBf2eEdhCDei-Xl4C-1QS1_H-I3OaHmkD4smNWgZFXIfpM8HRZlu6EumgPZX-XajOEWNob2_WPAnuBS3Bmry8s/s200/gimnasio1.jpg" title="Gym." width="200" /></a><br />
<div style="text-align: justify;">
<span style="color: #073763;">That’s the way most schools, hospitals or residential homes use to do their sanitary controls. Other application is for people who go to a </span><b style="color: #073763; text-align: justify;">gym </b><span style="color: #073763;"> and lose heart because they see that most people are in better shape than them... This doesn’t happen to us, right Ibra? In this case they should realize that when they take a look at the people who are at the gym, they are extracting a biased sample, since it’s likely that people they see are those who spend more hours there, and consequently are in better physical condition.</span><br />
<div style="color: #073763;">
<br /></div>
<div style="color: black;">
<span style="color: #073763;">We can also apply this paradox to other things. That way, it’s a very useful technique for children who <b>collect </b>photocards of players like us. They may apply the friendship paradox to complete their albums in a better way. Usually, they try to change their repeated cards among his friends. But they almost never achieve to complete their albums.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfxFNvKzQ-2ZCiovq1TdjFrsts0hP1_GVdeYsYO1t2dAgQzY2nVV3q7zus1OX_-PwGbAIWG1cF9IafC9l-Lrha0Aec2-1cQ_2cJmnh4aDcOKHAHxTQ6Tak97NrV-hNRWXb0-sXNDXHHDk/s1600/cromos.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Football players photocards." border="0" height="128" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfxFNvKzQ-2ZCiovq1TdjFrsts0hP1_GVdeYsYO1t2dAgQzY2nVV3q7zus1OX_-PwGbAIWG1cF9IafC9l-Lrha0Aec2-1cQ_2cJmnh4aDcOKHAHxTQ6Tak97NrV-hNRWXb0-sXNDXHHDk/s640/cromos.gif" title="Football players photocards." width="640" /></a></div>
<div style="color: black; text-align: start;">
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<div style="color: black; text-align: start;">
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<div style="color: black;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgF4opOlRvwa5teqVGtOZ1MGxYEUsrvtoPKcpupzvbpUNpKhL_5t65QpH-yizcclMe6NWauGweDC8ug1XkvItpBKRZfTOBfwEkddrLC2mZnRMDZ68632zJwvs2fgq0yjotTM2LReSm34h8/s1600/intercambiodecromos.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Changing football players photocards." border="0" height="133" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgF4opOlRvwa5teqVGtOZ1MGxYEUsrvtoPKcpupzvbpUNpKhL_5t65QpH-yizcclMe6NWauGweDC8ug1XkvItpBKRZfTOBfwEkddrLC2mZnRMDZ68632zJwvs2fgq0yjotTM2LReSm34h8/s320/intercambiodecromos.JPG" title="Changing football players photocards." width="320" /></a><span style="color: #073763;">It would be better than they approach a few children randomly selected, and ask them to submit to their friends, because then they’ll meet the most popular children of the school, who have more contacts and are more likely to have those missing stickers. Even though they should take care of not becoming infected of any contagious illness by them…</span><br />
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<span style="color: #073763;"><br /></span></div>
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<div style="color: #073763;">
And you can also understand why, when we go in vacation to the beach, we find that it's usually crowded. What about those empty beaches in the brochures? Once again, it’s very likely that the beach we go is full of people, and not completely deserted, because of this paradox.</div>
</div>
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<div style="text-align: justify;">
</div>
<span style="color: #660000;">So I shouldn’t worry about having so few friends, right?</span><br />
<br />
<div style="text-align: justify;">
<span style="color: #073763;">Not at all. You can see that it happens everywhere. Not only in the Paris Saint Germain, but in all teams: Real Madrid, <span class="unidas">AC Milan</span>, Manchester United, <span class="unidas">L.A. Galaxy</span>, Santos, <span class="unidas">FC Barcelona</span>... And it happens to most players: Cristiano Ronaldo, Neymar, Falcao, Messi ...</span></div>
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<div style="text-align: justify;">
<span style="color: #660000;">Oh, really…? </span>-Ibra asked flashing a shy smile-. <span style="color: #660000;">I think after these explanations, and as far as it’s lunch time, I should invite you to have lunch at a restaurant a know...</span></div>
<br />
<div style="text-align: justify;">
<span style="color: #073763;">That’s right. But, if you are interested in the <b>friendship paradox</b>, you can write down these other links: <a href="http://mindyourdecisions.com/blog/2012/09/04/why-your-friends-have-more-friends-than-you-the-friendship-paradox/#.UmbcwvnIbGA" target="_blank">'Why your friends have more friends than you' on Mind your decisions</a>, <a href="http://www.psych.nyu.edu/jost/Zuckerman%20&%20Jost%20(2001)%20What%20Makes%20You%20Think%20You're%20So%20Popular1.pdf" target="_blank">'What makes you think you're so popular?' of Zuckerman and Jost</a>, </span><a href="http://web.williams.edu/Mathematics/sjmiller/public_html/hudson/The%20Friendship%20Paradox-%20Jackson%20Lu%204.pdf" target="_blank">'The friendship paradox' of Jackson (Guannan) Lu</a>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxDpqfIi79cTuq8AWoGriJWy_yBzjs8L0Dujk04AF5FPYAiHowWKczkecR-VPmJA2bpHpqXXGDYc5fdXVf3CXmQZ0BcM_hcDa3XmO7LzOpR0HNs_YXgVcDyhYsvYU3XF3PryLQjjZUfsg/s1600/restaurantecena1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Restaurant in Paris." border="0" height="426" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxDpqfIi79cTuq8AWoGriJWy_yBzjs8L0Dujk04AF5FPYAiHowWKczkecR-VPmJA2bpHpqXXGDYc5fdXVf3CXmQZ0BcM_hcDa3XmO7LzOpR0HNs_YXgVcDyhYsvYU3XF3PryLQjjZUfsg/s640/restaurantecena1.jpg" title="Restaurant in Paris." width="640" /></a></div>
<br />WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-86694513841088035752013-09-03T15:26:00.000+01:002013-09-03T15:28:48.169+01:00Herds of goats and group decisions<a href="http://matematicasyfutbol.blogspot.com.es/2013/09/rebanos-de-cabras-y-decisiones-de-grupo.html" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq2wq5rjLaANT4l-sjeQg9UTchvqOSy36d_j8_VYlCaL-ZkzhY0ta22s-gCo946Ss-8f_rkY9hK-ZExg36uhE2Ajp7lX3OTLKNnvSaTVgKiu3C0Sm4zRwAFLwytg5TaAyhdjurO7FQngc/s1600/versionespanyola.gif" width="60" /></a>
<a href="http://matesifutbol.blogspot.com.es/2013/09/ramats-de-cabres-i-decisions-de-grup.html" style="float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYxdbwexXkGFKRd5fqYMvnJMNLgIRhh0uM66729vdeZ-Abndi7cdXiv5TEkLcy_Twtq1EPBrZls7dwg-RMfCUTkMCeWhXgOfLVvWxIziVzGJUsKuv8wK0fBWurrnUfMl4HCLhWoAhNAlo/s1600/versiocatalana.gif width="60" /></a>
<div class="rojo negrita" style="float: left; margin-left: 2em;">
FIRST HALF</div>
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<p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg357Ow9rMYcu-XTwHEgv1JD4aqrUPjZak-jFdkjOwPSoY27eXnwTt3o7kIiX93l-3stkUHK3TWp8YECfGOPsJkPmj1X2pYR4O6NoxCVZTNjrqlP1U63vYT6xfj-3Fxkrnjawd1UgQlQgo/s1600/cabra1.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;">
<img alt="Happy goat, after eating all the grass of the football pitch" border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg357Ow9rMYcu-XTwHEgv1JD4aqrUPjZak-jFdkjOwPSoY27eXnwTt3o7kIiX93l-3stkUHK3TWp8YECfGOPsJkPmj1X2pYR4O6NoxCVZTNjrqlP1U63vYT6xfj-3Fxkrnjawd1UgQlQgo/s1600/cabra1.jpg" width="400" title="Happy goat, after eating all the grass of the football pitch" /></a>
<span class="negrita">Emergency meeting</span> at <span class="rojo">FC Barcelona</span> headquarters.</p>
<p>Last Saturday some <span class="pino negrita">wild goats</span> entered the <span class="negrita">training fields</span> of FC Barcelona, and ate all the grass of the pitches.</p>
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<p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhl3a10uuf_tQ__G8WBrsbjeGUTY19WTjaEYj9RgVbfv-41s1mDK5Cj61nEx3KE3B855NuWuJ-Gu_5p1ctcjKr0ATdg0dIqYnVnM1QNxXZ1SOB_725wvoNRCQuUrMGH7ZGk6ylB5AZDwgU/s1600/rebanyo1.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;">
<img alt="Some goats eating the grass of the football fields" border="0" height="265" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhl3a10uuf_tQ__G8WBrsbjeGUTY19WTjaEYj9RgVbfv-41s1mDK5Cj61nEx3KE3B855NuWuJ-Gu_5p1ctcjKr0ATdg0dIqYnVnM1QNxXZ1SOB_725wvoNRCQuUrMGH7ZGk6ylB5AZDwgU/s1600/rebanyo1.jpg" title="Some goats eating the grass of the football fields" width="320" /></a>
One week before <span class="marron">Spanish league</span> starts, the team has no place for training.</p>
<p>That way, the <span class="marino negrita">president</span>, <span class="marino">Sandro Rossell</span>, the <span class="verde negrita">sports director</span> <span class="verde">Andoni Zubizarreta</span>, the <span class="violeta negrita">physical trainer</span>, <span class="violeta">Elvio Paolorosso</span>, and the <span class="burdeos negrita">coach</span>, <span class="burdeos negrita">Tata Martino</span>, have a meeting today.</p>
<p>They must decide <span class="negrita">where</span> they will train all this week.</p>
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<p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJDB7xDRAGRXPmWlj8tOZ5HWpDEnCk5kv6nGXR9w2Q1JjSGarRv8Vz-K5evB__zSHZ0AaYwZZmAS54NGWG4yOPNmPnvMWTp7qfSHg3k3l3xJkwdBTTBH0QIPLQX_QwX3mptDrvQWKTruI/s1600/rossell1.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;">
