Western Michigan University mathematician Allen J. Schwenk discovered this oddity in 2000: Consider three fair six-sided dice of different colors, marked with the following numbers:
- Red: 2, 2, 2, 11, 11, 14
- Blue: 0, 3, 3, 12, 12, 12
- Green: 1, 1, 1, 13, 13, 13
- The red die beats the green die 7/12 of the time.
- The blue die beats the red die 7/12 of the time.
- The green die beats the blue die 7/12 of the time.
We’ve seen that before. But look at this:
- A pair of green dice beats a pair of red dice 693/1296 of the time.
- A pair of red dice beats a pair of blue dice 675/1296 of the time.
- A pair of blue dice beats a pair of green dice 693/1296 of the time.
The favored color in each pairing has changed! Schwenk writes, “I call this a perverse reversal.”
(And a bonus: It turns out that a pair of Schwenk dice of any one color is an even match against a mixed pair of the other two colors.)
(Allen J. Schwenk, “Beware of Geeks Bearing Grifts,” Math Horizons 7:4 [April 2000], 10-13, via Jennifer Beineke and Lowell Beineke, “Some ABCs of Graphs and Games,” in Jennifer Beineke and Jason Rosenhouse, eds., The Mathematics of Various Entertaining Subjects, 2016.)