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
$P(\text{second player wins}) = 7/12 \gt 1/2$.
If the first player picks die $B$ and the second player picks die $C$ we get
$P(\text{second player wins}) = 7/12 \gt 1/2$.
If the first player picks die $C$ and the second player picks die $A$ we get
$P(\text{second player wins}) = 25/36 \gt 1/2$.
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 $N = 30$?
solutions/s376.md.