Crazy dice
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Crazy dice refers to to a stand mathematical problem or puzzle in elementary combinatorics, involving a re-labeling 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 2 fair six-sided dice (by taking the sum of the two rolls). The below table enumerates the number of such ways of rolling a given value n:
| n | # of ways |
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
| 5 | 4 |
| 6 | 5 |
| 7 | 6 |
| 8 | 5 |
| 9 | 4 |
| 10 | 3 |
| 11 | 2 |
| 12 | 1 |
A question arises as to whether or not there are any 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 re-labeling, via the labeling
- D1 = {1,2,2,3,3,4}, D2 = {1,3,4,5,6,8}.
A pair of dice with this labeling are called a set of crazy dice. It is straight-forward to verify that the various possible sums occur with the same frequencies as given by the above table.
This article incorporates material from Crazy dice on PlanetMath, which is licensed under the GFDL.
