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？