<img alt="President of FC Barcelona, Sandro Rossell" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJDB7xDRAGRXPmWlj8tOZ5HWpDEnCk5kv6nGXR9w2Q1JjSGarRv8Vz-K5evB__zSHZ0AaYwZZmAS54NGWG4yOPNmPnvMWTp7qfSHg3k3l3xJkwdBTTBH0QIPLQX_QwX3mptDrvQWKTruI/s1600/rossell1.jpg" title="President of FC Barcelona, Sandro Rossell" /></a>
The <span class="marino">president</span> starts by proposing an idea:</p>
<p><span class="marino">- What about making a week of <span class="negrita">training sessions</span> in <span class="azul negrita">Andorra</span>?</span></p>
<p>The <span class="verde">sports director</span> says it’s a good idea. </p>
<p>The <span class="violeta">physical trainer</span> also thinks it's a good decision. </p>
<p>And the <span class="burdeos">coach</span>, given that everyone agrees with the idea of training in <span class="azul negrita">Andorra</span>, either demurs. </p>
<div class="separator" style="clear: both; text-align: center;"></div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbnM8sV_DOUequR7exECRcSYkCeE_NHWJOBITyDchDvnwJ4pY6sQLGGeHS-y1aiA09PKJbhXtlGyJPlkCTDXOpXuRhfNYjO80Wq7KO99ErVXDglqsv7r0ouR51ae1uWFXoWdlqliHUqfI/s1600/zubizarreta1.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;">
<img alt="Andoni Zubizarreta, sports director of FC Barcelona" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbnM8sV_DOUequR7exECRcSYkCeE_NHWJOBITyDchDvnwJ4pY6sQLGGeHS-y1aiA09PKJbhXtlGyJPlkCTDXOpXuRhfNYjO80Wq7KO99ErVXDglqsv7r0ouR51ae1uWFXoWdlqliHUqfI/s1600/zubizarreta1.jpg" title="Andoni Zubizarreta, sports director of FC Barcelona"/></a>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi99FNI0qcfXEVAC2QS4DR1Fd8h7Ws8iiwj7xteGi8lY5_pBC_34fU4YHF2ARTZC-6WgBMBxZOOqVs8Z6ZtG0I-5Bk9ZhYyhL0oYOr9wBKN3VPQql82hOoaHfAnJl6NHcvaBwTiJix6hoo/s1600/elvio1.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-right: 1em;">
<img alt="Elvio Paolorosso, physical trainer of FC Barcelona" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi99FNI0qcfXEVAC2QS4DR1Fd8h7Ws8iiwj7xteGi8lY5_pBC_34fU4YHF2ARTZC-6WgBMBxZOOqVs8Z6ZtG0I-5Bk9ZhYyhL0oYOr9wBKN3VPQql82hOoaHfAnJl6NHcvaBwTiJix6hoo/s1600/elvio1.jpg" title="Elvio Paolorosso, physical trainer of FC Barcelona" /></a>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP4CK8CvzycQDu9-4sXUsu4Wi7DRZa57gkk0UUw5tblHojEjpN7cwJRro488lQ5Exl2lIkykh-9KNQqp7RyMGxOuo6bSwjl5eqAbnq2a83BSK5A_jRyrTIQI3k8jlD1HFc2nJI79eMTWg/s1600/tata1.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-right: 1em;">
<img border="0" alt="Gerardo 'Tata' Martino, coach of FC Barcelona" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP4CK8CvzycQDu9-4sXUsu4Wi7DRZa57gkk0UUw5tblHojEjpN7cwJRro488lQ5Exl2lIkykh-9KNQqp7RyMGxOuo6bSwjl5eqAbnq2a83BSK5A_jRyrTIQI3k8jlD1HFc2nJI79eMTWg/s1600/tata1.jpg" title="Gerardo 'Tata' Martino, coach of del FC Barcelona"/></a>
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<div class="separator" style="clear: both; text-align: center;"></div>
<p>So, acting <span class="negrita">unanimously</span>, they decide that this week <span class="rojo negrita unidas">FC Barcelona</span> will train in <span class="azul negrita">Andorra</span>.</p>
<p>Everyone agrees, but... do you think they’ve made the <strong>right decision</strong>?</p>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkZNjtAol92vhyphenhyphenzF8zPYQBJoWZPhgKjAjLQWtah0o5An2tD_vbR4BFGOi9RuMj21ey8lkQZWBzOQZhzGHdtdTPG4bs74FQqjHb743a1Xp2TrlYl2sLfPQfZKvWlmc7Z1MCgwRXo1zfMcI/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkZNjtAol92vhyphenhyphenzF8zPYQBJoWZPhgKjAjLQWtah0o5An2tD_vbR4BFGOi9RuMj21ey8lkQZWBzOQZhzGHdtdTPG4bs74FQqjHb743a1Xp2TrlYl2sLfPQfZKvWlmc7Z1MCgwRXo1zfMcI/s640/homomathematicus.jpg" width="640" /></a></div>
<br />
<p class="rojo negrita">SECOND HALF</p>
<p class="lector">If all participants have expressed what they really thought was best for the team, the decision should be correct, right? </p>
<p>Well, in this case, they haven't taken the best decision.</p>
<p class="lector">Do you think they voted against their minds?</p>
<p>No, everyone has really voted according to what they thought was best for the club.</p>
<p class="lector">So, were all <span class="negrita">wrong</span>? It’s assumed that if they’re working in those posts, it’s because their high capacity to make such decisions correctly, right?</p>
<p>It’s true. They have enough capacity to perform the job. And none of them was wrong. However, all together took a wrong decision.</p>
<p class="lector">How can it be possible? </p>
<p>Let's see what was <span class="negrita">thinking</span> each one of them, and what they said.</p>
<p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqZ5hQVTi1F22bXZINJV3SCePGtxB-2wdVU0MmfoB_0WECByc0BMnoo93sDWCEWZSbBS2ZFH9feRnmsaG62SP6TPLEWO8y5w0wKnNHrcYmgIa6kquaRxPydLwRtIfj0AkTXSC5ROtjVNo/s1600/rossellpiensa.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;">
<img alt="Thoughts of Sandro Rossell, president of FC Barcelona" border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqZ5hQVTi1F22bXZINJV3SCePGtxB-2wdVU0MmfoB_0WECByc0BMnoo93sDWCEWZSbBS2ZFH9feRnmsaG62SP6TPLEWO8y5w0wKnNHrcYmgIa6kquaRxPydLwRtIfj0AkTXSC5ROtjVNo/s1600/rossellpiensa.gif" title="Thoughts of Sandro Rossell, president of FC Barcelona" width="320" /></a>
Let's start with the <span class="marino negrita">president</span>, Sandro Rossell. </p>
<p>He thought the <span class="verde">sports director</span> would prefer to put a distance between the team and the Barcelona media reporters, so that the players could train more relaxed to face the new season. And Andorra is an excellent place to achieve this purpose. </p>
<p>Sandro also believed that the <span class="violeta">physical trainer</span> would like a high-altitude preparation, as this would provide players a superior physical background, with which they'll cope with greater guarantees the season. </p>
<p>And he thought that the new <span class="burdeos">coach</span> would agree on having all the team concentrated for one more week, as there have been several changes of players during the summer, and this could promote a better feeling between them and the new coach. </p>
<p>However, the unexpected organization of the stay outside Barcelona was going to cause several problems to him. He personally would have <span class="negrita">preferred</span> they had made the training sessions at the <span class="negro negrita">Mini Estadi</span>, a few meters from Camp Nou, but he realized that all the others would prefer the option of Andorra, and therefore suggested this idea. </p>
<p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip6W4lknvffbdNYTnYM84Z1-7FTUqmlGhe_sduECbRUynYEqiiq5ZKb-dAFcumwrF4daC9sPMwVLTs0d-CJ_oYu0NNhBEDAtzbm_Fc1ltGvm2cbTfTtJh1_ll9Erxps6FD5DEglrEudcw/s1600/zubipiensa.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;">
<img alt="Thoughts of Andoni Zubizarreta, sports director of FC Barcelona" border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEip6W4lknvffbdNYTnYM84Z1-7FTUqmlGhe_sduECbRUynYEqiiq5ZKb-dAFcumwrF4daC9sPMwVLTs0d-CJ_oYu0NNhBEDAtzbm_Fc1ltGvm2cbTfTtJh1_ll9Erxps6FD5DEglrEudcw/s1600/zubipiensa.gif" title="Thoughts of Andoni Zubizarreta, sports director of FC Barcelona" width="320" /></a>
Let's see what happened to Andoni Zubizarreta, <span class="verde">sports director</span> of the club.</p>
<p>Rossell had already expressed his preference, so Andoni thought that the <span class="marino">president</span> wanted the stay in Andorra because it could be a good promotion for the club. </p>
<p>Also, he thought the <span class="violeta">physical trainer</span> would welcome the fact that the players could breathe the fresh mountain air, not the pollution of the city. </p>
<p>And surely the <span class="burdeos">coach</span> should enjoy the stay in this small country for a week, as he knew of his interest in mountain tourism. </p>
<p>Therefore, Zubizarreta thought everyone would like rather training in Andorra, and so gave his vote. Nevertheless, with the <span class="negrita">money</span> they were going to spend on Andorra, he could have signed a central defender that the team needed. </p>
<p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCKrfpm_y7DYRLCtGByVTScEOZqRL_9x28wBVzMnNJYxpW8yicd5XQoHqnBWEKhW_Fr7PJ1vPXemv_J0YhruP-VOxAoyxkMVWi-s44EMwaObhReM2t0Zc5KX6KzMHxt6y0siGzQn906Y4/s1600/alviopiensa.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;">
<img alt="Thoughts of the physical trainer of del FC Barcelona, Alvio Paolorosso" border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCKrfpm_y7DYRLCtGByVTScEOZqRL_9x28wBVzMnNJYxpW8yicd5XQoHqnBWEKhW_Fr7PJ1vPXemv_J0YhruP-VOxAoyxkMVWi-s44EMwaObhReM2t0Zc5KX6KzMHxt6y0siGzQn906Y4/s1600/alviopiensa.gif" title="Pensamientos del preparador físico del FC Barcelona, Alvio Paolorosso" width="320" /></a>
What happened with the <span class="violeta">physical trainer</span>, Alvio Paolorosso? </p>
<p>He was working for the club for a few days, so he didn’t know an alternative place for training. Anyway, it was clear that going to Andorra should be a favorable option, as both the <span class="marino">president</span> and the <span class="verde">sports director</span> had expressed their wishes for the team of training there. </p>
<p>In addition, the <span class="burdeos">coach</span> certainly would like to get away from the bustle of Barcelona. Tata Martino had been very busy with his move from Argentina, so he would appreciate positively this little ‘holidays’ in the mountains.</p>
