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 |
Read more about this topic: Sicherman Dice