Sicherman Dice - Mathematical Justification

Mathematical Justification

Let a canonical n-sided die be an n-hedron whose faces are marked with the integers such that the probability of throwing each number is 1/n. Consider the canonical cubical (six-sided) die. The generating function for the throws of such a die is . The product of this polynomial with itself yields the generating function for the throws of a pair of dice: . From the theory of cyclotomic polynomials, we know that

where d ranges over the divisors of n and is the d-th cyclotomic polynomial. We note also that

.

We therefore derive the generating function of a single n-sided canonical die as being

and is canceled. Thus the factorization of the generating function of a six-sided canonical die is

The generating function for the throws of two dice is the product of two copies of each of these factors. How can we partition them to form two legal dice whose spots are not arranged traditionally? Here legal means that the coefficients are non-negative and sum to six, so that each die has six sides and every face has at least one spot. (That is, the generating function of each die must be a polynomial p(x) with positive coefficients, and with p(0) = 0 and p(1) = 6.) Only one such partition exists:

and

This gives us the distribution of spots on the faces of a pair of Sicherman dice as being {1,2,2,3,3,4} and {1,3,4,5,6,8}, as above.

This technique can be extended for dice with an arbitrary number of sides.