<p>Therefore, he also gave his vote in favor of Andorra. However, after the grueling promotional tour around the world the team had made in recent weeks, he thought it might be better to train at sea level, at the <span class="negro negrita">Mini Estadi</span>, and not to subject the team to an altitude training which could <span class="negrita">exhaust</span> them. </p>
<p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-sKuA3ZTN4mvYyrEXAo5848mdnQOxl7-GTekBoLMy5mDVVIEMEcOj4tLSEpAW4ed3GiumDtexzL1jW585FdJC6gSnBLOYxkou48q2iECuIiqAIOfW6PeRhKNvOoN87f8qXnDZxIM_XXE/s1600/tatapiensa.gif." imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;">
<img alt="Thoughts of Gerardo Tata Martino, coach of FC Barcelona" border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-sKuA3ZTN4mvYyrEXAo5848mdnQOxl7-GTekBoLMy5mDVVIEMEcOj4tLSEpAW4ed3GiumDtexzL1jW585FdJC6gSnBLOYxkou48q2iECuIiqAIOfW6PeRhKNvOoN87f8qXnDZxIM_XXE/s1600/tatapiensa.gif" title="Thoughts of Gerardo Tata Martino, coach of FC Barcelona" width="320" /></a>
And finally, let's go over the thoughts of the <span class="burdeos">coach</span> Gerardo 'Tata' Martino. </p>
<p>It was clear that the other three preferred to go to Andorra, instead of training at the Mini Estadi. </p>
<p>He was convinced that the <span class="marino">president</span> had in mind to get a good economical return on the team's stay in Andorra, and therefore had suggested that destination, and that the <span class="verde">sports director</span> believed advisable not to modify the schedule of trainings of the youth teams at the Mini Estadi.</p>
<p>Gerardo also assumed that the <span class="violeta">physical trainer</span> would love to change boring training sessions on the pitch for funny training mountain routes. </p>
<p>However, Tata Martino, after several weeks of touring around the world, wanted to come into contact with a <span class="negrita">well-prepared</span> grass like the <span class="negro negrita">Mini Estadi</span>, instead of the Andorran sports fields, which probably would not be in perfect conditions. </p>
<p>But given that everyone else seemed to prefer the other option, both for what they’ve voted and for what he believed they were thinking, he chose to vote in the same direction. </p>
<p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3FH9dd9IsfOF5tvkuiJXhc2GL0QhNzYyTYEtiA4pKIIL1_1wXaNEGS_bt6tgi7S9vWFdSDcytnNlYA0LgxARG4m4MpWLSlvYBxfvtPJFLe0acb0J7bybRE4NOk2GsgA8n7k7q9tcmk1Q/s1600/miniestadi1.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;">
<img height="255" alt="Camp Nou and Mini Estadi" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3FH9dd9IsfOF5tvkuiJXhc2GL0QhNzYyTYEtiA4pKIIL1_1wXaNEGS_bt6tgi7S9vWFdSDcytnNlYA0LgxARG4m4MpWLSlvYBxfvtPJFLe0acb0J7bybRE4NOk2GsgA8n7k7q9tcmk1Q/s1600/miniestadi1.jpg" title="Camp Nou and Mini Estadi" width="320" /></a>
So everyone would have preferred to train at the <span class="negrita">Mini Estadi</span> rather than making a concentration of 7 days in <span class="azul negrita">Andorra</span>... </p>
<p>In addition, we must add that the all have '<span class="negrita">sacrificed</span>' their <strong>personal interests</strong> for what they thought it was the <strong>general interest</strong> of the team, so their frustration can be even greater.</p>
<p class="tecnico">This is an example of what is called <strong>'Abilene Paradox'</strong>. This paradox, which affects the fields of psychology, sociology and business, was described by <strong>Jerry B. Harvey</strong>, from the George Washington University, in 1988. </p>
<p class="tecnico">It's about <strong>bad decisions</strong> taken often by <strong>groups</strong>, either neighborhood communities, families, friends, teams, parliaments, boards of directors, or entire societies. And it may affect to small decisions like choosing a restaurant to have dinner, or economic decisions affecting a large number of countries. </p>
<p class="tecnico">In all these cases, both small and large groups of people end up doing something that none of them want individually. </p>
<p>Why is this? </p>
<p class="tecnico">Sometimes it’s because all the people follows <strong>the opinion of the leader / the majority / the 'herd'</strong>, either to avoid problems, to not discuss, to not become the killjoy, or because it’s not well seen <strong>going against the general opinion</strong>. </p>
<p class="tecnico">This way, people somehow feel <span class="negrita">‘pressured’</span>, consciously or unconsciously, to go for what is supposed most of the people thinks. </p>
<div class="separator" style="clear: both; text-align: center;"></div>
<p class="tecnico"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidLk2LS0hSd7XXyuVuj4O1nsC5xt7UC6fMpf9fFNnGs67A7ibJXqE4wFL_X91cbB2IXRsHLiZgAZhGksFaJHYLXxcWOAnd26TTNsKw4pmtpL73L-H8ziZ-w1DgLyxkffHDwcggEc4BjDk/s1600/andorra1.jpg" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: -1em;">
<img alt="Andorra" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidLk2LS0hSd7XXyuVuj4O1nsC5xt7UC6fMpf9fFNnGs67A7ibJXqE4wFL_X91cbB2IXRsHLiZgAZhGksFaJHYLXxcWOAnd26TTNsKw4pmtpL73L-H8ziZ-w1DgLyxkffHDwcggEc4BjDk/s1600/andorra1.jpg" title="Andorra" /></a>
So, nobody is willing to take the first step to express an <strong>objection</strong> to the <strong>'group think'</strong>, at least at the time of the decision. Although it’s not uncommon that at the slightest setback, everyone runs to criticize the decision, stating that they already knew what was going to happen, and that they should had voted another choice. </p>
<p class="tecnico">In fact, the latest researches make scientists believe that the human <strong>brain</strong> is <strong>programmed</strong> to change its point of view if it doesn’t fit the norm. </p>
<p class="tecnico">Moreover, studies suggest that <strong>'Groupthink'</strong> does not only apply to the time of voting, but even after a while, the group components end just thinking the way everybody believe they should think. In this manner, the human brain protects itself from the contradiction of thinking one way and acting contrary. </p>
<p>And which is the way to avoid falling into this paradox? </p>
<p class="tecnico">Well, the only way is <strong>promoting</strong> within the group the <strong>exchange of information</strong> between members and the <strong>respect for all opinions</strong>, so that alternative and effective solutions may arise. </p>
<p>If you are interested in the topic, you can check more about the Abilene Paradox on <a href="http://en.wikipedia.org/wiki/Abilene_paradox" target="_blank">Wikipedia</a> and on <a href="http://mystrategicplan.com/resources/how-to-identify-groupthink-an-introduction-to-the-abilene-paradox/" target="_blank">Mystrategicplan.com</a></p>WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-17973362708698097912013-08-12T16:43:00.001+01:002023-05-26T16:58:26.994+01:00Which team will win the UEFA Champions League?<!--Page resume-->
<span style="display: none;">The next Champions League final allows us to study one of the basic properties of Probability, and to learn how to earn a little money by betting for its winner.
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9RKinnhxF8UY3R47NLyMyAwRFltBTdxeKzqb6sk5yS55CNe9Cc1qXcyflQJ_e9R7ZFVIEZ_Ge0oqHr1pJ6YLfaPYy2MrRBARm8-7-79SyX97jKrCBIY7nDwLWYGM8dENFS44l-lgAqP0/s1600/endevi3.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;">
<img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9RKinnhxF8UY3R47NLyMyAwRFltBTdxeKzqb6sk5yS55CNe9Cc1qXcyflQJ_e9R7ZFVIEZ_Ge0oqHr1pJ6YLfaPYy2MrRBARm8-7-79SyX97jKrCBIY7nDwLWYGM8dENFS44l-lgAqP0/s1600/endevi3.jpg" width="150" /></a>
<a href="http://matematicasyfutbol.blogspot.com.es/2013/08/adivinos-quien-ganara-la-champions.html" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq2wq5rjLaANT4l-sjeQg9UTchvqOSy36d_j8_VYlCaL-ZkzhY0ta22s-gCo946Ss-8f_rkY9hK-ZExg36uhE2Ajp7lX3OTLKNnvSaTVgKiu3C0Sm4zRwAFLwytg5TaAyhdjurO7FQngc/s1600/versionespanyola.gif" width="60" /></a>
<a href="http://matesifutbol.blogspot.com.es/2013/08/equip-guanyara-champions.html" style="float: left; margin-bottom: 1em; margin-right: 1em;"><img 60="" border="0" height="60" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYxdbwexXkGFKRd5fqYMvnJMNLgIRhh0uM66729vdeZ-Abndi7cdXiv5TEkLcy_Twtq1EPBrZls7dwg-RMfCUTkMCeWhXgOfLVvWxIziVzGJUsKuv8wK0fBWurrnUfMl4HCLhWoAhNAlo/s1600/versiocatalana.gif" width="60" /></a>
</span>
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<div class="rojo negrita" style="float: left; margin-left: 2em;">
FIRST HALF</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwo82CWoCmVvTRCDxHrPGGHnrs2qDFQT91HQ1qIuzghUUzxDJBaQOpJHsmrj9xaSUSNex-wI6dd3qD4txs76ZFoG_cfP6y9wBq8jyzA5YoanOkLr62dP9YEq-NMx2IFBbknp23t4rtbiQ/s1600/estadiodaluz1.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQW4_fsS1kqR4K41ZRCyYMIfVVwKOhcjXJXsQRCvnuAE0jXpLvhMnzqjrXCwmWlBK_1-IkdLVI-kom8kWJwvY0FRaBsOvrObiAYt6i6XpQumjxmuZ1bRPsiyk63mGWnYusyVhfBLXBHXkuBPxB14sfu_lRhvXAI2jHkHhTeRZy101KuJhGXVihXXwP/s960/Estadio%20Ataturk%20de%20Estambul.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="672" data-original-width="960" height="224" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQW4_fsS1kqR4K41ZRCyYMIfVVwKOhcjXJXsQRCvnuAE0jXpLvhMnzqjrXCwmWlBK_1-IkdLVI-kom8kWJwvY0FRaBsOvrObiAYt6i6XpQumjxmuZ1bRPsiyk63mGWnYusyVhfBLXBHXkuBPxB14sfu_lRhvXAI2jHkHhTeRZy101KuJhGXVihXXwP/s320/Estadio%20Ataturk%20de%20Estambul.jpg" width="320" /></a><span class="negrita" style="text-align: justify;">June, 10th</span><span style="text-align: justify;"> </span><strong style="text-align: justify;">2023</strong><span style="text-align: justify;">. Whithin two hours, the </span><strong style="text-align: justify;">final</strong><span style="text-align: justify;"> of the </span><span class="pino negrita" style="text-align: justify;">UEFA Champions League 2022-2023</span><span style="text-align: justify;"> will start.</span></div>
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It's going to take place on Atatürk Olympic Stadium, among Inter Milan and Manchester City.</div>
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<span style="text-align: justify;">It's expected to be a very interesting and </span><span class="unidas" style="text-align: justify;">hard-fought</span><span style="text-align: justify;"> </span><strong style="text-align: justify;">football match</strong><span style="text-align: justify;">, and there's not a </span><span class="negrita" style="text-align: justify;">clear</span><span style="text-align: justify;"> </span><strong style="text-align: justify;">favourite</strong><span style="text-align: justify;"> for the game.</span><br />
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<span style="text-align: justify;">I'm going to watch it on TV with some of my friends, and we have decided to bet on it just to guess which team will win the </span><strong style="text-align: justify;">match</strong><span style="text-align: justify;">.</span><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXuij1h1Ep2he2KwHpPpL2cduqRyBkdnwlVSJoW-Khded4k9nGKv9JtSUd_K1Hh5w18ekwMWDokoHLVaP7Sf7L8Tr4BEaHGiHxRxnKzW6fZxANltgjVdXLr51QzGZTvu-AmMcOHPMjdGMbl1vOYcvOkExeUMY6H1p2pdH7uBN--PPKLYPPF_PtamWr/s414/Escudo%20Inter%20de%20Mil%C3%A1n.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="414" data-original-width="380" height="255" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXuij1h1Ep2he2KwHpPpL2cduqRyBkdnwlVSJoW-Khded4k9nGKv9JtSUd_K1Hh5w18ekwMWDokoHLVaP7Sf7L8Tr4BEaHGiHxRxnKzW6fZxANltgjVdXLr51QzGZTvu-AmMcOHPMjdGMbl1vOYcvOkExeUMY6H1p2pdH7uBN--PPKLYPPF_PtamWr/w234-h255/Escudo%20Inter%20de%20Mil%C3%A1n.gif" width="234" /></a><span style="font-size: x-large;">vs. </span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFW-Gx2kzKWVuI2bWu7-eMsE1TKfaPZj8KLpGW6RNdynrkWc34FM9y2KQmv4jtNy7NsAq7Vn7IZkwlx6uD64n9AyxKVAwsY1Ucs-y-iiYrUusqoULqWdAf6Yr1EaDNI5n8rlMJA8jSFth6eKF5BCe1Gc3BwMycWq5JkvDnCmNIpGBKu57iKCJpmwY6/s650/Escudo%20Manchester%20City.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="650" data-original-width="625" height="260" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFW-Gx2kzKWVuI2bWu7-eMsE1TKfaPZj8KLpGW6RNdynrkWc34FM9y2KQmv4jtNy7NsAq7Vn7IZkwlx6uD64n9AyxKVAwsY1Ucs-y-iiYrUusqoULqWdAf6Yr1EaDNI5n8rlMJA8jSFth6eKF5BCe1Gc3BwMycWq5JkvDnCmNIpGBKu57iKCJpmwY6/w250-h260/Escudo%20Manchester%20City.gif" width="250" /></a></div>
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In order to have a slight <span class="negrita">advantage</span> over my friends, I'm looking online for some <strong>fortune-tellers</strong> specialized in <strong>sports</strong> themes, so they predict me which team will be the winner.</div>
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And these are the options I've found: <br />
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1 – <span class="pino negrita">Tess the e-Guess:</span> hits <span class="marino negrita">80%</span> of the time; the consult costs <span class="rojo negrita">10£</span>.<br />
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2 – <span class="lila negrita">Saffir the Seer:</span> he has a percentage of <span class="marino negrita">60%</span> hits, and seeks <span class="rojo negrita">5£</span> per consult.<br />
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3 – <span class="azul negrita">Pierre Neverhitsanything:</span> hardly charges <span class="marino negrita">1£</span> per visit, and his percentage of hits is <span class="marino negrita">10%</span>.<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJdIBoHqrEQES49q4vU_spPhpEFYmrhImj7zHx9GmBkCNE7-nXZxIqw3hyq8ugNhMHW45xuIQG-AN1_sCxOwkmlY-_nubXJagiu9-On2midLEPfkfuFa5i-6f6SnsGhUpyYjVBIZCKcqk/s1600/endevi1.jpg" style="margin-left: auto; margin-right: auto;"><img alt="Tess the e-Guess consulting her magic cup" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJdIBoHqrEQES49q4vU_spPhpEFYmrhImj7zHx9GmBkCNE7-nXZxIqw3hyq8ugNhMHW45xuIQG-AN1_sCxOwkmlY-_nubXJagiu9-On2midLEPfkfuFa5i-6f6SnsGhUpyYjVBIZCKcqk/s1600/endevi1.jpg" title="Tess the e-Guess consulting her magic ball" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">Tess the e-Guess</span></td></tr>
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<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNHOuFFtm_9jRRd3fLGBc0wWRwottLZcJ5r7D1DT4_ojRjdJwtYHOjtfvDrDIgcONJc99taU1DktKDA_u2naoH-G7ObwursKZmK2gEU7E1Yk6ldD5bTBEGJxQ5v46sPFFXNZPyiqJJr1I/s1600/endevi2.jpg" style="margin-left: auto; margin-right: auto;"><img alt="Saffir the Seer consulting his magic cup" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNHOuFFtm_9jRRd3fLGBc0wWRwottLZcJ5r7D1DT4_ojRjdJwtYHOjtfvDrDIgcONJc99taU1DktKDA_u2naoH-G7ObwursKZmK2gEU7E1Yk6ldD5bTBEGJxQ5v46sPFFXNZPyiqJJr1I/s1600/endevi2.jpg" title="Saffir the Seer consulting his magic cup" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 13px;"><span style="font-size: small;">Saffir the Seer</span><br />
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<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9RKinnhxF8UY3R47NLyMyAwRFltBTdxeKzqb6sk5yS55CNe9Cc1qXcyflQJ_e9R7ZFVIEZ_Ge0oqHr1pJ6YLfaPYy2MrRBARm8-7-79SyX97jKrCBIY7nDwLWYGM8dENFS44l-lgAqP0/s1600/endevi3.jpg" style="margin-left: auto; margin-right: auto;"><img alt="Pierre Neverhitsanything consulting his magic cup" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9RKinnhxF8UY3R47NLyMyAwRFltBTdxeKzqb6sk5yS55CNe9Cc1qXcyflQJ_e9R7ZFVIEZ_Ge0oqHr1pJ6YLfaPYy2MrRBARm8-7-79SyX97jKrCBIY7nDwLWYGM8dENFS44l-lgAqP0/s1600/endevi3.jpg" title="Pierre Neverhitsanything consulting his magic cup" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 13px;"><span style="font-size: small;">Pierre Neverhitsanything</span><br />
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I’m not sure which fortune-teller I should consult to. Can you help me?<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkZNjtAol92vhyphenhyphenzF8zPYQBJoWZPhgKjAjLQWtah0o5An2tD_vbR4BFGOi9RuMj21ey8lkQZWBzOQZhzGHdtdTPG4bs74FQqjHb743a1Xp2TrlYl2sLfPQfZKvWlmc7Z1MCgwRXo1zfMcI/s1600/homomathematicus.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkZNjtAol92vhyphenhyphenzF8zPYQBJoWZPhgKjAjLQWtah0o5An2tD_vbR4BFGOi9RuMj21ey8lkQZWBzOQZhzGHdtdTPG4bs74FQqjHb743a1Xp2TrlYl2sLfPQfZKvWlmc7Z1MCgwRXo1zfMcI/s640/homomathematicus.jpg" width="640" /></a></div>
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SECOND HALF</div>
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Finally, I’ve decided to call my good fellow Joe Vitruvius, who has come home immediately in order to help me.</div>
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<span style="color: #073763;">I suppose you've thought that you might be worth talking to Tess, because, although she’s the most expensive fortune-teller (10£), she also ensures a <b>higher success rate</b></span> – Joe says.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8yu0WaVWKheginV9tcHUt3B-Ki5JAlpM3euf8z4GWTpx_y9nKj7PIrcdW6fil0At8r_YOmkF52RT4nc_Ujbfo4jU_tYJbHhrjBT_l1aaF_JeczwIdhENhVOkwSg9AZPnOWJXN7Xo5Igw/s1600/policias1.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8yu0WaVWKheginV9tcHUt3B-Ki5JAlpM3euf8z4GWTpx_y9nKj7PIrcdW6fil0At8r_YOmkF52RT4nc_Ujbfo4jU_tYJbHhrjBT_l1aaF_JeczwIdhENhVOkwSg9AZPnOWJXN7Xo5Igw/s1600/policias1.jpg" width="146" /></a>That’s true. I thought that if I make a high bet, I'll be compensated for the higher <b>cost </b>of the query.<br />
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<span style="color: #073763;">Do you remember when we studied <b>Probability</b>?</span><br />
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Uff, just a little.<br />
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<span style="color: #073763;">Well, we should refresh our memory a bit. Let’s see. We have that we can calculate the probability of an event as the ratio between the favourable cases by the total possible alternatives we can have.</span><br />
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Yes, I can remember this.<br />
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<span style="color: #073763;">Here we get the example of <b>Pierre Neverhitsanything</b>. You told me that he has a rate of success of about 10%. Do you know exactly what this means?</span><br />
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Yes, indeed. That means that if he makes 100 predictions, he will guess 10 of them.<br />
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<span style="color: #073763;">OK. In mathematical terms, we would say that if we define the <b>event</b> A=”to hit the forecast”, we get that the <b>probability </b>of the event A is P(A) = 10% = 0.1</span><br />
<span style="color: #073763;"><br /></span>
