115 lines
5.9 KiB
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115 lines
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<h2>Nontransitive sets of dice</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 376</h3><span style="width:300px;color:#666;">Published on Sunday, 18th March 2012, 01:00 am; Solved by 185</span></div>
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Consider the following set of dice with nonstandard pips:
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Die A: 1 4 4 4 4 4<br />
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Die B: 2 2 2 5 5 5<br />
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Die C: 3 3 3 3 3 6<br />
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</p>
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<p>
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A game is played by two players picking a die in turn and rolling it. The player who rolls the highest value wins.
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<p>
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If the first player picks die A and the second player picks die B we get<br />
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P(second player wins) = <sup>7</sup>/<sub>12</sub> > <sup>1</sup>/<sub>2</sub>
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</p>
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<p>
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If the first player picks die B and the second player picks die C we get<br />
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P(second player wins) = <sup>7</sup>/<sub>12</sub> > <sup>1</sup>/<sub>2</sub>
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</p>
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<p>
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If the first player picks die C and the second player picks die A we get<br />
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P(second player wins) = <sup>25</sup>/<sub>36</sub> > <sup>1</sup>/<sub>2</sub>
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</p>
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<p>
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So whatever die the first player picks, the second player can pick another die and have a larger than 50% chance of winning.<br />
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A set of dice having this property is called a <b>nontransitive set of dice</b>.
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</p>
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<p>
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We wish to investigate how many sets of nontransitive dice exist. We will assume the following conditions:<ul>
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<li>There are three six-sided dice with each side having between 1 and <var>N</var> pips, inclusive.</li>
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<li>Dice with the same set of pips are equal, regardless of which side on the die the pips are located.</li>
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<li>The same pip value may appear on multiple dice; if both players roll the same value neither player wins.</li>
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<li>The sets of dice {A,B,C}, {B,C,A} and {C,A,B} are the same set.</li>
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</ul>
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</p>
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<p>
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For <var>N</var> = 7 we find there are 9780 such sets.<br />
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How many are there for <var>N</var> = 30 ?
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