The IBM Ponder This problem for July 2013 throws an 8 sided die 3 times, and can get 120 possible different positive integer sums. If all the faces have positive integer sides, what is the lowest possible value for the highest face? For an $n$ sided die, the maximal number of different sums is a tetrahedral number. Here are my best results for d3-d7.
d3: 1, 2, 5 -- 10 sums
d4: 1, 2, 8, 12 -- 20 sums
d5: 1, 2, 16, 19, 24 -- 35 sums
d6: 1, 3, 12, 27, 43, 46 -- 56 sums
d7: 1, 2, 8, 51, 60, 79, 83 -- 84 sums
The d8 is the contest problem, so don't post that answer. I'm curious about d9, d10, and so on. I used a sieve method, but it doesn't scale up well. There is a related question in polynomials
Total[Sign[CoefficientList[Expand[(1 + x^2 + x^8 + x^12)^3 ], x]]]
The above 4 term binary polynomial has 20 coefficients in its cube, which is maximal.
Is there an elegant way to find d9, d10, d11, d12, and so on?
George Sicherman (of Sicherman dice fame) sent me his best results for d9 and d10
1 2 19 93 133 162 200 204 210
1 4 5 14 47 156 216 270 332 347
These involved a large search, no nice method for finding these is known yet.