May 22, 2010 •

Reversible Non-transitive Dice

Non-transitive dice are a set of three or more dice where die A beats die B, B beats C, and C beats A: “the relation 'is more likely to roll a higher number' is not transitive,” similar to the game Rock, Paper, Scissors.

An example of such “trick” dice are the following three six-sided dice with the numbers 1-18 written on each of the 18 sides. Get some blank cubes and write the following numbers on each side:

1,2,9,14,15,16 on the first die (call this “1” for its lowest number)
3,4,5,10,17,18 on the second (call this “3”)
6,7,8,11,12,13 on the third (“6”)

The Game

According to Wikipedia,

Warren Buffett once attempted to win a game of dice with Bill Gates using nontransitive dice. “Buffett suggested that each of them choose one of the dice, then discard the other two. They would bet on who would roll the highest number most often. Buffett offered to let Gates pick his die first. This suggestion instantly aroused Gates’s curiosity. He asked to examine the dice, after which he demanded that Buffett choose first.”

The game was up at this point. You can't trick your friends if they are clever enough to see that you get the last move, the upper hand. They may not have the right intuition about probabilities but they may see that something is awry if they can never choose last.

Reversibility

I discovered a variation and three separate sets of actual dice (below) that allow the trickster to switch things up slightly. The dice have a special property: if both players roll their die twice and sum, the probabilities of winning reverse (if ties are thrown out and re-rolled).

New Dice Game

These reversible dice allow for a new dice game.

It is already astonishing and unintuitive that a set of 3 dice exist that are not transitive with respect to winning probabilities, but even more astonishing that rolling the “better” die twice against two rolls of the “worse” die (and summing) allows the “worse” die to win the majority of the time (if ties are re-rolled).

The Dice

These sets of dice:

(1,2,9,14,15,16) (3,4,5,10,17,18) (6,7,8,11,12,13)
(1,2,11,12,13,18) (3,4,5,14,15,16) (6,7,8,9,10,17)
(1,6,7,8,17,18) (2,9,10,11,12,13) (3,4,5,14,15,16)

are non-transitive in two directions (independently—in other words, you only need one set of three dice, preferably the first).

Die (3,4,5,10,17,18) beats (1,2,9,14,15,16) when rolled 1x: odds of winning: 21/36, losing: 15/36.
Die (1,2,9,14,15,16) beats (3,4,5,10,17,18) when rolled 2x: winning 675/1296, losing: 609/1296, tying 12/1296.
Die (1,2,9,14,15,16) beats (6,7,8,11,12,13) when rolled 1x: winning 21/36, losing 15/36.
Die (6,7,8,11,12,13) beats (1,2,9,14,15,16) when rolled 2x: winning 635/1296, losing 619/1296, tying 42/1296.
Die (6,7,8,11,12,13) beats (3,4,5,10,17,18) when rolled 1x: winning 21/36, losing 15/36.
Die (3,4,5,10,17,18) beats (6,7,8,11,12,13) when rolled 2x: winning 635/1296 losing 619/1296, tying 42/1296.

The rest of the odds for the second set of three dice:

[3,4,5,14,15,16] and [1,2,11,12,13,18]: 1x: 21, 15 / 36.
[1,2,11,12,13,18] and [3,4,5,14,15,16]: 2x: 641, 621 / 1296.
[1,2,11,12,13,18] and [6,7,8,9,10,17]: 1x: 21, 15 / 36.
[6,7,8,9,10,17] and [1,2,11,12,13,18]: 2x: 633, 611 / 1296.
[6,7,8,9,10,17] and [3,4,5,14,15,16]: 1x: 21, 15 / 36.
[3,4,5,14,15,16] and [6,7,8,9,10,17]: 2x: 699, 561 / 1296.

The last set of three dice:

[1,6,7,8,17,18] and [3,4,5,14,15,16]: 1x: 21, 15 / 36.
[3,4,5,14,15,16] and [1,6,7,8,17,18]: 2x: 641, 621 / 1296.
[2,9,10,11,12,13] and [1,6,7,8,17,18] 1x: 21, 15 / 36.
[1,6,7,8,17,18] and [2,9,10,11,12,13]: 2x: 633, 611 / 1296.
[3,4,5,14,15,16] and [2,9,10,11,12,13]: 1x: 21, 15 / 36.
[2,9,10,11,12,13] and [3,4,5,14,15,16]: 2x: 699, 561 / 1296.

where [dieA] and [dieB] are rolled 1x or 2x: dieA wins, dieB wins / totalGames.