Sicherman dice are pairs of dice numbered [1, 2, 2, 3, 3, 4] and [1, 3, 4, 5, 6, 8] respectively. They have the nice feature that the distribution of results when you roll them and add them together is the same as for a normal pair of six sided dice. But some games care about rolling doubles, so they don't work the same as regular dice for that purpose.

This suggests a puzzle: color the faces of a pair of Sicherman dice so that rolling a matching color pair works the same as rolling doubles on normal dice. (That is, there is exactly one matching color pair for each even sum: 2, 4, 6, 8, 10, and 12.) How many distinct ways are there to do this? (Up to isomorphism, in other words, the choice of particular colors isn't what's important.)

Some faces might not match any face on the other die. It's convenient to designate one color to be the color that never matches, so we don't have spurious extra solutions where we choose various ways of splitting the never-matching faces into different colors. So let's use black for that. All faces that match no others are required to be black, and black faces are considered to not match each other, in case there happen to be non-matching faces on both dice.

Given this formulation of the problem, I have found four solutions. I think I have them all, but I'm not entirely certain. I hope to get physical copies of some of these solutions made; last year I backed a Kickstarter to produce custom engraved dice, and some of these will probably make it into my order.
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