Sicherman cube

Sicherman dice (after the inventor George Sicherman) are a pair of playing dice , which are labeled so that each diced with this pair of sum appears at the same frequency as a pair of ordinary dice. Otherwise, this property does not provide any further labeling of cubes with positive integers. The Sicherman cubes were made popular by Martin Gardner in 1978. Instead of the numbers 1 to 6, one of the dice is labeled 1, 2, 2, 3, 3, 4, the other with 1, 3, 4, 5, 6, 8:

Sicherman cube + 1 2 2 3 3 4th 1 2 3 3 4th 4th 5 3 4th 5 5 6th 6th 7th 4th 5 6th 6th 7th 7th 8th 5 6th 7th 7th 8th 8th 9 6th 7th 8th 8th 9 9 10 8th 9 10 10 11 11 12

Ordinary dice + 1 2 3 4th 5 6th 1 2 3 4th 5 6th 7th 2 3 4th 5 6th 7th 8th 3 4th 5 6th 7th 8th 9 4th 5 6th 7th 8th 9 10 5 6th 7th 8th 9 10 11 6th 7th 8th 9 10 11 12

So in a game with two ordinary dice, in which only the sum of the numbers rolled is used, you can also use Sicherman dice without changing the probability distribution . However, a double does not occur equally often.

Proof that no further labeling has this property can be provided with the help of the generating function and the unique prime factorization into and circle division polynomials . Even if you have three or more dice, you can obtain all the solutions by replacing one or more pairs of ordinary dice with Sicherman dice. ${\displaystyle x\mapsto (x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6})^{2}}$ ${\displaystyle x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}=x(1+x)(1+x+x^{2})(1-x+x^{2})}$ ${\displaystyle x}$

literature

• Martin Gardner: Penrose Tiles to Trapdoor Ciphers, WH Freeman & Co, New York (1989), ISBN 0-7167-1987-8
• Duane Broline: Renumbering the Faces of Dice, Mathematics Magazine 52 (1979), pp. 312-315
• Joseph Gallian: Cyclotomic Polynomials and Nonstandard Dice, Discrete Mathematics 27 (1979), pp. 245-259

Web links

• Eric W. Weisstein: Sicherman Dice, WolframMathWorld (English)