Suppose you have a set of 3 dice: A, B, and C. Dice A beats B most of the time, B beats C most of the time, and C beats A most of the time. How is this possible?

Here's how:

  • Die A has numbers 2, 2, 4, 4, 9, 9.
  • Die B has numbers 1, 1, 6, 6, 8, 8.
  • Die C has numbers 3, 3, 5, 5, 7, 7.

A > B for 5 out of 9;

B > C for 5 out of 9;

C > A for 5 out of 9.

A v.s. B, here's the scenario when A wins:

  • Die A: 2, 2, 4, 4, 9, 9.
  • Die B: 1, 1, 6, 6, 8, 8.
  • Die A: 2, 2, 4, 4, 9, 9.
  • Die B: 1, 1, 6, 6, 8, 8.
  • Die A: 2, 2, 4, 4, 9, 9.
  • Die B: 1, 1, 6, 6, 8, 8.

B v.s. C, here's the scenario when B wins:

  • Die B: 1, 1, 6, 6, 8, 8.
  • Die C: 3, 3, 5, 5, 7, 7.
  • Die B: 1, 1, 6, 6, 8, 8.
  • Die C: 3, 3, 5, 5, 7, 7.
  • Die B: 1, 1, 6, 6, 8, 8.
  • Die C: 3, 3, 5, 5, 7, 7.

C v.s. A, here's the scenario when C wins:

  • Die C: 3, 3, 5, 5, 7, 7.
  • Die A: 2, 2, 4, 4, 9, 9.
  • Die C: 3, 3, 5, 5, 7, 7.
  • Die A: 2, 2, 4, 4, 9, 9.
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