A standard pair of dice consists of two identical cubes, each with the integers from 1 to 6 occurring once each. When the dice are thrown, the total on the faces can be any integer from 2 to 12; where the frequency of occurrence are 1 for 2 or 12, 2 for 3 or 11, and so forth up to a frequency of 6 for the total 7. A non-standard pair of dice has a positive integer on each face, the totals on the faces can be any integer from 2 to 12, and the frequencies of occurrence are the same as on a standard dice, yet the numbering is not identical to a standard pair of dice. Show that a non-standard pair of dice exists, and it is unique.

I originally saw this problem in an old Martin Gardner Scientific American column, and I posted it to a Problem of the Month column that I used to run on the Cambridge College Web site. Since that column no longer exists, I thought I would reprint it here.

I really like the problem because:

(1) The result is quite surprising.

(2) It can be solved by an average middle-school student, requiring only some logic and persistence.

(3) There is an extremely neat advanced solution.

For the elementary solution, one way to start is to determine the largest number that can occur on any face.

The advanced solution was suggested to me by the probabilist and friend John Lamperti, and uses generating functions. To see that solution, click here.

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