<span style="color: #073763;">Now let’s see the probability that Pierre fails with his prediction.</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9RKinnhxF8UY3R47NLyMyAwRFltBTdxeKzqb6sk5yS55CNe9Cc1qXcyflQJ_e9R7ZFVIEZ_Ge0oqHr1pJ6YLfaPYy2MrRBARm8-7-79SyX97jKrCBIY7nDwLWYGM8dENFS44l-lgAqP0/s1600/endevi3.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Image of the fortune-teller Pierre Neverhitsanything consulting its magical cup" border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9RKinnhxF8UY3R47NLyMyAwRFltBTdxeKzqb6sk5yS55CNe9Cc1qXcyflQJ_e9R7ZFVIEZ_Ge0oqHr1pJ6YLfaPYy2MrRBARm8-7-79SyX97jKrCBIY7nDwLWYGM8dENFS44l-lgAqP0/s1600/endevi3.jpg" title="Image of the fortune-teller Pierre Neverhitsanything consulting its magical cup1" width="240" /></a>It’s easy. It will be 90%.<br />
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<span style="color: #073763;">That's right. One of the <b>basic properties</b> of the <b>Probability theory</b>, states that for one event (A), the probability of its opposite or <b>complementary event</b> [written as P(not A) or P(A')] is equal to 1 minus the probability of A:</span><br />
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<div style="text-align: center;">
<i><b>P(A')= 1 – P(A)</b></i></div>
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<span style="color: #073763;">In our case, if we set A’=”to fail the prediction”, then P(A’)= 1 – 0,1 = 0,9 = 90%. And we’ve just solved the problem. We need no more data to know which fortune-teller is the best choice we can take.</span><br />
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I don’t understand it. What do you mean? I can’t see the relation between this formula and the choice of the best fortune-teller. We should bear in mind the costs of the queries. The formula should be more complicated.<br />
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<span style="color: #073763;">Nevertheless, in this case, the cost of every option is completely irrelevant for the problem's resolution. You must realize that we only want to know the winner, not the final score.</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_9M3SxKjPNT3DVkQxaOfUZ4NqannLsm1FGsRmSBaF35L5vUXQ8jUEzQb2N69tdTukL_2r1B3_jX-9tInv-GY45_WDV8ylxjsIe16TUg7JykHcNlbqbVEB-dMrhomC2GLF1IwQ412WXVk/s1600/MosesMabhidaStadium12.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="131" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_9M3SxKjPNT3DVkQxaOfUZ4NqannLsm1FGsRmSBaF35L5vUXQ8jUEzQb2N69tdTukL_2r1B3_jX-9tInv-GY45_WDV8ylxjsIe16TUg7JykHcNlbqbVEB-dMrhomC2GLF1IwQ412WXVk/s1600/MosesMabhidaStadium12.jpg" width="320" /></a><span style="color: #073763;"><br /></span>
<span style="color: #073763;">So, it's so simple as to consult Pierre Neverhitsanything, listen to what he says, and do just the opposite. That way, we'll get a <b>chance </b>to guess the winner of 90%. And moreover, at a <b>much cheaper</b> price than the other two soothsayers!</span><br />
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If you like this story, don't hesitate about sharing it with your friends through the links above. And you can see much more stories about math and football on the links on the left side.<br />
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<div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><br />WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-47540635205603984872013-07-23T18:35:00.000+01:002013-07-24T21:50:34.010+01:00Football and happiness.<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDuvQHoI1JAf9zJwiI6s5uETWZnYvxrE7psN13Q4hnaiCHVl_ctzU_VSQnnA7Z5eaWERW-1os_CDcfNuhUMQ7L1k1INTJxeer7aMw6OsuftN6dJG74077N6CrPatZJcf968jRGeqo3DVE/s1600/escudobayern2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;" alt="Bayern Munich badge" title="Bayern Munich badge"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDuvQHoI1JAf9zJwiI6s5uETWZnYvxrE7psN13Q4hnaiCHVl_ctzU_VSQnnA7Z5eaWERW-1os_CDcfNuhUMQ7L1k1INTJxeer7aMw6OsuftN6dJG74077N6CrPatZJcf968jRGeqo3DVE/s1600/escudobayern2.gif" alt="Bayern Munich badge" title="Bayern Munich badge"/></a> <b><span style="font-size: x-large;">vs</span></b>. <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhw-6dIl0zexJrtpf5srY3Fra7hWsIPVHPLBpCuyEGjQ854t18A1iVt2jYi6wRc2tIhAzvVOZRYqTvBrJnhmypDtvdtDG3rm2O1uGut5eWduxPPTMb0OqVMu1rdYbtMt9gmDLYuFbtKTl0/s1600/escudoborussiadortmund.gif" alt="Borussia Dortmund badge" title="Escudo del Borussia de Dortmund" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhw-6dIl0zexJrtpf5srY3Fra7hWsIPVHPLBpCuyEGjQ854t18A1iVt2jYi6wRc2tIhAzvVOZRYqTvBrJnhmypDtvdtDG3rm2O1uGut5eWduxPPTMb0OqVMu1rdYbtMt9gmDLYuFbtKTl0/s1600/escudoborussiadortmund.gif" alt="Borussia Dortmund badge" title="Borussia Dortmund badge"/></a></div>
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<p><span class="negrita burdeos">Weidenfeller</span>, <span class="negrita burdeos">Grosskreutz</span> and <span class="negrita burdeos">Schieber</span> are <span class="negrita">Borussia Dortmund</span> players. </p>
<p>They wear on their backs number <span class="super negrita burdeos">1</span>, <span class="super negrita burdeos">19</span> and <span class="super negrita burdeos">23</span>, respectively. </p>
<p>They’ve just lost the Champions League final at Wembley, but they are happy. </p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilpATZGd3mWiS40QQ-ES8LaCYkIxghyphenhyphenB6CxvjqsbvIJYro8i7alNIBJSZ6iQtg1zesd1u5Ofxyor9sDr6iJRaXsZ-tLNRgGYEaLc-X8ovB6gSgQaVDxRMPGnibjnfBc7xeSM_lim0KrfY/s1600/weidenfeller1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilpATZGd3mWiS40QQ-ES8LaCYkIxghyphenhyphenB6CxvjqsbvIJYro8i7alNIBJSZ6iQtg1zesd1u5Ofxyor9sDr6iJRaXsZ-tLNRgGYEaLc-X8ovB6gSgQaVDxRMPGnibjnfBc7xeSM_lim0KrfY/s1600/weidenfeller1.gif" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRsqZQrFLEDVUiiCGE4hKItbhoSJNkMbIwLgwsaloIrreaJat-cXTG8OO9_l_RCs5KP8pffqNCtsbPO6Qn6AZ0opcrOyNp8ueZrABqnOVnbxp7E3sABn0M2ucGgdlMdQYZZU_P3kh_LJI/s1600/grosskreutz19.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRsqZQrFLEDVUiiCGE4hKItbhoSJNkMbIwLgwsaloIrreaJat-cXTG8OO9_l_RCs5KP8pffqNCtsbPO6Qn6AZ0opcrOyNp8ueZrABqnOVnbxp7E3sABn0M2ucGgdlMdQYZZU_P3kh_LJI/s1600/grosskreutz19.gif" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVQJMow5wJhIlN03MYPXs8kZS0NqwL22Dnb1bRb5ztoO_hN0rR7lnoVx90fitMUiD_4mdlx_BvY7Rb11DHxVaImVEQkdIsK4K-3tXCSO0bI8-J31-S-PeU9yHh3z9Hoo6jc94BTKpzSjY/s1600/schieber23.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVQJMow5wJhIlN03MYPXs8kZS0NqwL22Dnb1bRb5ztoO_hN0rR7lnoVx90fitMUiD_4mdlx_BvY7Rb11DHxVaImVEQkdIsK4K-3tXCSO0bI8-J31-S-PeU9yHh3z9Hoo6jc94BTKpzSjY/s1600/schieber23.gif" /></a></div>
<p><span class="negrita rojo">Lahm</span>, <span class="negrita rojo">Boateng</span> and <span class="negrita rojo">Javi Martinez</span> are <span class="negrita">Bayern Munich</span> players. </p>
<p>They wear on their backs number <span class="super negrita rojo">21</span>, <span class="super negrita rojo">17</span> and <span class="super negrita rojo">8</span>, respectively. </p>
<p>They just won the Champions League final at Wembley, but they’re not happy. </p>
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVMuJKE5dUTlFY4nWbj7oKez45OY1ZteriL11ARyaahSJ95KfPrz2bun3e1NkxMPJ3ojs_Qk5UtLu5CPCfDd7nv-2JsSxOTIDaemWkxSqAj0a1b5zwn9cpovRVBbgd_gDI3kPPoU2tJ4c/s1600/lahm21.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVMuJKE5dUTlFY4nWbj7oKez45OY1ZteriL11ARyaahSJ95KfPrz2bun3e1NkxMPJ3ojs_Qk5UtLu5CPCfDd7nv-2JsSxOTIDaemWkxSqAj0a1b5zwn9cpovRVBbgd_gDI3kPPoU2tJ4c/s1600/lahm21.gif" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEig3XjA5j66HrbbN3cKXn1DyQI53GOmQRkO5DJhAGAcRhihL4M_ymteM0jIxTI-arwZ-i5WHyBwCuBHz6h1Nm71K_hzT3GstVDKOiD6NeHF-BMSXhyphenhyphenM3kvBzr6dw16CHuwFC5I7nTmT2z0/s1600/boateng17.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEig3XjA5j66HrbbN3cKXn1DyQI53GOmQRkO5DJhAGAcRhihL4M_ymteM0jIxTI-arwZ-i5WHyBwCuBHz6h1Nm71K_hzT3GstVDKOiD6NeHF-BMSXhyphenhyphenM3kvBzr6dw16CHuwFC5I7nTmT2z0/s1600/boateng17.gif" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuGo1Ew9I6aafLoxxsClEs6f2VhtsvX6WtsLx4nCKhNbTOVpa_S39BidKgKPx7odvoymTjUzLhI7uh42EvbtojRyzjcKomQFZh9fRnXtvI7HQjaKp6SEExZyBKwiBx1ImGwVuSKYQn58k/s1600/martinez8.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuGo1Ew9I6aafLoxxsClEs6f2VhtsvX6WtsLx4nCKhNbTOVpa_S39BidKgKPx7odvoymTjUzLhI7uh42EvbtojRyzjcKomQFZh9fRnXtvI7HQjaKp6SEExZyBKwiBx1ImGwVuSKYQn58k/s1600/martinez8.gif" /></a><br /></div>
<p>Can you imagine what happens here? </p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkZNjtAol92vhyphenhyphenzF8zPYQBJoWZPhgKjAjLQWtah0o5An2tD_vbR4BFGOi9RuMj21ey8lkQZWBzOQZhzGHdtdTPG4bs74FQqjHb743a1Xp2TrlYl2sLfPQfZKvWlmc7Z1MCgwRXo1zfMcI/s1600/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="252" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkZNjtAol92vhyphenhyphenzF8zPYQBJoWZPhgKjAjLQWtah0o5An2tD_vbR4BFGOi9RuMj21ey8lkQZWBzOQZhzGHdtdTPG4bs74FQqjHb743a1Xp2TrlYl2sLfPQfZKvWlmc7Z1MCgwRXo1zfMcI/s640/homomathematicus.jpg" width="640" /></a></div>
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<p>Perhaps, you’ll think: <span class="lector">Well, I can understand <span class="negrita burdeos">Borussia</span> players: I guess that reaching a continental final is a thing to be proud of, although having lost it.</span> </p>
<p>No doubt, this is a very important sporting achievement, but this sort of things is better valued after a while than some minutes after losing the final. </p>
<p><span class="lector">Maybe Borussia players are considering their incentive bonus for having reached the final ... </span></p>
<p>Although the premium is not inconsiderable, certainly they’d change the bonus for winning the title. </p>
<p><span class="lector">Possibly they are now thinking about their holidays ... </span></p>
