Balanced Non-Transitive Dice

TL;DR

This paper investigates balanced non-transitive triples of dice, proving their existence for all sizes, and develops an efficient algorithm to verify their properties, with extensions to larger sets of dice.

Contribution

It establishes the existence of balanced non-transitive dice triples for all sizes and introduces an $O(n^2)$ verification algorithm, extending the concept to larger sets.

Findings

Balanced non-transitive dice triples exist for all $n \\geq 3$

An $O(n^2)$ algorithm can verify balanced non-transitive triples

Extensions to larger sets of dice are possible

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.