Re yesterday’s puzzle, you’ll find answers in the comments. (We are blessed with some very smart commenters here at The Big Questions!!)
Commenter Roger Schlafly pointed this Wikipedia article where I was surprised and delighted to see a reference to a paper co-written by my old friend Dave Rusin. I did not remember that Dave had anything to do with this problem, but in retrospect I bet I knew this at one time.
I managed to dig out some notes I jotted down on this subject many many years ago. I have not doublechecked these results, and I can’t completely vouch for the careful accuracy of my younger self, so take these for what they’re worth. But here’s what I once claimed to have proved:
The reason there is exactly one pair of nonstandard six-sided dice is that six is the product of two distinct primes. For the same reason, there is exactly one pair of nonstandard n-sided dice when n is 10, or 15, or 21, or …. For any product of three distinct primes, there are at most 40 nonstandard pairs.
I also found (in what appears to be my handwriting) this chart, which I reproduce with the same caveats:
| Number of sides | Number of nonstandard pairs |
| 1 | 0 |
| 2 | 0 |
| 3 | 0 |
| 4 | 1 |
| 5 | 0 |
| 6 | 1 |
| 7 | 0 |
| 8 | 3 |
| 9 | 1 |
| 10 | 1 |
| 11 | 0 |
| 12 | at most 13 |
| 13 | 0 |
| 14 | 1 |
| 15 | 1 |
| 16 | 9 |
| 17 | 0 |
| 18 | at most 13 |
| 19 | 0 |
| 20 | at most 13 |
| 21 | 1 |
| 22 | 1 |
| 23 | 0 |
| 24 | at most 94 |
| 25 | 1 |
| 26 | 1 |
| 27 | 3 |
| 28 | at most 13 |
| 29 | 0 |
| 30 | at most 40 |
| 31 | 0 |
| 32 | 25 |
Corrections welcome!
