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Intransitive dice
Intransitive dice are sets of dice where the usual ranking breaks down. With three dice, A can beat B more than half the time, B can beat C more than half the time, and yet C can beat A more than half the time. Winning a roll is not a fixed order of strength, so no single die is best. Some sets go further, so that every die has another die that beats it more often than not. Because people expect rankings to be transitive, games built on these dice can be rigged in ways that feel surprising. The effect arises from how the probabilities of each face combine, not from any trick in the dice themselves.
Each die's faces have different values, so the chance that one die beats another depends on the full distribution of faces, not on a single number. Because these pairwise comparisons do not have to agree, a loop of wins can appear. Choosing faces carefully allows a designed cycle where each die beats the one before it.
The dice show that majority preference can be cyclical, which is the same logic behind voting paradoxes and rock-paper-scissors. Game designers and statisticians use such sets to illustrate how comparisons that look transitive may not be. It is a vivid reminder that probability does not always follow everyday ideas of who is stronger.
The same cycle appears in ranked voting, where a majority can prefer A to B, B to C and C to A. It is also the logic behind rock-paper-scissors.
From Wikipedia, the free encyclopedia · text CC BY-SA 4.0
Illustrative photo · Shghasemi, CC BY-SA 4.0, via Wikimedia Commons
