Here are a few mostly functionally equivalent things:
The first is a pair of perfectly normal dice. The second is a “Merged d6”, a 36-sided die I bought through a crowdfunding campaign. Each of the sides is labeled with a sum of the results of rolling two normal dice. One of each of the even numbers between two and twelve is colored green. These let you simulate rolling doubles, as is required for games like Monopoly: each roll of two of the same number is represented. (The small print size of the numbers and the fact that it takes a moment to figure out which number is on top make this somewhat less practical than the normal dice, but I collect dice for mathematical interest rather than practicality.)
The blue and green dice are Sicherman dice. They are numbered a bit oddly. One of them has faces numbered 1, 2, 2, 3, 3, 4. The other’s faces are numbered 1, 3, 4, 5, 6, 8. The distribution of sums of die rolls is nevertheless the same as that of a normal pair of dice. In fact, it is the only non-standard numbering for a pair of dice with this property where the numbers are all positive integers.
Unfortunately, Sicherman dice don’t work for games that distinguish doubles rolls. Could we design a version of the Sicherman dice that does work for such games? The specially colored faces of the Merged d6 suggested a direction to take. We might color the faces of the Sicherman dice differently, and call a roll a doubles roll if the colors match. How many different ways are there to color the faces so as to produce exactly one match for every even number between two and twelve?
I posed this question on Google+. Joe DeVincentis found that there are sixteen essentially different solutions, of which two are the most interesting to me for designs that I might be interested in having made at some point:
1 2 2 3 3 4 — 1 3 4 5 6 8
This has the advantage of only requiring three colors, and all of the faces that never match are on the same die. It also has an easy mnemonic that you could use even without coloring the faces: one on low die, odd on high die; four on low die, even on high die. Variations where some subset of the never-matching faces are a different color from the others are considered essentially similar here.
1 2 2 3 3 4 — 1 3 4 5 6 8
Here the colors can be ordered from low to high, and have the same order on each die.
The rest of the solutions follow. For clarity, I’ve colored all of the numbers that never match black even when they exist on both dice. We may define a rule that black is not considered to match itself.