The mind-bending world of nontransitive dice

This post originally appeared in my newsletter, The Magnet.

Jack Vance's Tales of the Dying Earth trilogy is set billions of years in the future when our sun is a red orb on the verge of blinking out. Earth's few thousand remaining inhabitants practice different kinds of magic, including a "strange abstract lore" called "mathematics":

"Within this instrument… resides the Universe. Passive in itself and not of sorcery, it elucidates every problem, each phase of existence, all the secrets of time and space… where equations fall away to elements like resolving chords, and where always prevails a symmetry either explicit or multiplex, but always of a crystalline serenity."

Nontransitive dice feel like magic to me. I first heard mention of them in William Poundstone's 2005 book Fortune's Formula, where he recounted a 1968 meeting between Warren Buffet and the mathematician/hedge fund manager Edward O. Thorp:

Thorp and wife met Buffett and wife for a night of bridge at the Buffett's home in Emerald Bay, a community a little down the coast from Irvine. Thorp was impressed with Buffett's breadth of interests. They hit it off when Buffett mentioned nontransitive dice, an interest of Thorp's. These are a mathematical curiosity, a type of "trick" dice that confound most people's ideas about probability.

That was all Poundstone had to say about them, but I was intrigued and looked them up. At first glance, nontransitive dice look like ordinary dice, but you can see how they are numbered differently:

What makes these dice special? Here's a scenario that reveals their surprising property:

Show your friend three colored dice. Point out that they are not like regular dice.

Ask your friend to select a die. If they choose black, pick red. Roll the dice 20 or 30 times and keep score. You will win about ⅔ of the rolls.

Now give your friend a chance for a rematch. Your friend will probably ask for the red die since it beat the black die. Give it to them, and select the yellow die for yourself. You'll win about 7/12 of the rolls.

There's nothing remarkable about this so far. But what happens next is the part that "confounds most people's ideas about probability."

Your friend will likely think, "If the red die beats the black die, and the yellow die beats the red die, then the yellow die must be the strongest die." They will choose yellow and you'll select the black die. And once again, you will win about 7/12 of the rolls.

Red beats black. Black beats yellow. Yellow beats red!

In fact, no matter which of the three dice your friend picks, you can always pick one that will beat it! It's like playing Rock Paper Scissors and knowing in advance which one your friend will use. How can this be possible?

Take a look at the figure above. It shows the outcomes for all 36 possible rolls between each pair of dice. Red beats black 25 out of the 36 possibilities. Yellow beats red 21 out of 36. Black beats yellow 21 out of 36. Even though this figure explains why the dice behave as they do, they still feel like magic to me.

I made a 3D model of the dice and printed them on my 3D printer (they are in the photo at the top). You can also make your own using square craft cubes or order pre-made sets online.

Many different kinds of nontransitive dice have been invented over the years. Wikipedia has a good article describing them.

(Portions of this were adapted from my book, Maker Dad)

See also: Tenzi is a great family dice game