Revisiting Dice Relabeling using Cyclotomic Polynomials

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.