**From New Scientist #1487, 19th December 1985** [link]

During last Christmas’s intellectual activity — backgammon, ludo, and so on — Pam complained about the unfairness of using a pair of ordinary dice.

“With *this* pair,” she said, “you can only throw a total from 2 up to 12. But I should like to be able to throw any whole number from 2 up to *much* more than that. And with *this* pair I am much more likely to throw some totals that others — I get 7, for instance, six times as often as I get 12. What I want is a pair of dice which will throw every possible total with equal probability. And finally, there’s a 4 on each of these dice, and I object to square numbers. I don’t mind 1 — I don’t think of 1 as really square — but I don’t like 4 and I would equally object to 9 or 16 or 25 or 36.”

So, I have designed a special pair of dice for Pam’s Christmas present this year, which, I am glad to say, meets her wishes entirely. They are six-sided dice of the ordinary shape, with a positive whole number on each face, and they are equally likely to throw any total from 2 to 37 inclusive.

What are the numbers on the faces of each die, please?

There are now 950 *Enigma* puzzles available on the site.

[enigma339a] [enigma339]

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*Related*

This Python 3 program runs in 46ms.

Solution:One die has faces (1, 2, 7, 8, 13, 14) the other has faces (1, 3, 5, 19, 21, 23).There are also 6 other pairs of dice that can make all the numbers from 2 to 37, but that include non-unity square numbers on their faces:

And if we allow 0, instead of just positive numbers we can get three further solutions that don’t use non-unity squares:

and 11 further solutions that do use non-unity squares: