Sicherman dice - Wikipedia

Sicherman dice /ˈ s ɪ k ər m ən / are the only pair of 6-sided dice that are not normal dice, bear only positive integers, and have the same probability distribution for the sum as normal dice.

A standard exercise in elementary combinatorics is 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 {\displaystyle n} :

Crazy dice is a mathematical exercise 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. The Sicherman dice are crazy dice that are re-labeled with only positive integers. (If the integers need not be positive, to get the same probability distribution, the number on each face of one die can be decreased by k and that of the other die increased by k, for any natural number k, giving infinite solutions.)

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.

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