68 lines
2.0 KiB
Markdown
68 lines
2.0 KiB
Markdown
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---
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id: 5900f4e51000cf542c50fff7
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title: 'Problem 376: Nontransitive sets of dice'
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challengeType: 5
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forumTopicId: 302038
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dashedName: problem-376-nontransitive-sets-of-dice
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---
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# --description--
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Consider the following set of dice with nonstandard pips:
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$$\begin{array}{} \text{Die A: } & 1 & 4 & 4 & 4 & 4 & 4 \\\\ \text{Die B: } & 2 & 2 & 2 & 5 & 5 & 5 \\\\ \text{Die C: } & 3 & 3 & 3 & 3 & 3 & 6 \\\\ \end{array}$$
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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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If the first player picks die $A$ and the second player picks die $B$ we get
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$P(\text{second player wins}) = \frac{7}{12} > \frac{1}{2}$
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If the first player picks die $B$ and the second player picks die $C$ we get
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$P(\text{second player wins}) = \frac{7}{12} > \frac{1}{2}$
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If the first player picks die $C$ and the second player picks die $A$ we get
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$P(\text{second player wins}) = \frac{25}{36} > \frac{1}{2}$
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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. A set of dice having this property is called a nontransitive set of dice.
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We wish to investigate how many sets of nontransitive dice exist. We will assume the following conditions:
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- There are three six-sided dice with each side having between 1 and $N$ pips, inclusive.
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- Dice with the same set of pips are equal, regardless of which side on the die the pips are located.
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- The same pip value may appear on multiple dice; if both players roll the same value neither player wins.
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- The sets of dice $\\{A, B, C\\}$, $\\{B, C, A\\}$ and $\\{C, A, B\\}$ are the same set.
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For $N = 7$ we find there are 9780 such sets.
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How many are there for $N = 30$?
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# --hints--
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`nontransitiveSetsOfDice()` should return `973059630185670`.
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```js
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assert.strictEqual(nontransitiveSetsOfDice(), 973059630185670);
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```
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# --seed--
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## --seed-contents--
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```js
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function nontransitiveSetsOfDice() {
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return true;
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}
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nontransitiveSetsOfDice();
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```
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# --solutions--
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```js
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// solution required
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```
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