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Problem 376


Problem 376


Nontransitive sets of dice

Consider the following set of dice with nonstandard pips:

Die A: 1 4 4 4 4 4
Die B: 2 2 2 5 5 5
Die C: 3 3 3 3 3 6

A game is played by two players picking a die in turn and rolling it. The player who rolls the highest value wins.

If the first player picks die A and the second player picks die B we get
P(second player wins) = 7/12 > 1/2

If the first player picks die B and the second player picks die C we get
P(second player wins) = 7/12 > 1/2

If the first player picks die C and the second player picks die A we get
P(second player wins) = 25/36 > 1/2

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.

We wish to investigate how many sets of nontransitive dice exist. We will assume the following conditions:

  • There are three six-sided dice with each side having between 1 and N pips, inclusive.
  • Dice with the same set of pips are equal, regardless of which side on the die the pips are located.
  • The same pip value may appear on multiple dice; if both players roll the same value neither player wins.
  • The sets of dice {A,B,C}, {B,C,A} and {C,A,B} are the same set.

For N = 7 we find there are 9780 such sets.
How many are there for N = 30 ?


非传递性骰子集合

考虑如下的非标准骰子集合;

骰子A:1 4 4 4 4 4
骰子B:2 2 2 5 5 5
骰子C:3 3 3 3 3 6

两名玩家玩一个游戏,他们轮流选择一枚骰子,然后掷出,点数高者获胜。

如果先手玩家选择骰子A,而后手玩家选择骰子B,我们有
P(后手玩家获胜) = 7/12 > 1/2

如果先手玩家选择骰子B,而后手玩家选择骰子C,我们有
P(后手玩家获胜) = 7/12 > 1/2

如果先手玩家选择骰子C,而后手玩家选择骰子A,我们有
P(后手玩家获胜) = 25/36 > 1/2

因此,无论先手玩家选择哪个骰子,后手玩家总能找到一枚骰子,使自己有超过50%的概率获胜。
拥有这一性质的骰子集合被称为非传递性骰子集合

我们想要研究存在多少种非传递性骰子集合。首先,我们假设以下条件成立:

  • 有三枚六面骰子,每一面上的点数在1至N之间,包括1和N。
  • 只要骰子上的点数均相同,无论点数标在哪个面上,都被视为相同的骰子。
  • 多个骰子上可以有同样的点数;如果两名玩家掷出同样的点数则均不获胜。
  • 骰子集合{A,B,C}、{B,C,A}和{C,A,B}是相同的集合。

对于N = 7,我们能找到9780种这样的集合。
对于N = 30,有多少种这样的集合?