<p>That’s a good reason to be happy, but ... what about <span class="negrita rojo">Bayern</span> players? </p>
<p>They’ve just won the Champions League, they will receive a great bonus, and they'll go on vacation too... </p>
<p><span class="lector">It's true. So which is the explanation? </span></p>
<p>Very simple: because <span class="negrita burdeos">Borussia Dortmund</span> players carry <strong>happy numbers</strong>, and <span class="negrita rojo">Bayern Munich</span> players wear <strong>unhappy numbers</strong> on their shirts. </p>
<p><span class="lector">Are there <strong>happy</strong> and <strong>unhappy numbers</strong>? </span></p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgK6Acot6ZSULMCTjGKdP2aEl7YR8H0oSldfR9dBrfIfD2j90EEZirLY_QovgUgTEMj-LOOZ9fB_48V2bb7YJJ-mCiMYLugFLoDbjXzn1ed5EI5JemQqte79VogxZjWObZ84oRKqYQ1lgg/s1600/uno.gif" alt="happy number 1 feliz" title="happy number 1" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgK6Acot6ZSULMCTjGKdP2aEl7YR8H0oSldfR9dBrfIfD2j90EEZirLY_QovgUgTEMj-LOOZ9fB_48V2bb7YJJ-mCiMYLugFLoDbjXzn1ed5EI5JemQqte79VogxZjWObZ84oRKqYQ1lgg/s320/uno.gif" alt="happy number 1" title="happy number 1" width="154" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgH8Cp41osxm4KARTwphPNxFl0GYXY6CFDapLS1KS-u4j2YkYb7Rd6W5UrMNxmMiTj_zYXs3LTst3aghwItSc3PPpCmUCZLpzIEEZc66hPpx_LXsPmCaWNFol5tt5Qk1tzQkmHiXXJ5NgQ/s1600/dos.gif" alt="unhappy number 2" title="unhappy number 2" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgH8Cp41osxm4KARTwphPNxFl0GYXY6CFDapLS1KS-u4j2YkYb7Rd6W5UrMNxmMiTj_zYXs3LTst3aghwItSc3PPpCmUCZLpzIEEZc66hPpx_LXsPmCaWNFol5tt5Qk1tzQkmHiXXJ5NgQ/s320/dos.gif" alt="unhappy number 2" title="unhappy number 2" width="210" /></a></div>
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<p>Yes, there are. </p>
<p><span class="lector">And what things distinguishes one happy number from an unhappy one? </span></p>
<p>Well, in Mathematics, <strong class="rojo">happy numbers</strong> are defined as follows: choose one number and calculate <strong>the sum of the squares of its digits</strong>. With the <strong>result</strong>, <strong>repeat the process</strong> until the sum <strong>equals 1</strong>, or until we enter an <strong>infinite cycle</strong> that will never reach number 1. </p>
<p>Following this process, the numbers that end in number 1, are called '<strong>happy numbers</strong>', while the numbers that enter in a loop are known as '<strong>unhappy</strong> (or sad) <strong>numbers</strong>'. </p>
<p>Let's see an example of happy number: </p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7Vu3PtC8N83D5KUJopFRky0wIPjLmKl8rY-7JdVccmd8SOKMQyTWKEPlcAm5LQFULWGp2Qnfb9ZWtWznZnEMlFLHs9rrud4O4Iqq0k3fN49mBRhlUVcO5GfV3WqZqLxGVlbdqgI-idjA/s1600/formulafeliz.JPG" alt="procedure to check happy numbers" title="procedure to check happy numbers" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7Vu3PtC8N83D5KUJopFRky0wIPjLmKl8rY-7JdVccmd8SOKMQyTWKEPlcAm5LQFULWGp2Qnfb9ZWtWznZnEMlFLHs9rrud4O4Iqq0k3fN49mBRhlUVcO5GfV3WqZqLxGVlbdqgI-idjA/s1600/formulafeliz.JPG" alt="procedure to check happy numbers" title="procedure to check happy numbers" /></a></div>
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<p>And now let's see an example of unhappy (or sad) number: </p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfXhqt-f8SR534sL7OkG39HrFeHflbr79GVahurAEAXw1ZOUOtaxZwQNKEWs7v4pD2x21tGSTHg2cvEU24zpBf3AEwBnAa3o2zciqy8eBviIK0i7zlBrzfdd0ZlKAmchVNxF4zMxc80ak/s1600/11.jpg" alt="procedure to check unhappy numbers" title="procedure to check unhappy numbers" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfXhqt-f8SR534sL7OkG39HrFeHflbr79GVahurAEAXw1ZOUOtaxZwQNKEWs7v4pD2x21tGSTHg2cvEU24zpBf3AEwBnAa3o2zciqy8eBviIK0i7zlBrzfdd0ZlKAmchVNxF4zMxc80ak/s1600/11.jpg" alt="procedure to check unhappy numbers" title="procedure to check unhappy numbers" /></a></div>
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<p>You can see that, after obtaining number 4, we've entered in a loop of numbers</p>
<p class="tecnico">In mathematical form, we stand that: Given a number n = n<sub>0</sub>, define a sequence n<sub>1</sub>, n<sub>2</sub>, ... where n<sub>i+1</sub> is the sum of the squares of the digits of n<sub>i</sub>. Then n is happy if and only if there exists i such that n<sub>i</sub>=1. </p>
<p class="tecnico">Thus, the first happy numbers are: </p>
<div class="separator" style="clear: both; text-align: center;"><span class="super negrita">1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, ...</span></div>
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiY1IqjlKzw7nNl1aJOvncmx8Wyd32r8-zMGywBqfe1HznN0M_x2FNAkk0zyI_Vv6qUlCDEm67eXCqb7zO8tAe-6xY_1U4ibFKA15cUdt_RoljJ5u2kIsoiJJrGici2yZkJqfUq53TdJvg/s1600/siete.gif" alt="happy number 7" title="happy number 7" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiY1IqjlKzw7nNl1aJOvncmx8Wyd32r8-zMGywBqfe1HznN0M_x2FNAkk0zyI_Vv6qUlCDEm67eXCqb7zO8tAe-6xY_1U4ibFKA15cUdt_RoljJ5u2kIsoiJJrGici2yZkJqfUq53TdJvg/s320/siete.gif" alt="happy number 7" title="happy number 7" width="227" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7bvN09wKh35-ZQyVyJR35zx1JiNjZQ586tDbQwKjTQedBgl6QfmKAcF0fLYV6zAWa3kM7M0KC81i925hcWewd8VKzenvH-FkafX3UQKHW5MGcRJWQcARLC2xSlmW4VSG9qHA1mCxCThA/s1600/cinco.gif" alt="unhappy number 5" title="unhappy number 5" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7bvN09wKh35-ZQyVyJR35zx1JiNjZQ586tDbQwKjTQedBgl6QfmKAcF0fLYV6zAWa3kM7M0KC81i925hcWewd8VKzenvH-FkafX3UQKHW5MGcRJWQcARLC2xSlmW4VSG9qHA1mCxCThA/s320/cinco.gif" alt="unhappy number 5" title="unhappy number 5" width="192" /></a><br />
<p class="tecnico">Also, if a number is happy, then all members of the sequence of results generated are also happy. And if a number is unhappy, the whole sequence of numbers generated by applying the procedure will be also unhappy. </p>
<p class="tecnico">Thus, in the second example, we can determine not only number 11 is unhappy, but also numbers 2, 4, 16, 37, 58, 89, 145, 42 and 20 are unhappy ... </p>
<p class="tecnico">Indeed, from a happy number we’ll obtain a lot of additional happy numbers, by inserting zeroes among its digits, or changing the order of its digits: 2, 4, 16, 37, 58, 89, 145, 42 and 20.</p>
<p> And, why are so called, and what is their use? </ p>
<p class="tecnico">It's not known the origin of its name. Perhaps they are happy because they finally reach their goal (the number 1), while the unhappy are doomed to wander forever through space mathematical ... </ p>
<p class="tecnico">Their usefulness..., so far nothing but mathematical entertainment to practice with the exponentiation, but perhaps in the future we will find some practical application. Want to try? </ P>
<p>If you want to learn more about happy numbers, you can see on the <a href="http://en.wikipedia.org/wiki/Happy_number" target="_blank">Wikipedia</a> or on the web <a href="http://www.articlesforeducators.com/article.asp?aid=1" target="_blank">Articles for education</a><br /></p>
WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com1tag:blogger.com,1999:blog-8244131673148079224.post-4582066607695410162013-07-15T21:55:00.000+01:002014-02-24T21:58:05.834+00:00Llorente and the optimal stopping theory.<a href="http://www.matifutbol.com/en/decision.eng.html" target="_blank"><span style="color: red;">Llorente's hard decision. Optimal stopping theory.</span></a><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxkVPQzQIUtUZJzr67I1LBHgZJ_ueCI6S7OcED6fLVhnfu7NYMNGK2tstcBCyLIjpScvQzkwvBQ7WxbI-OSrgNtKig6e3dbqm0aNKITUVpzpK7skAAAzXHXjKRcSthYQVE8iPeQnL6kY8/s1600/4casosesp.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxkVPQzQIUtUZJzr67I1LBHgZJ_ueCI6S7OcED6fLVhnfu7NYMNGK2tstcBCyLIjpScvQzkwvBQ7WxbI-OSrgNtKig6e3dbqm0aNKITUVpzpK7skAAAzXHXjKRcSthYQVE8iPeQnL6kY8/s1600/4casosesp.jpg" height="200" width="133" /></a>15/07/2013<br />
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The problem of Fernando Llorente about which team he will choose to sign for, introduces us to the theory of taking decisions and the optimal stopping theory.<br />
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Another puzzle of logic and strategy, which show us how we can take the best decissions in our daily life.<br />
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Click <a href="http://www.matifutbol.com/en/decision.eng.html" target="_blank">here</a> to see our story.<br />
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WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-23658539300350037692013-06-17T15:44:00.002+01:002013-06-18T15:28:16.601+01:00Press, stats and football charts.<p class="rojo negrita">FIRST HALF</p>
<p> Madrid press is clear: it's the time for the relay in the hegemony of <span class="rojo unidas">FC Barcelona</span>.</p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_J7EcoxKnPg0PDBMKFPOzcAbVVXaLaWMFr-TPzNgcey95aqlvCtsxrd51x7pzsXI-BWFydoP5BfN988FW3F_dHYK89ZRf_SyEEYoIRmx2YhnIZNy-6fycCb04_ylh85fhsDaWJKYorFg/s1600/periodicomarca1.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_J7EcoxKnPg0PDBMKFPOzcAbVVXaLaWMFr-TPzNgcey95aqlvCtsxrd51x7pzsXI-BWFydoP5BfN988FW3F_dHYK89ZRf_SyEEYoIRmx2YhnIZNy-6fycCb04_ylh85fhsDaWJKYorFg/s320/periodicomarca1.gif" width="224" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjaTAJwsS5R5ys6It09bRkQFekPfoImYG2k8Nu75BVOCQ_jXkXKywRdWX6pnjrvyvl5kstKIQ84zmvCQJkHJ3ZXjM8Ldv78FRGgP3CIxeugm_LJSicBKk3ee_aNk8JNXX2eT4fh04q1R6U/s1600/portadaas1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjaTAJwsS5R5ys6It09bRkQFekPfoImYG2k8Nu75BVOCQ_jXkXKywRdWX6pnjrvyvl5kstKIQ84zmvCQJkHJ3ZXjM8Ldv78FRGgP3CIxeugm_LJSicBKk3ee_aNk8JNXX2eT4fh04q1R6U/s320/portadaas1.gif" width="240" /></a></div>
