Sicherman dice

3 years ago by in Arithmetic, Arithmetic

Sicherman dice are the only pair of 6-sided dice that are not normal dice, bear only positive integers, and have the same probability distribution for the sum as normal dice.

The faces on the dice are numbered 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8.

Mathematics

Crazy dice is a standard mathematical problem or puzzle 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.

It is a standard exercise in elementary combinatorics 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 n:

n 2 3 4 5 6 7 8 9 10 11 12
# of ways 1 2 3 4 5 6 5 4 3 2 1

A question arises whether there are other ways of re-labeling the faces of the dice with positive integers that generate these sums with the same frequencies. The surprising answer to this question is that there does indeed exist such a way. These are the Sicherman dice.

The table below lists all possible totals of dice rolls with standard dice and Sicherman dice. One Sicherman die is coloured for clarity: 1–2–2–3–3–4, and the other is all black, 1–3–4–5–6–8.

2 3 4 5 6 7 8 9 10 11 12
Standard dice 1+1 1+2
2+1
1+3
2+2
3+1
1+4
2+3
3+2
4+1
1+5
2+4
3+3
4+2
5+1
1+6
2+5
3+4
4+3
5+2
6+1
2+6
3+5
4+4
5+3
6+2
3+6
4+5
5+4
6+3
4+6
5+5
6+4
5+6
6+5
6+6
Sicherman dice 1+1 2+1
2+1
3+1
3+1
1+3
1+4
2+3
2+3
4+1
1+5
2+4
2+4
3+3
3+3
1+6
2+5
2+5
3+4
3+4
4+3
2+6
2+6
3+5
3+5
4+4
1+8
3+6
3+6
4+5
2+8
2+8
4+6
3+8
3+8
4+8

Mathematical Justification

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. Consider the canonical cubical (six-sided) die. The generating function for the throws of such a die is x + x^2 + x^3 + x^4 + x^5 + x^6. The product of this polynomial with itself yields the generating function for the throws of a pair of dice: x^2 + 2 x^3 + 3 x^4 + 4 x^5 + 5 x^6 + 6 x^7 + 5 x^8 + 4 x^9 + 3 x^{10} + 2 x^{11} +x^{12}. From the theory of cyclotomic polynomials, we know that

x^n - 1=\prod_{d\,\mid\,n}^n \Phi_d(x).\;

where d ranges over the divisors of n and \Phi_d(x)\, is the d-th cyclotomic polynomial. We note also that

\frac{x^n -1}{x-1}=\sum_{i=0}^{n-1} x^i=1 + x + \cdots + x^{n-1}.

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

x + x^2 + \cdots + x^n=\frac{x}{x-1} \prod_{d\,\mid\,n}^n \Phi_d(x)

\Phi_1(x)=x - 1\, and is canceled. Thus the factorization of the generating function of a six-sided canonical die is

x\,\Phi_2(x)\,\Phi_3(x)\,\Phi_6(x)=x\;(x+1)\;(x^2 + x + 1)\;(x^2 - x +1)

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 legalmeans 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:

x\;(x + 1)\;(x^2 + x + 1)=x + 2x^2 + 2x^3 + x^4

and

x\;(x + 1)\;(x^2 + x + 1)\;(x^2 - x + 1)^2=x + x^3 + x^4 + x^5 + x^6 + x^8

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.

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