# 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 ?