Sicherman dice


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

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. The table shows the number of such ways of rolling a given value :
n23456789101112
# of ways12345654321

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. The Sicherman dice are crazy dice that are re-labeled with only positive integers.
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.
23456789101112
Standard dice1+11+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 dice1+12+1
2+1		1+3
3+1
3+1		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

History

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.
The numbers can be arranged so that all pairs of numbers on opposing sides sum to equal numbers, 5 for the first and 9 for the second.
Later, in a letter to Sicherman, Gardner mentioned that a magician he knew had anticipated Sicherman's discovery. For generalizations of the Sicherman dice to more than two dice and noncubical dice, see Broline, Gallian and Rusin, Brunson and Swift, and Fowler and Swift.

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 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, and
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. with positive coefficients, and with p = 0 and p
Only one such partition exists:
and
This gives us the distribution of spots on the faces of a pair of Sicherman dice as being and, as above.
This technique can be extended for dice with an arbitrary number of sides.