Shuffled

Playing cards have been around, in their current 52-card deck incarnation, for roughly 500 years.  Let’s assume, incorrectly, that the current world population — roughly 6 billion people — has been constantly at that level over the course of those five centuries.  Doing a bit of easy math, over the course of that time period, we’ve experienced 3 trillion person-years in our fictitious example.  Or, to make it an even bigger number, roughly 9×10¹⁹person-seconds.   That’s a nine, followed by 19 zeroes.

If every human alive for those 500 or so years shuffled a deck of cards, one deck per second, we’d end up with that many decks shuffled.  That’s a huge number.  90,000,000,000,000,000,000.

And almost certainly, each one of those decks, post-shuffle, would be unique.

The number of possible permutations — arrangements, basically — of a deck of cards is 52! (“fifty two factorial”), or approximately 8×10⁶⁷.   That’s such a long number, it breaks the margins of this page, so forgive the space:

8,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000.

Now that is a huge number.  How huge?  It is, quite literally, beyond astronomical — according to a team of Australian astronomers, the number of stars in the observable universe is, comparatively, smaller, at 7×10²² of them.

In order to achieve even a 50% chance of a permutation repeating, one would need to perform 9×10³³ shuffles.  And the entire world, again, can perform “merely” 9×10¹⁹.   In other words, every time you shuffle a deck of cards, you are almost certainly creating an arrangement which the universe has never before seen.

Bonus fact: 90,000,000,000,000,000,000 is pronounced “ninety quintillion.”

From the Archives: Oneteen and Twoteen: How the names for the numbers 11 and 12 came to be.

Related: The only thing potentially rarer?  (No, not really.)  The cult classic book, “Cards as Weapons,” by Ricky Jay.  From the product description on Amazon: “The author of the critically acclaimed ‘Learned Pigs and Fireproof Women,’ a nationally known sleight-of-hand expert, movie actor and magician extraordinaire, presents a stylish parody of self-defense books that demonstrates how ordinary playing cards can be used as a means of protection. Photos.”  The only problem: The book’s print run was tiny; it typically sells for over $100.