Verifying Grime Dice Experimentally

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

I have verified Grime dice with a monte carlo simulation, and calculated probabilities combinatorially, now I’m going to try to verify them experimentally.

James Grime has a 2-die cycle of who wins:

James Grime’s 2-die game who-wins relationships

There’s 2 cycles in that diagram. I’m interested in the outer cycle, Blue → Red → Magenta → Yellow → Olive → Blue

And a single-die cycle:

single die Grime dice winning cycle

Again, I’m interested in the outer cycle of the single-die game: Blue → Olive → Yellow → Magenta → Red → Blue

The two cycles are in opposite orders.

If I roll the dice 21 times, I’ll be able to pick out a winning cycle each way.

Procedure

I’m going to run the procedure with 5 dice (one of each color), and with 10 dice (two of each color), so I can see both cycles.

  1. Put 5 (10) dice in a cup.
  2. Shake cup 4 times with hand clamped over the top to prevent dice from spilling.
  3. Dump cup on the white lid of a Sterilite bin. It’s got a small lip preventing dice from ending up on the floor.
  4. Record each color of dice score (sum of 2 for the 10 dice experiment).

Repeat steps (1) through (4) 20 times, for a total of 21 rolls.

I’ll compare pairs of colors in the Blue → Red → Magenta → Yellow → Olive → Blue cycle. I won’t compare every combination. The winner of (for example) Blue and Red will be the die (or dice) with at least 11 of 21 victories. At that point I can draw diagrams of experimentally determined winner→loser cycles.

Experimental setup

Gime dice experimental setup

10 Dice

Blue Red Magenta Yellow Olive Blue
9 13 7 11 10 9
4 8 12 11 5 4
4 8 12 6 10 4
9 8 12 11 10 9
9 8 7 6 10 9
14 8 12 6 10 14
4 13 7 6 5 4
14 13 12 11 10 14
9 8 7 11 10 9
14 8 7 6 10 14
9 8 12 16 10 9
9 13 7 6 10 9
9 8 7 11 5 9
9 8 7 11 10 9
4 8 2 6 10 4
4 13 7 16 5 4
14 8 7 11 10 14
9 13 12 11 5 9
9 8 7 6 5 9
14 8 12 11 5 14
9 13 12 11 10 9
  • Blue vs Red: blue, 12:9
  • Red vs Magenta: red, 15:6
  • Magenta vs Yellow: magenta, 13:8
  • Yellow vs Olive: yellow, 15:6
  • Olive vs Blue: olive, 13:8

2-die color winning cycle

Grime dice winning cycle, 10-die experiment

That matches reasonably well with calculated probabilities:

Grime dice outer cycle

I can not convince Circo to locate the colors consistently, but if you mentally rotate the graphs by a color, you can see the colors appear in the same order, and the colors win in the same direction.

5 Dice

Blue Red Magenta Yellow Olive Blue
7 4 1 3 5 7
2 4 1 8 5 2
7 4 6 3 5 7
2 4 6 8 5 2
2 4 6 8 5 2
2 9 1 3 5 2
2 9 6 3 5 2
7 4 1 3 5 7
2 4 6 3 5 2
2 4 6 3 0 2
2 4 1 3 5 2
7 9 6 3 5 7
7 4 6 8 9 7
7 9 1 3 5 7
7 4 6 3 5 7
2 4 1 3 0 2
2 4 1 3 5 2
7 4 1 8 5 7
7 4 6 8 5 7
2 4 6 8 5 2
7 9 1 8 5 7
  • Blue vs Red: red, 14:7
  • Red vs Magenta: red, 12:9
  • Magenta vs Yellow: yellow, 16:5
  • Yellow vs Olive: olive, 12:9
  • Olive vs Blue: blue, 11:10

Single die color winning cycle

Grime dice winning cycle, 5-die experiment

This is not good agreement with calculated probabilities, or monte carlo simulation. Red beats both blue and magenta. The house usually, but not always, wins, I guess.