Sicherman dice

They were invented in 1978 by George Sicherman of Buffalo, New York.

A standard exercise in elementary combinatorics is to calculate the number of ways of rolling any given value with a pair of fair six-sided dice (by taking the sum of the two rolls).

The table shows the number of such ways of rolling a given value

: Crazy dice is a mathematical exercise in elementary combinatorics, involving a re-labeling of the faces of a pair of six-sided dice to reproduce the same frequency of sums as the standard labeling.

(If the integers need not be positive, to get the same probability distribution, the number on each face of one die can be decreased by k and that of the other die increased by k, for any natural number k, giving infinitely many solutions.)

One Sicherman die is colored for clarity: 1–2–2–3–3–4, and the other is all black, 1–3–4–5–6–8.

The Sicherman dice were discovered by George Sicherman of Buffalo, New York and were originally reported by Martin Gardner in a 1978 article in Scientific American.

Let a canonical n-sided die be an n-hedron whose faces are marked with the integers [1,n] such that the probability of throwing each number is 1/n.

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

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.

A pair of Sicherman dice. Opposite faces sum to 5 on the left die, and 9 on the right.
Comparison of sum tables of normal (N) and Sicherman (S) dice. If zero is allowed, normal dice have one variant (N') and Sicherman dice have two (S' and S"). Each table has 1 two, 2 threes, 3 fours etc.