http://plus.maths.org/content/non-transitiv-dice

This article describes a rigged dice game with two or three die, but they’re not ordinary die for they don’t have the values of 1,2,3,4,5, 6. Instead Die A has the values 3, 3, 3, 3, 3, 6: Die B has 2, 2, 2, 5, 5, 5 and Die C has 1, 4, 4, 4, 4, 4. The article explains the mathematical concept of conditional probability upon non-transitive die to show how in the long run, typically more than 20 rolls, Die A always beats Die B, Die B always beats Die C, and Die C always beats Die A.

These die are different from transitive die whose conditional probability dictating the winner does not follow a circular pattern. For transitive die, Die A always beats Die B, Die B always beats Die C, but Die C does not beat Die A. Unlike transitive die, the key winning property of non-transitive dice is their circular pattern.

The article also shows multiple games you can play with your opponents and still outsmart them every time. Such as games with 3 people, and also reversing the pattern of dice to make it seem like wins should turn to defeat, even though the probable win streak still follows a circle pattern.

There are also different types of non-transitive dice such as Efron Dice and Grime Dice. Efron dice uses 4 sets of die, A, B, C, and D where dice A beats B, B beats C, C beats D, and D beats A, they all do so with a probability of 2/3. There are also 2 pairs of dice opposing each other on the circle. Die B actually beats D, but die A and C have a 50-50 chance of winning, with neither dominating. The Efron arrangement does not have the reverse flip property pattern when the number of dice is doubled.

For Grime Dice, there are 5 non-transitive dice, A, B, C, D, E ; which uses values from 0 to 9. Using these dice is a way to play against 2 opponents, let the 2 other players make their choice of the dice first, yours last, whether it is a 1 die or 2 dice game mode.

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