Computer-Assisted Results for $n$-Sided Non-Transitive Dice Taking Face Values in $[n]$

Source abstract

We consider non-transitivity among dice with $n$ sides restricted to taking face values in the set $\{1,\ldots,n\}$. We give a computer-assisted proof that $n=6$ is the smallest $n\ge3$ which admits a non-transitive cycle of maximum length $n$ with uniform and minimal winning advantage. For $n=6$, we identify the two unique non-transitive cycles of length $6$. We present computational evidence supporting a conjecture that $n=6$ is the largest $n\ge3$ which admits a non-transitive cycle of maximum length $n$ with uniform and minimal winning advantage.