# Dice Transitivity

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.

### DETAILS

Oskar van Deventer developed the set
{{2,14,17},{7,10,16},{5,13,15},{3,9,21},{4,11,18},{6,8,19},{1,12,20}}.
Eric Harshbarger developed the "Go First 4" dice. Whether there is a "Go First 5" set on 60-sided dice is unsolved.
{{4,4,4,4,0,0},{3,3,3,3,3,3},{6,6,2,2,2,2},{5,5,5,1,1,1}}.
James Grime developed the set
{{2,2,7,7,7},{1,6,6,6,6},{4,4,4,4,9},{3,3,3,8,8},{5,5,5,5,5}}.
References
[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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