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<p> After recent years, <span class="violeta">Real Madrid</span> is getting closer to <span class="rojo unidas">FC Barcelona</span> level and, with the data studied, in the last year it has surpassed it.</p>
<p> To strengthen their argument, Madrid journalists use the following data: the incomes from the last two years of the two clubs, and the individual and collective titles achieved by the two teams.</p>
<p>Regarding <span class="negrita">incomes</span>, <span class="violeta">Real Madrid</span> has achieved in <span class="marron">2013</span> an increasing of <span class="negrita">80%</span> from the previous year, while <span class="rojo unidas">FC Barcelona</span> have only grown a <span class="negrita">40%</span>.</p>
<p> And in regard to individual and collective <span class="negrita">titles</span>, <span class="violeta">Real Madrid</span> has achieved a <span class="negrita">40%</span> less of titles than last season, while <span class="rojo unidas">FC Barcelona</span> won <span class="negrita">10%</span> less than the previous year.</p>
<p> If we transfer this data to a <span class="super negrita">table</span>, and assign number <span class="negrita">100</span> to the value of each variable in <span class="negrita"><span class="verde">2012</span></span>, we have the following figures: </p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhognZ8if-Qw0Qk6h6xRjjZ5CLMYq8yKiC3kBZ__7oZ5SinNKHK30eb7QOyY1ZiO8UyDspAPLLXZyYCgZNFrxIi4nQZaYmVHTlE-pEXWMwTf_FtTFTe7XLMK0uJiBln88-orrsGHyoFUn4/s1600/tablas2012eng.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="182" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhognZ8if-Qw0Qk6h6xRjjZ5CLMYq8yKiC3kBZ__7oZ5SinNKHK30eb7QOyY1ZiO8UyDspAPLLXZyYCgZNFrxIi4nQZaYmVHTlE-pEXWMwTf_FtTFTe7XLMK0uJiBln88-orrsGHyoFUn4/s640/tablas2012eng.jpg" width="640" /></a></div>
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<p> Now let's move these data to a <span class="super negrita">chart</span>, so that we can analyze them in a more noticeable way: </p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivcPn5Gkau6D5rLwXLl8gt8iMUmk64qHEjAGMq7omMy_sWN4aWUkn5lJELPywhT4mnNMbyi_V-ur8y57ggu82iNiT7VBw4XDarOcBKe8f3c3cfsIXHUhrHQ6wPq18NFaA1PZAorwCbbYw/s1600/comparativa+1.eng.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="290" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivcPn5Gkau6D5rLwXLl8gt8iMUmk64qHEjAGMq7omMy_sWN4aWUkn5lJELPywhT4mnNMbyi_V-ur8y57ggu82iNiT7VBw4XDarOcBKe8f3c3cfsIXHUhrHQ6wPq18NFaA1PZAorwCbbYw/s1600/comparativa+1.eng.jpg" width="640" /></a></div>
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<p> If you look at the <span class="negrita">average</span> of the two variables, we see a more positive and increasing trend in the data from <span class="violeta">Real Madrid</span> than in the graph of <span class="rojo unidas">FC Barcelona</span>.</p>
<p>We can realize that the <span class="negrita">improvement</span> of the variables average is more pronounced in the case of <span class="violeta">Real Madrid</span>. So, the average rate rises from 100 to <span class="negrita">115</span> in case of <span class="rojo unidas">FC Barcelona</span>, and from 100 to <span class="negrita">120</span> in case of <span class="violeta">Real Madrid</span>. </p>
<p>The conclusion, in view of these charts, is what news media from Madrid tell: <span class="rojo unidas">FC Barcelona</span> is in a cycle order, and <span class="violeta">Real Madrid</span> is in sharp rise. </p>
<p>What do you think will be the opinion of the sports <span class="negrita">media</span> from <span class="negrita">Barcelona</span>?</p>
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<a href="http://www.matifutbol.com/im/homomathematicus.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="250" src="http://www.matifutbol.com/im/homomathematicus.jpg" width="640" /></a></div>
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<p class="rojo negrita">SECOND HALF</p>
<p> Some of you will think: <span class="lector">Barcelona's newspapers will say that these data are not certain, and will stand that <span class="rojo unidas">FC Barcelona</span> is still better than <span class="violeta">Real Madrid</span>.</span></p>
<p>May be, but the truth is that the press in Barcelona should not use this argument, since it's easy to verify that the data used are correct.</p>
<p class="lector">Yes, but all data are relative percentages. Surely that <span class="rojo unidas">FC Barcelona</span>'s absolute data are much higher than those of <span class="violeta">Real Madrid</span>.</p>
<p> That's right. May be that absolute numbers from which <span class="rojo unidas">FC Barcelona</span> starts are higher than <span class="violeta">Real Madrid</span>, and that despite the improvement in <span class="violeta">Real Madrid</span> data, <span class="rojo unidas">FC Barcelona</span> is still above in absolute figures. </p>
<p> In fact, one of the shortcomings in many of the <span class="negrita">statistics</span> that we find everyday is that there's no information about the <span class="negrita">original data</span> on which they are based. Two countries can have the same IPC, but we must know the standard of living in each of them, and the base level from which we start, for comparison purposes. Two teams may have won 50% more titles than last year, but we can't equal one team that won 10 titles last year and wins 15 this season, with one that had won 2 titles last season and wins 3 titles this year</p>
<p>However, this doesn't mean that Madrid newspapers offer false information. They have not said anything about <span class="negrita">absolute figures</span>, all they have done is focus on <span class="negrita">trends</span> of both teams in relative values. And, apparently, they are right in their argument.</p>
<p class="lector"> Sure. What happens is that among hundreds of ratios that could be handled, they've chosen those two (economy and titles) which better support their argument. Sure the other ratios were beneficial to the <span class="rojo unidas">FC Barcelona</span>...</p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRR7n7SwC0JjIWoT28ASlnK5CZuxD54OHZ70T5zduGJuKfCuMRahzovUdEfvhTGtsP2VdT-e2sWzGTq6u00D5efq2VJh-4NZpRxUr2bJiuAYkfCpH0qR2lAZo5i_6H5Y-E5NvcN70WSi8/s1600/copasydinero.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="195" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRR7n7SwC0JjIWoT28ASlnK5CZuxD54OHZ70T5zduGJuKfCuMRahzovUdEfvhTGtsP2VdT-e2sWzGTq6u00D5efq2VJh-4NZpRxUr2bJiuAYkfCpH0qR2lAZo5i_6H5Y-E5NvcN70WSi8/s400/copasydinero.gif" width="400" /></a></div>
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<p> Well, we know that each one uses the data that conform better his argumentation (including the author of this story). However, it seems that the <span class="negrita">data</span> used are fairly <span class="negrita">representative</span> of the <span class="negrita">success</span> of a team.</p>
<p> To be strict, there should have been a <span class="negrita">previous study</span> about which variables are the most representative of a club success. And we should start by <span class="negrita">defining</span> what we mean by <span class="negrita">success</span> of a club. </p>
<p>Nowadays, perhaps we could agree that now the most representative indicators of the success of a football club are the <span class="negrita"> sports results</span> and the <span class="negrita"> economic performance </span>. Although we should not forget their <span class="negrita">social responsibility</span> (support grassroots sport, collaboration with NGOs, etc..) and their impact on <span class="negrita">social networks</span>, all of them increasingly relevant aspects. </p>
<p class="lector"> Yes, but although these two factors (titles and economics) are the most representative, need not have the same <span class="negrita">relevance</span>. Surely if we weigh the two indices in a different way to that done (in our example both are equal, <span class="negrita">50%</span> of the total), we'll come to different results ... </p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUaldjmCN0W4lRUbVI1y6GLlrvBF2MWvLLfvMZokj3qEIP7U74bpHq7f9kNsmhnvrh5LZBlqkdMc3F9p86tw3P758tHc6iDdgpEuBozjCK2-_KFP9ALynBTP9Uzpaew8gT0g6hSQOrMpw/s1600/balanzatitulosydinero.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="214" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUaldjmCN0W4lRUbVI1y6GLlrvBF2MWvLLfvMZokj3qEIP7U74bpHq7f9kNsmhnvrh5LZBlqkdMc3F9p86tw3P758tHc6iDdgpEuBozjCK2-_KFP9ALynBTP9Uzpaew8gT0g6hSQOrMpw/s320/balanzatitulosydinero.gif" width="320" /></a></div>
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<p>Indeed, according to the importance we give to each index, we can draw <span class="negrita"> different conclusions</span> from those reached by the newspapers from Madrid.</p>
<p> However, even with the indices chosen, and the weighting established for them, we can still question the conclusions of Madrid papers.</p>
<p class="lector">Also, here we are mixing two variables: money and sportive results, which doesn’t have nothing to do each other....</p>
