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crazy dice
It is a standard exercise in elementary combinatorics to calculate the number of ways of rolling any given value with 2 fair 6-sided dice (by taking the sum of the two rolls). The below table enumerates the number of such ways of rolling a given value :
| # 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 somewhat (un?)natural question is to ask 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
and 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.
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