On the loading of dice (Q1296763)

The starting point of this paper are two naive questions in connection with the forging of ordinary dice, namely ``Is it possible to forge the dice such that the distribution of the sum is uniform on \(2,\dots, 12\)?'' and ``Can there exist properly forged dice such that the distribution of their sum is precisely as in the case of unloaded dice?'' Both problems are treated in the more general context of distributions on \(0, 1,\dots, n\) and \(0,\dots,m\) in relation with the associated convolution on \(0,\dots, n+m\). The numbers \(n\) and \(m\) where forging is possible are completely classified. In this characterization roots of unity, certain polynomials (palindrome polynomials) and unique additive representations of integers by certain prescribed numbers play an important role.