Abnormal normal dice

A pair of Sicherman dice have numbers ranging from 1 to 8 on their faces, but roll two of them together and they produce the same totals as a pair of normal dice.

Normal dice
Photo by CHUTTERSNAP on Unsplash

Roll a normal cubic die and you have an equal chance of getting a 1, 2, 3, 4, 5, or 6. Roll two normal dice and add the result, and you get a kind of triangular probability distribution. There’s only one way to get 2 (1+1) or 12 (6+6), but there’s six different ways to get a total of 7 (1+6, 2+5, 3+4, 4+3, 5+2, and 6+1):

Dice probabilities
Tim Stellmach, Public domain, via Wikimedia Commons

In 1978, a guy named George Sicherman invented some abnormal dice. They come in pairs. One Sicherman die has the following numbers: 1, 3, 4, 5, 6 and 8. The other die has these numbers: 1, 2, 2, 3, 3, 4. Taken separately, the dice can produce a range of numbers from 1 to 8. But taken together, these two dice have a rather remarkable property: they behave exactly like normal dice.

The highest total you can get from a pair of Sicherman dice is 12: 8+4. The lowest is 2: 1+1. There are six different ways to get a total of 7: 3+4, 4+3, 4+3, 5+2, 5+2, and 6+1. In fact, the probability of getting any total is identical to normal dice. Pretty neat.

You can achieve the same result with a few different number combinations – but all of the other combos use either zeros or negative numbers as well. When it comes to positive whole numbers, the Sicherman combination is unique.

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