Roll a standard pair of six-sided dice, and note the sum. There is one way of obtaining a 2, two ways of obtaining a 3, and so on, up to one way of obtaining a 12. Find all other pairs of six-sided dice such that:

The set of dots on each die is not the standard {1,2,3,4,5,6}.
Each face has at least one dot.
The number of ways of obtaining each sum is the same as for the standard dice.

Question

Roll a standard pair of six-sided dice, and note the sum.  There is one way of obtaining a 2, two ways of obtaining a 3, and so on, up to one way of obtaining a 12.  Find all other pairs of six-sided dice such that:

The set of dots on each die is not the standard {1,2,3,4,5,6}.
Each face has at least one dot.
The number of ways of obtaining each sum is the same as for the standard dice.

zarkon · Answer

Unique factorization of Z[x]...