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"Haven’t I Seen This Before?": How Random is a Hand of Klondike

By Mike - 10/23/2022

The standard deck of 52 playing cards is nothing short of an engineering marvel. It is light, compact and can be used to entertain in many different ways. You can build a house of cards, as well as play games of Klondike, FreeCell, Spider Solitaire and even War, all with that same deck that fits in your back pocket.

Something you might not have considered, though, are the possibilities contained within that single deck of cards. Have you ever given a second thought to the cards you hold in your hand when you play a game of Klondike Solitaire? Have you ever stopped to think, “Haven’t I seen this arrangement of cards before?” Probably not. Odds are that few people have, but the actual math behind your run-of-the-mill deck of cards is truly astounding.

The possible number of arrangements for a standard deck of cards is described as 52! (fifty-two factorial). That means 52x51x50… etc. until you’ve reached the last card. If you can’t quite wrap your mind around that concept, this article and the associated video are going to blow your mind, but here is what that actual number looks like:

80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000

That’s 8.0658175e+67 or an 8 followed by 67 digits. For reference, there are roughly 8 billion (8,000,000,000) people on earth, the circumference of our planet is 24,000 miles and the distance from Earth to the Sun is about 92,000,000 miles. 52! is such an unfathomably massive number that there are more ways in which a standard deck of playing cards can be arranged than there are atoms on our planet*. You know, the smallest unit of matter that makes up everything we see and touch EVERYWHERE?!?

The above video has an excellent exercise that will help you visualize the enormity of 52! but we are going to take a moment to apply this information to our favorite way of using a deck of cards: playing a hand of Klondike Solitaire.

As you no doubt know, to begin a game of Klondike, you’ll need to create the tableau, which consists of 28 cards laid out in front of you. Using the same concept above, this gives us 28!, or 3.0488834e+29. So, while the actual tableau for a game of Klondike is a little more palatable in terms of randomness, at less than half of 52!, you still need to draw those cards from the original deck.

It doesn’t matter how thoroughly you shuffle those cards or where you draw them from within the deck. The odds that you’ve seen a particular tableau in its exact order in a previous game, no matter how long ago, are infinitesimally small. And, if you think you have, you should probably play the lottery!

* The estimated number of atoms on Earth is: 133,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (133 followed by 48 digits).