52!

I want to show you a trick. I’ll need you to supply me with a deck of cards. Make sure it is well shuffled, please.

Thank you. Here you see I have your deck of cards in my hands. No monkey business. It’s just the way you gave it to me. Shuffle it again if you like. Now here is your question: what are the odds that this exact ordering of cards has previously existed in all the time since the development of the standard card deck? I’m including every time anyone in the world has ever shuffled a deck. It’s probably quite low. Would you guess something like a tenth of a percent chance? A hundredth of a percent?

Some hands of poker are common enough to have their own names, like Wild Bill Hickok’s “Dead Man’s Hand.” Two pair, aces over eights. Perhaps your deck’s twin first appeared in a riverboat poker room on the Mississippi in 1869. Who knows? Maybe old Wild Bill himself was dealt from a doppelganger deck. But probably not. And to say probably not is an understatement. The extraordinarily high likelihood is that your particular deck ordering has never before existed in the history of the universe, and get this, will never recur before the last sun in the last galaxy darkens and drains into the inky void. Even though the science of combinatorics assures that this is so, it still seems odd, doesn’t it? It seems odd because humans find it hard to reason about combinatorics. Don’t feel bad. I’m not sure why, but that’s just how we’re built.

Here, let’s do the math. It won’t hurt. Much.

Instead of a deck of cards, let’s play a guessing game. I’ll think of a number from 1 to 52. Can you guess it? It wouldn’t take you too many tries to guess my number, 37. Let’s say that number corresponds to the Jack of Hearts. What would the guessing game look like with two cards? Now there are 52 x 51 = 2652 possibilities. How long would it take you to guess my number if it was between 1 and 2652? A while. Okay now (you knew this part was coming) how many possibilities are there with 52 cards? There are 52 x 51 x 50 x … x 2 x 1 possible decks. This is denoted as 52 factorial, or 52! And hoo boy, is this number a doozy!

What are the odds you could guess my number now? It’s between 1 and (checks calculator) 8 x 1067? What’s that? You want to see it unrolled in all its glory? Okay fine. Stand back. Could we zoom out the camera a bit, please?

Your odds are 1 in

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

Happy now? I’ll spare you the stacks of words that rhyme with “zillion.” Just be aware that this is a freaking big number. In practical terms, it means your chances of guessing my number are zero. Not a chance. Even if you drove to the moon and back 15 times in a Winnebago filled with frictionless monkeys. The odds that your deck has existed before is so close to zero that it’s pointless to call it anything except zero.

Image by Midjourney

I don’t know about you, but this makes me feel optimistic. The universe is so full of possibility – think of all the magic hiding in a single deck of cards! Right there in the palm of your hand! From 26 letters we can deal not only the works of Shakespeare (typing monkeys notwithstanding) but all past and all future works in English. From 88 piano keys comes a never-ending geyser of tunes. There are only 92 stable elements, yet from this deck we can deal a universe that, all by itself, invented you. And you’re pretty special, so that’s saying something. Sometimes I brandish a deck of cards just to feel, just to wield the incredible life-giving power of combinatorics. There is no end to novelty! You don’t live in a desert. The exits are not blocked and the labyrinth goes on forever. In every conceivable direction there is naught but bewildering delight!

Here. Let me show a card trick. The best kind of magic.