<p>It’s true. This is a typical 'trick' that many analysts use. Sometimes we see how completely heterogeneous figures are added: in our case, economic and sportive data. And it happens also in the sports performance index, in which a collective title, such as the achievement of the league title, worths the same as an individual title, for instance, a trophy awarded to a defense for being the player that has scored more headed goals in a Cup competition.</p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPFqmuUnM1AsaR7AVHBDE1ERmwUefwbzVd2MvNwA_ebdONctj1pycdEXNH6I5kv8l-_-K2QzCdCVuSwGbWZX_TECbzrsHMsGaCq_CpLLiNaV3pEEJa5QdX-8Si0e46jqWNFMnkJQ8h2Hk/s1600/copasycopitas.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="188" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPFqmuUnM1AsaR7AVHBDE1ERmwUefwbzVd2MvNwA_ebdONctj1pycdEXNH6I5kv8l-_-K2QzCdCVuSwGbWZX_TECbzrsHMsGaCq_CpLLiNaV3pEEJa5QdX-8Si0e46jqWNFMnkJQ8h2Hk/s640/copasycopitas.gif" width="640" /></a></div>
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<p>However, even with many caveats, we'll take for good the analysis of Madrid newspapers.</p>
<p class="lector"> So we must admit that the tendency of <span class="violeta">Real Madrid</span> has been better than that of <span class="rojo unidas">FC Barcelona</span> in the last season, right?</p>
<p> Well, it remains to check a non negligible <span class="negrita">detail</span> that makes Barcelona newspapers believe that <span class="rojo unidas">FC Barcelona</span> is still better than <span class="violeta">Real Madrid</span>. </p>
<p>And which is this detail?</p>
<p> Well, it's as simple as choosing appropriately the <span class="negrita">base year</span> o reference year. </p>
<p> In the articles on Madrid newspapers, they rely their conclusions upon the results of each team in <span class="verde">2012</span>. </p>
<p> This is, they assign the index base <span class="negrita">100</span> to the results of each team in <span class="verde">2012</span>, and they calculate the variations in <span class="marron">2013</span> on it. </p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhognZ8if-Qw0Qk6h6xRjjZ5CLMYq8yKiC3kBZ__7oZ5SinNKHK30eb7QOyY1ZiO8UyDspAPLLXZyYCgZNFrxIi4nQZaYmVHTlE-pEXWMwTf_FtTFTe7XLMK0uJiBln88-orrsGHyoFUn4/s1600/tablas2012eng.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="182" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhognZ8if-Qw0Qk6h6xRjjZ5CLMYq8yKiC3kBZ__7oZ5SinNKHK30eb7QOyY1ZiO8UyDspAPLLXZyYCgZNFrxIi4nQZaYmVHTlE-pEXWMwTf_FtTFTe7XLMK0uJiBln88-orrsGHyoFUn4/s640/tablas2012eng.jpg" width="640" /></a></div>
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<p> Let's see how, with the same data, its interpretation radically changes, just only transferring the <span class="negrita"> reference year</span>.</p>
<p> We fix now as <span class="negrita">base 100</span> the value of each index in <span class="marron">2013</span>, and we recalculate the table, remembering that <span class="violeta">Real Madrid</span> had achieved a <span class="negrita">80%</span> increase in revenue and <span class="negrita">40%</span> fewer titles, and that the <span class="rojo unidas">FC Barcelona</span> increased <span class="negrita">40%</span> its revenue, and its titles decreased <span class="negrita">10%</span>.</p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigFzG7zkZDvpetW8bMn2x7rLs6SvDePv6mK7xGdN1xgcrLoG_jZGc8TaUPmVXARV7A8xdZzK3QeofPJVHNQWStjOkAEWZ8aGcc5LEih5kwscIIXaQA9qlz2IAWgDQ1QPe9A9DbxVSjctA/s1600/tablas2013eng.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="182" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigFzG7zkZDvpetW8bMn2x7rLs6SvDePv6mK7xGdN1xgcrLoG_jZGc8TaUPmVXARV7A8xdZzK3QeofPJVHNQWStjOkAEWZ8aGcc5LEih5kwscIIXaQA9qlz2IAWgDQ1QPe9A9DbxVSjctA/s640/tablas2013eng.jpg" width="640" /></a></div>
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<p>Let’s see these data in a graphic:</p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7hw1HFk7vkR52kAgq1mzvMhJFRj0eNCFCWFLob3PLW7lNt-jVEAzBeZ_hIl1WeUCGnuYyPsTtdAx49QrqyR60j9U40qQkcYewgmd4av4prG76T-dsL7CN1a4SsmfeVWdFG5o94zEzlJI/s1600/comparativa2.eng.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="294" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7hw1HFk7vkR52kAgq1mzvMhJFRj0eNCFCWFLob3PLW7lNt-jVEAzBeZ_hIl1WeUCGnuYyPsTtdAx49QrqyR60j9U40qQkcYewgmd4av4prG76T-dsL7CN1a4SsmfeVWdFG5o94zEzlJI/s640/comparativa2.eng.jpg" width="640" /></a></div>
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<p> We can see how with the same data, and changing only the year in which we set the <span class="negrita">100% rate</span>, the conclusions are radically different: the <span class="rojo unidas">FC Barcelona</span> maintains the same positive trend, but the trend of the average of the indexes clearly becomes negative for <span class="violeta">Real Madrid</span> with this new perspective. </p>
<p> So Barcelona newspapers can use the same statistic as Madrid’s media to make on them their cover titles.</p>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhITYQwR_X-iBAL3Rwzuk1SvL76Ap-ahKPjcuPlAa_V9CA-vgFLqdD44Wu-UQTTlxWR0DseKWxIXigtKbhokFXtaZsvKh0tzUMz_OJbDs-oPhNIMqc0iK_d9niT1gkYX1M14s0k3h-aUWI/s1600/periodicosport1.gif" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhITYQwR_X-iBAL3Rwzuk1SvL76Ap-ahKPjcuPlAa_V9CA-vgFLqdD44Wu-UQTTlxWR0DseKWxIXigtKbhokFXtaZsvKh0tzUMz_OJbDs-oPhNIMqc0iK_d9niT1gkYX1M14s0k3h-aUWI/s320/periodicosport1.gif" width="265" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9D9X-3h-NbV3svV53b4TZkAdiTRQo8AwBYIyx_Cuc9zJegdlYl6CEAdjAsOHuSF76tqIUxyANAx-3HRcHN2hsibC_5vQaEW6X9gEqhbJSCJ1ckFSCxmrCdDlj_1i6Rx_0YunSYaoYd1A/s1600/periodicomundo1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9D9X-3h-NbV3svV53b4TZkAdiTRQo8AwBYIyx_Cuc9zJegdlYl6CEAdjAsOHuSF76tqIUxyANAx-3HRcHN2hsibC_5vQaEW6X9gEqhbJSCJ1ckFSCxmrCdDlj_1i6Rx_0YunSYaoYd1A/s320/periodicomundo1.gif" width="247" /></a></div>
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<p class="tecnico">With our example, we can say that the simple choice of the value or the time we take to <span class="negrita">index, base or coordinate origin</span> issue can lead to very different conclusions about the same data. Or we could also say that we can handle most statistical data in such way that they fit the target we want to prove.</p>
<p class="tecnico">This story is a tribute to <span class="negrita tecnico">Durrell Huff</span> and his book <span class="negrita tecnico">‘How to lie with Statistics’</span>. Some months ago, a follower of Matifutbol, whom I want to thank, recommended me to read this extraordinary book.
<p class="tecnico">Such books throw light on some aspects of Statistics, which should be known by everybody, in order that people could handle correctly all these data, indexes and graphics that surround us nowadays.</p>WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-21652889267029933692013-04-15T16:03:00.000+01:002014-02-16T17:17:28.978+00:00Mathematical black holes<span style="color: red;"><a href="http://www.matifutbol.com/en/blackhole.html" target="_blank">A black (and red) hole</a></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAo6qmAvQ4cEjTSuynIEgZIyNqSXPGOY8xmaUSFjRsPDoStfLMpZD3Zpu3ExWWf-hBfH9PfbiFwD27Yi1KeRPkRx56FTkku2InZHSPiQN1pka7fG6_8m6-3Fh6ba1x6ZoXCLUMaSmrF4E/s1600/berlusconi2.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAo6qmAvQ4cEjTSuynIEgZIyNqSXPGOY8xmaUSFjRsPDoStfLMpZD3Zpu3ExWWf-hBfH9PfbiFwD27Yi1KeRPkRx56FTkku2InZHSPiQN1pka7fG6_8m6-3Fh6ba1x6ZoXCLUMaSmrF4E/s1600/berlusconi2.jpg" /></a><br />
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15/04/2013<br />
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Thanks to a prize consisting on the proceeds of a football match, we learn about some numbers that attract the others, which can't escape from them.<br />
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Mathematical sequences which lead to mathematical black holes<br />
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Click <a href="http://www.matifutbol.com/en/blackhole.html" target="_blank">here</a> to see our story.<br />
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WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0tag:blogger.com,1999:blog-8244131673148079224.post-83880263501609964632013-04-05T15:34:00.000+01:002013-12-19T15:35:21.100+00:00Close-packing<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvEvomAuZKfceAtkEHPpNcdnniR8CK_kzETtyhlB6Sed67Np8EGOhZPuo-HeLWnntzxJrUneI9-HXxUF2sr8MNOE90-ocChHs1H7hQQ0qpxxNW0JPRRnCTkeyaHc8sAmWPiGDWxN_LgX8/s1600/terrenojuegollenobalones.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvEvomAuZKfceAtkEHPpNcdnniR8CK_kzETtyhlB6Sed67Np8EGOhZPuo-HeLWnntzxJrUneI9-HXxUF2sr8MNOE90-ocChHs1H7hQQ0qpxxNW0JPRRnCTkeyaHc8sAmWPiGDWxN_LgX8/s320/terrenojuegollenobalones.jpg" width="320" /></a><a href="http://www.matifutbol.com/en/balls.html" target="_blank"><span style="color: red;">Let's give away some balls.</span></a><br />
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05/04/2013</div>
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Organizers of Brazil 2014 FIFA World Cup want to give away some official balls.<br />
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We face a close-packing problem in order to know if we can fill Maracana stadium with enough balls to satisfy the Championship organizers' wishes.<br />
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Click <a href="http://www.matifutbol.com/en/balls.html" target="_blank">here</a> to see the story.</div>
WTF1968http://www.blogger.com/profile/04231956917589692209noreply@blogger.com0