I love this problem, which I found on the Internet many years ago. I suppose you could find a solution by Googling, but that’s of course no fair.
A standard pair of six-sided dice induce a probability distribution on the outcomes 1 through 12: The probability of rolling a 1 is 0, of rolling a 2 is 1/36, of rolling a 3 is 1/18, etc. Is there any nonstandard pair of six-sided dice that induces exactly the same probability distribution? If so, how many such pairs are there?
(A non-standard pair of six-sided dice might have, say, the numbers 1,2,2,3,8,9 on one cube and the numbers 2,3,4,4,4,4 on the other.)
What about the same problem for seven-sided dice, or eight-sided, or n-sided?
