TY - JOUR

T1 - Cyclotomic polynomials and nonstandard dice

AU - Gallian, Joseph A.

AU - Rusin, David J.

PY - 1979/12

Y1 - 1979/12

N2 - 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.

AB - 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.

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U2 - 10.1016/0012-365X(79)90161-4

DO - 10.1016/0012-365X(79)90161-4

M3 - Article

AN - SCOPUS:49249141558

SN - 0012-365X

VL - 27

SP - 245

EP - 259

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 3

ER -