A decades-old proof showed that seven shuffles are enough to mix up a deck of cards. But it requires you to cut the deck with the precision of a professional magician. A new proof gets around that obstacle.

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(opens a new tab) 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.”

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