Is every card shuffle unique?


While it’s possible that two packs of cards may have been shuffled into the same order, the odds of that having happened are actually tiny and yes, it’s hugely likely that each properly shuffled deck is indeed a unique variation of those 52 cards.

While it seems improbable, given how many packs of cards there are being shuffled in the world every day of every month of every year since playing cards were invented (about 700 years ago), the maths behind it back up the notion that proper shuffles result in unique orders.

Of course, technically, you could take a fresh deck (that is already in order, as boxfresh decks tend to be) and split the deck just once and claim that is a shuffle and yes, the resulting sequence has probably occurred before. But a full proper shuffle is almost certainly unique each time. To start, however, the concept of what constitutes a “proper shuffle” is something that magicians and mathematicians have pondered for years. But in 1990 Dr Persi Diaconis, a former professional magician-turned-professor of statistics at Harvard University, concluded that it takes just seven ordinary, imperfect shuffles to mix a deck of cards thoroughly.

Fewer was not sufficient while more shuffles did not significantly improve the mixing of cards. “The usual shuffling produces a card order that is far from random,’’ he told The New York Times. Most people shuffle cards three or four times, while five times is considered excessive.’’

He further claimed that he knew of people who went to casinos and made money off the back of decks being undershuffled, although now — over two decades later — the vast majority of casinos will use sophisticated shuffling machines rather than relying on croupiers to mix up the deck after a hand is finished. So now we’ve largely established what actually constitutes a proper shuffle, how many permutations of a shuffled deck are there?

It’s actually a good way of illustrating factorials and huge growth. If you have one card there is only one possible order. For two cards it’s 2×1=2 and for three cards it’s 3x2x1=6. There’s a mathematical way to write 3x2x1 and that’s 3! — a factorial. As the number of cards increases, the factorial becomes huge.

So 10! is 3,628,800 permutations and by the time you get up to 20! the result is 2,432,902,008,176,640,000. For those who prefer words, that’s two and a half billion, billion. So for a full deck you’ve got 52 choices for the first card, times 51 choices for the second card, times 50 choices for the third card and so on. So it’s 52 x 51 x 50 x 49 x 48… and so on until the final x 2 x 1 (or 52!) or and the result is 8×1067 possible orderings. Or for those who prefer to see the full number, it’s…


To give you an idea of how big this number is in experiential terms, if a new permutation of 52 cards were written out every second starting 13.8 billion years ago (when the Big Bang is thought to have occurred), that writing would still be going on today and for millions of years to come. Or to look at it another way, there are more permutations of 52 cards then there are estimated atoms on Earth. So yes, it’s very nearly certain that there have never been two properly shuffled decks alike in the history of the world, and there very likely never will be.

Think about that next time you bemoan your hand at poker. It’s not just a result of a bad shuffle, but a uniquely bad one, but on the plus side, at least you won’t experience it ever again.

For Esquire magazine


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