Cyclotomic polynomials and nonstandard dice

In this paper, we consider a broad generalization of a problem which first appeared in Scientific American. The original problem was to find all possible ways to label n cubes with positive integers so that the n cubes, when thrown simultaneously, will yield the same sum totals with the same frequency as n ordinary dice labelled 1 through 6. We investigate the analogous problem for n dice, each with m labels. A simple, purely algebraic characterization of solutions to this problem is given, and the problem is solved for certain infinite families of the parameter m. Several results on the general problem are included, and a number of avenues for further research are suggested.