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

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.”

Mixing Cold Spots

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 became a professional mathematician. Card tricks continue to play a role in his research.

Caroline Gates

Cutoffs are similar to phase transitions in physics, such as the sudden crystallization of liquid water into solid ice at zero degrees Celsius. But cutoffs occur in the specific mathematical context of “ Markov chains ,” mathematical models that probabilistically describe how a system (like a deck of cards) moves between different configurations.

Cutoff phenomena, as their name suggests, happen in much the same way as Ernest Hemingway famously described going bankrupt: gradually, then suddenly. And while cutoffs are ubiquitous — they’re expected to occur in “most large, complex systems,” according to Sellke — it’s also hard to prove general theorems about them. “For most problems where one thinks there is a cutoff,” said Laurent Saloff-Coste , a mathematician at Cornell University who has collaborated with Diaconis, “one doesn’t know how to prove it.”

That’s why the “seven shuffles are enough” theorem was such a big deal. Bayer and Diaconis — who as a teenager ran away from home to apprentice with a magician specializing in card tricks , before becoming a renowned mathematician — didn’t just prove the existence of a precise cutoff in a real-world system. They provided a single formula for where that cutoff should be, and that formula worked for decks of any size.

Yet terms and conditions also apply. One: The riffle shuffle has to follow a realistic but strict model where cards are randomly interleaved from the left or right pile one by one. (Each card gets dropped from either the left or the right pile with a probability that’s proportional to the number of cards remaining in that pile. This means that the cards don’t simply alternate between left and right, which would result in a predictable structure; instead, the order might go “left, right, right, left, right, left, left.”)

Two: The deck has to be cut more or less in half before shuffling.

“All of our analysis depends on those details,” Diaconis said.

In 1999, Steven Lalley , a mathematician at the University of Chicago, attempted to loosen those constraints by seeking a cutoff proof for riffle shuffles that didn’t start with roughly evenly cut decks. “It seemed natural to me to ask — there are some people who tend to cut the deck a little higher or a little lower,” he said.

These less evenly cut decks have sets of cards that tend to stay in the same relative order even after multiple shuffl…