Revisiting Dice Relabeling using Cyclotomic Polynomials

TL;DR

This paper investigates the relabeling of dice without altering sum frequencies, focusing on cases where the number of dice is two and the die size is a product of three primes, using cyclotomic polynomials.

Contribution

It provides a new method for decomposing dice and extends previous results on relabeling, especially for prime power die sizes.

Findings

Derived formulas for relabeling when m is a product of three primes.

Developed a decomposition method for two dice of different sizes.

Refined existing results for prime power die sizes.

Abstract

We continue the exploration of a question of dice relabeling posed by Gallian and Rusin: Given $n$ dice, each labeled 1 through $m$, how many ways are there to relabel the dice without changing the frequencies of the possible sums? We answer this question in the case where $n = 2$ and $m$ is a product of three prime numbers. We also explore more general questions. We find a method for decomposing two $m$-sided dice into two dice of different sizes and give some preliminary results on relabeling two dice of different sizes. Finally, we refine a result of the aforementioned authors in the case where m is a prime power.