Miwin’s Dice

https://commons.wikimedia.org/wiki/File:Miwin_Wuerfel_Titan.gif
Image: Wikimedia Commons

Physicist Michael Winkelmann devised these nontransitive dice in 1975.

  • Die I has sides 1, 2, 5, 6, 7, 9.
  • Die II has sides 1, 3, 4, 5, 8, 9.
  • Die III has sides 2, 3, 4, 6, 7, 8.

Collectively the 18 faces bear the numbers 1 to 9 twice. The numbers on each die sum to 30 and have an arithmetic mean of 5.

But Die I beats Die 2, Die 2 beats Die 3, and Die 3 beats Die 1, each with probability 17/36.

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