Sicherman Dice - Mathematics

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 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

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