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Problem of the Week 797:

Weird Dice

Roll two ordinary dice and look at the sum of the numbers rolled. The number of ways of getting a sum equal to 2 is 1 (1+1). The number of ways of rolling a sum equal to 3 is two (1+2 and 2+1). The number of ways of rolling a sum equal to 4 is three (1+3, 3+1, 2+2), and so forth until the largest possible sum which is 12 and can only occur in one way (6+6).

Do there exist other pairs of 6 sided dice such that

  1. Every side has a positive number of dots.
  2. The set of dots on each die is not a permutation of the ordinary die, i.e., is not {1,2,3,4,5,6} in some order.
  3. The number of ways of rolling a sum with the other pair of dice is the same as the number of ways of rolling a sum with the ordinary dice, as explained above.

© Copyright 1996 Stan Wagon. Reproduced with permission.

The Math Forum

1 October 1998