For those of you unfamiliar with the concept of Sicherman dice, the idea is that you can have an alternate arrangement of numbers on the sides of a pair six-sided dice such that when the pair is rolled together, the probability curve for the total remains the same. Following the instructions in the justification section on the wiki page makes it pretty easy to find alternate configurations for larger n-sided dice.
My question is, is it possible to have a Sicherman trio of sorts, where three non-standard n-sided dice rolled together are equivalent to three standard n-sided diced rolled together. You can accomplish this trivially by taking Sicherman pairs and adding a standard die, but I'm not interested in that solution. I've tried with 8 and 10-sided dice, but there wasn't a non-trivial solution. Is there a better way than brute-force to find a n where a Sicherman Trio of n-sided dice is possible?
Dice sides must be positive integers.