Balanced Non-Transitive Dice

Abstract: We study triples of labeled dice in which the relation "is a better die than" is non-transitive. Focusing on such triples with an additional symmetry we call "balance," we prove that such triples of $n$-sided dice exist for all $n \geq 3$. We then examine the sums of the labels of such dice, and use these results to construct an $O(n^2)$ algorithm for verifying whether or not a triple of $n$-sided dice is balanced and non-transitive. Finally, we consider generalizations to larger sets of dice.