Calculating Grime Dice Probabilities

I got a set of non-transitive Grime dice from UK enterprise Maths Gear.

I have verified Grime dice with a monte carlo simulation, but it’s not that hard to calculate the actual probabilities.

The procedure is to generate all 36 sums of a particular dice color, then for each of the sums, compare to the 36 sums of the other colors. There’s 5 colors, and we’re concerned with combinations, not permutations, so 5C2 or 10 combinations. For each of the 10 combinations, there are 36 x 36 = 1296 comparisons. Count which color has a greater sum for each of the 1296 comparisons.

As an example, each yellow die has six sides with 3, 3, 3, 3, 8, 8 on the sides. The 36 sums for yellow are 16 sums of 6, 16 sums of 11, and 4 sums of 16.

Each red die has sides with 4, 4, 4, 4, 4, 9 pips. The 36 sums for 2 red dies are 1 sum of 18, 25 sums of 8 and 10 sums of 13.

red yellow red wins yellow wins
1 sum of 18 16 6s, 16 11s, 4 16s 36 0
25 sums of 8 16 6s, 16 11s, 4 16s 25*16 25*16 + 25*4
10 sums of 13 16 6s, 16 11s, 4 16s 10*16 + 10*16 10*4
Totals: 756 540

Red win 756 times, yellow wins 540 time, exactly what the probability program says.

I wrote a program to perform these combination and counts. It would be tedious to do this by hand. You’d have to make 12960 comparisons total, and keep it all straight.

I found that no color combination comparisons are equal: there are no ties for any roll of two pairs.

Combination
red:yellow 756 540
blue:red 765 531
red:magenta 896 400
red:olive 671 625
yellow:blue 720 576
magenta:yellow 768 528
yellow:olive 896 400
blue:magenta 720 576
olive:blue 765 531
magenta:olive 756 540

Grime dice outer cycle

Above, a reproduction of Grime’s 2-dice “winning” cycle. The arrows indicate which color probably beats which color, and they’re labeled with the probability of winning. This is the “outer” cycle of Grime’s diagram. You can see he chose the highest probabilities for the outer cycle.

Below, the “inner” pentacle. The highest probability is red beating yellow at 0.583, the same probability as red beating magenta in the outer cycle.

Grime dice inner pentacle

Poor, lonely olive, doesn’t beat any color in the inner pentacle.