Sicherman dice

Sicherman dice ( after the inventor George Sicherman ) are a pair of dice that are labeled so that each pair of diced with this sum as often as with a pair of ordinary dice turns up. This property provides otherwise, no further labeling of cubes of positive integers. The Sicherman dice were made by Martin Gardner known 1978. Instead of the numbers 1 through 6 is one of the dice 1, 2, 2, 3, 3, 4, the other is labeled 1, 3, 4, 5, 6, 8:

So you can in a game with two ordinary dice, in which only the sum of the numbers of a roll is used, use without changing the probability distribution also Sicherman dice. A Pasch arises, however, not equally common.

A proof that no other label has this property can be provided by means of the generating function and the unique prime factorization in x and cyclotomic polynomials. Even with three or more cubes to obtain all solutions by replacing one or more pairs of ordinary dice by Sicherman dice.