<p><div><div><div><div data-show-faces="false" data-height="21" data-width="100" data-layout="button_count" data-send="true" data-href="https://www.r-bloggers.com/le-monde-puzzle-1650/"></div></div><div></div></div><div></div></div>
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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2018/09/05/le-monde-puzzle-1050/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><strong><a rel="nofollow" href="https://xianblog.files.wordpress.com/2015/10/lemondapari.jpeg" target="_blank"><img src="https://xianblog.files.wordpress.com/2015/10/lemondapari.jpeg?w=450&h=299&fit=456%2C456" data-recalc-dims="1"></a>A</strong> penultimate <a rel="nofollow" href="https://xianblog.wordpress.com/2011/09/03/le-monde-puzzle-website/" title="Le Monde puzzle [website]" target="_blank">Le Monde mathematical puzzle</a>  before the new competition starts [<a rel="nofollow" href="https://xianblog.wordpress.com/2017/03/28/le-monde-puzzle-1000-1025/" target="_blank">again</a>!]</p>
<blockquote>
<p><em>For a game opposing 40 players over 12 questions, anyone answering correctly a question gets as reward the number of people who failed to answer. Alice is the single winner: what is her minimal score? In another round, Bob is the only lowest grade: what is his maximum score?<br>
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<p>For each player, the score S is the sum δ¹s¹+…+δ⁸s⁸, where the first term is an indicator for a correct answer and the second term is the sum over all other players of their complementary indicator, which can be replaced with the sum over all players since δ¹(1-δ¹)=0. Leading to the vector of scores</p>
<pre>worz &lt;- function(ansz){
  scor=apply(1-ansz,2,sum)
  return(apply(t(ansz)*scor,2,sum))}
</pre>
<p>Now, running by brute-force a massive number of simulations confirmed my intuition that the minimal winning score is 39, the number of players minus one [achieved by Alice giving a single good answer and the others none at all], while the maximum loosing score appeared to be 34, for which I had much less of an intuition!  I would have rather guessed something in the vicinity of 80 (being half of the answers replied correctly by half of the players)… Indeed, while in <a rel="nofollow" href="https://xianblog.wordpress.com/2018/08/29/ims-workshop/" target="_blank">SIngapore</a>, I however ran in the wee hours a quick simulated annealing code from this solution and moved to 77.</p>
<p><a rel="nofollow" href="http://www.affairedelogique.com/" target="_blank"><img width="214" src="https://i0.wp.com/www.affairedelogique.com/backoffice/img/fck/affairedelogique/pepere.jpg" height="214"></a>And the 2018 version of <a rel="nofollow" href="http://www.affairedelogique.com/espace_probleme.php?corps=probleme&num=1066" target="_blank">Le Monde maths puzzle</a> competition starts today!, for a total of eight double questions, starting with an optimisation problem where the adjacent X table is filled with zeros and ones, trying to optimise (max and min) the number of <em>positive</em> entries [out of 45] for which an even number of neighbours is equal to one. On the represented configuration, green stands for one (16 ones) and P for the positive entries (31 of them). This should be amenable to a R resolution [R solution], by, once again!, simulated annealing. Deadline for <a rel="nofollow" href="http://www.affairedelogique.com/espace_probleme.php?corps=probleme&num=1066" target="_blank">the reply on the competition website</a> is next Tuesday, midnight [UTC+1]</p>
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