Sicherman Dice Sum Factorization

worked-example

Consider standard six-sided dice, each with faces (1, 2, 3, 4, 5, 6). We can represent each die with a generating polynomial where the coefficient of x^k is the number of faces with value k.

Standard Die:
P(x) = x + x^2 + x^3 + x^4 + x^5 + x^6

P(x) can be factored as:
P(x) = x(1 + x)(1 + x^2 + x^4) = x(1 + x)(1 - x + x^2)(1 + x + x^2)

When rolling two standard dice, the sum distribution is given by:
S(x) = P(x) * P(x) = x^2(1 + x)^2(1 - x + x^2)^2(1 + x + x^2)^2

By rearranging these prime polynomial factors, we can create two different polynomials with positive integer coefficients (representing faces):

A(x) = x(1 + x)(1 + x + x^2) = x + 2x^2 + 2x^3 + x^4
B(x) = x(1 + x)(1 - x + x^2)^2(1 + x + x^2) = x + x^3 + x^4 + x^5 + x^6 + x^8

These polynomials correspond to the faces of Sicherman dice: Die A has faces (1, 2, 2, 3, 3, 4) and Die B has faces (1, 3, 4, 5, 6, 8). Since A(x) * B(x) = S(x), they reproduce exactly the same sums!

experiment

Explore this convolution interactively: https://riano.app/module/sicherman-dice-sum-factorization

1. Select "Standard" in the "Dice pair" control. Observe that a sum of 7 has a probability of ~0.1667 (6 out of 36 outcomes).
2. Change "Dice pair" to "Sicherman". Note that the histogram remains perfectly aligned with the standard distribution, and the probability of 7 is still ~0.1667 (6 out of 36), even though the individual means are 2.5 and 4.5.
3. Select "Changed last face" in the "Dice pair" control. This changes Die B's faces to (1, 3, 4, 5, 6, 7). The "Same Entire Sum Law" indicates "No", and a sum of 12 now has 0 probability, proving that a specific factorization is required.

limitations

This module explores combinations of dice under strict constraints: faces must carry positive integers, and the dice must be fair (each face equiprobable). If we drop the positive-integer limitation and allow zero or negative face values, there are infinitely many pairs of dice that give the same sum distribution. The model here applies to polynomial equivalence for discrete finite support and does not hold for generic continuous distributions without bounding assumptions.

references

The original mathematical result by Col. George Sicherman was popularized by Martin Gardner. For a primary source exploring its polynomial foundations:

• Gallian, J. A., & Rusin, D. J. (1979). Cyclotomic Polynomials and Nonstandard Dice.https://www.jstor.org/stable/2689786