Dodecahedral dice are elegant. Label such dice with multiple copies of the same label to create dodecahedral d2, d3, d4, and d6 dice, the divisors of 12. We probably want labels running from 1 through N as well as 0 through N-1, though the N faces could be labelled 0/N.
Similarly for icosahedral: d2, d4, d5, d10.
Similarly for the 120-face disdyakis triacontahedron: d2, d3, d4, d5, d6, d8, d10, d12, d15, d20, d24, d30, d40, d60.
Also potentially useful are dice labeled with multiples of a lower die number, allowing rolling two dice and simply summing them instead of having to multiply by a factor (corresponding to radix conversion). The most common example is one d10 labeled 0..9 and another labeled in multiples of 10, 00..90, which are summed to yield a percentage.
This quickly results in a crazy number of dice possibilities: a dN labeled in multiples of M, which, when rolled with a dM, can sum to every number up to N*M. (We need the 0..X-1 labels.) Perhaps avoid combinations which can be handled by a single die.
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