Posts Tagged ‘ combinatorics ’

Le Monde puzzle [#783]

July 20, 2012
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Le Monde puzzle [#783]

In a political party, there are as many cells as there are members and each member belongs to at least one cell. Each cell has five members and an arbitrary pair of cells only shares one member. How many members are there in this political party? Back to the mathematical puzzles of Le Monde (science

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[not] Le Monde puzzle (solution)

April 13, 2012
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[not] Le Monde puzzle (solution)

Following the question on dinner table permutations on StackExchange (mathematics) and the reply that the right number was six, provided by hardmath, I was looking for a constructive solution how to build the resolvable 2-(20,5,1) covering. A few hours later. hardmath again came up with an answer, found in the paper Equitable Resolvable Coverings by van

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Coincidence in lotteries

October 19, 2010
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Coincidence in lotteries

Last weekend, my friend and coauthor Jean-Michel Marin was interviewed (as Jean-Claude Marin, sic!) by a national radio about the probability of the replication of a draw on the Israeli Lottery. Twice the same series of numbers appeared within a month. This lotery operates on a principle of 6/37 + 1/8: 6 numbers are drawn

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Those dice aren’t loaded, they’re just strange

June 18, 2010
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Those dice aren’t loaded, they’re just strange

I must confess to feeling an almost obsessive fascination with intransitive games, dice, and other artifacts. The most famous intransitive game is rock, scissors, paper. Rock beats scissors.  Scissors beats paper. Paper beats rock. Everyone older than 7 seems to know this, but very few people are aware that dice can exhibit this same behavior,

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Random sudokus [p-values]

May 21, 2010
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Random sudokus [p-values]

I reran the program checking the distribution of the digits over 9 “diagonals” (obtained by acceptable permutations of rows and column) and this test again results in mostly small p-values. Over a million iterations, and the nine (dependent) diagonals, four p-values were below 0.01, three were below 0.1, and two were above (0.21 and 0.42).

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Random [uniform?] sudokus [corrected]

May 19, 2010
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Random [uniform?] sudokus [corrected]

As the discrepancy in the sum of the nine probabilities seemed too blatant to be attributed to numerical error given the problem scale, I went and checked my R code for the probabilities and found a choose(9,3) instead of a choose(6,3) in the last line… The fit between the true distribution and the

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Random [uniform?] sudokus

May 19, 2010
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Random [uniform?] sudokus

A longer run of the R code of yesterday with a million sudokus produced the following qqplot. It does look ok but no perfect. Actually, it looks very much like the graph of yesterday, although based on a 100-fold increase in the number of simulations. Now, if I test the adequation with a basic chi-square

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Random sudokus [test]

May 17, 2010
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Random sudokus [test]

Robin Ryder pointed out to me that 3 is indeed the absolute minimum one could observe because of the block constraint (bon sang, mais c’est bien sûr !). The distribution of the series of 3 digits being independent over blocks, the theoretical distribution under uniformity can easily be simulated: #uniform distribution on the block diagonal

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