Dice Transitivity

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Roll dice and with the higher die winning. In terms of advantage, the possibilities are (equality), or .


Nontransitive dice have the property , and .

"Go First 3" and "Go First 4" dice have the property of equality (no advantage) with no shared numbers. When picking the first player in a game, each player can roll a die and have an equal chance of getting the highest roll.

In the graphs, hover over a letter to see the die values. Arrows indicate which die has an advantage.


Contributed by: Ed Pegg Jr (December 2020)
Open content licensed under CC BY-NC-SA



Oskar van Deventer developed the set


Eric Harshbarger developed the "Go First 4" dice. Whether there is a "Go First 5" set on 60-sided dice is unsolved.

Bradley Efron developed the set


James Grime developed the set



[1] E. Pegg Jr, "Tournament Dice," Math Games. (Sep 24 2020) www.mathpuzzle.com/MAA/39-Tournament%20Dice/mathgames_07_ 11_05.html.

[2] E. Harshbarger. "Go First Dice." (Sep 24, 2020) www.ericharshbarger.org/dice/go_first_dice.html.

[3] M. Gardner, "Nontransitive Paradoxes," Time Travel and Other Mathematical Bewilderments, New York: W. H. Freeman, 1988.

[4] M. Gardner, "Nontransitive Dice and Other Probability Paradoxes," Wheels, Life, and Other Mathematical Amusements, New York: W. H. Freeman, 1983.

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