Seven Perfect Shuffles Randomize a Deck of Cards. But How Many Sloppy Ones?

Samuel Velasco and Chris Young/Quanta Magazine; all inset illustrations from The Expert at the Card Table by S.W. Erdnase.

In 1992, mathematicians famously proved that seven “riffle shuffles” — the kind where a player splits a deck of cards into two piles, then uses their thumbs to interleave them back together in a zipperlike motion — are enough to mix up the deck.

When Dave Bayer and Persi Diaconis came up with this proof, they also revealed something surprising about what happens along the way: At first, the cards stay relatively orderly. But with that seventh shuffle, the deck suddenly tips into a highly unstructured state. This kind of behavior, called a cutoff phenomenon, is of interest beyond cards, and many dynamical systems — including “spin glasses” in condensed matter physics — are believed to exhibit it.

Unfortunately, Bayer and Diaconis’ proof — referred to by some as a mathematical miracle — only works if you adhere to some rigid constraints about how to cut and shuffle the deck. If you shuffle more like a middle schooler than a magician, the result doesn’t hold.

Now three mathematicians have finally extended the finding to less precise shuffles. Mark Sellke, a Harvard University statistician currently on leave to work at OpenAI, along with Jialu Shi and Jiamin Wang (graduate students at the University of Cambridge and Princeton University, respectively), proved that a cutoff phenomenon exists for riffle shuffling even when you don’t cut the deck into two nice, even piles.

Diaconis was effusive about the update to his work. “It’s a fresh idea, and it’s remarkable that something like that would work as effectively as it does,” he said. “It’s a brilliant piece of mathematics.”

To call the humble riffle shuffle “complicated” sells it absurdly short. The number of possible arrangements for an ordinary deck of cards is 52 factorial — that is, 52 × 51 × 50 × … × 3 × 2 × 1, or (roughly speaking) an 8 followed by 67 zeros, close to the estimated number of atoms in our galaxy. Another way to put the figure into context: Every time you shuffle a deck of cards, you produce a configuration that has almost certainly never existed before, and never will again.

But mathematical interest in card shuffling goes beyond its combinatorial complexity. Back in 1981, Diaconis and Mehrdad Shahshahani discovered cutoff phenomena in the context of card shuffling — after which mathematicians started to uncover them all over the place.

Persi Diaconis ran away from home when he was 14 years old to work with a magician. He returned to school 10 years later and b