Sicherman Dice

The subject of Sicherman dice began with a question posed and answered by Colonel George Sicherman which was originally reported in the February 1978 issue of Scientific American. Sicherman wanted to determine if there was another way to label a pair of sixsided dice, using only positive integers such that the probabilities of the sums would be the same as a standard dice. Sicherman discovered that, in fact, there is one other way. These are the Sicherman dice, one of which is labeled 1, 2, 2, 3, 3, and 4 and the other is labeled 1, 3, 4, 5, 6, and 8. The tables below show the sums of the regular dice and Sicherman dice, respectively. Looking at these tables it can be seen that for the all of the possible sums of the two dice, 2,3,4,...,12, the probabilities are 1 36 , 2 36 , 3 36 , 4 36 , 5 36 , 6 36 , 5 36 , 4 36 , 3 36 , 2 36 , and 1 36 .