Prompt Images

Okay, so grab a deck of cards if you happen to have one handy. Any deck with 52 cards will do.

Got ‘em? Great. Go ahead and shuffle them for a bit. As well as you possibly can. The key to this trick is to make sure the shuffle is more or less random.

### I have an amazing card trick to show you.

To be clear, I’m no magician. I only recently learned myself from someone on Twitter.

Cool.

### Before I show you the trick, though, we need to talk about the number of ways you can rearrange 52 cards.

Indulge me for a moment? I promise it’s worth the pay off.

Suppose you have N objects. These objects could be cards, or hamsters, or Kardashians. Here’s a question:

How many different ways can you order those N objects?

By “order” I mean arrange them in some way: line them up or stack them. For example, if you consider the three letters ABC, you can order them in six different ways:

• ABC
• ACB
• BAC
• BCA
• CAB
• CBA

As N gets bigger, though, it’s tedious to write out a list of all possible orders to figure out how many possibilities there are.

Thankfully, there’s a fairly simple formula for “the number of ways N objects can be ordered.” I will try to explain where this formula comes from below, but the TL;DR is that the number of ways you can arrange N objects is equal to:

N*(N-1)*(N-2)*… and so on… *3*2*1

For example, let our “objects” be the Kardashian sisters Kim, Khloe, Kourtney, and Kylie. In this case N=4. How many different ways can we order them?  Following our formula above it’s 4*3*2*1=24.

If you want to understand where this formula comes from, consider how you’d build it from scratch.

### First, you ask: How many ways are there to pick the first Kardashian?

Well there are four of them. So there are four possible choices for the first Kardashian. How many ways are there to pick the second Kardashian? Well, if we’ve already chosen one of them, then there are only 3 left. So there are three ways to pick the second Kardashian. Once we’ve picked the first two, there are only two options for our third Kardashian. Once we pick the third Kardashian, there’s only one left.

That is how we arrive at 4 (ways) *3 (ways) *2 (ways) *1 (way) = 24 different ways to order the Kardashians.

This same logic can be applied for any N, so this is how we arrive at the more general formula above.

### A convenient shorthand for formulas like 3*2*1 and 5*4*3*2*1 is something mathematicians call “factorial” notation.

Basically, take any number N and add an exclamation on the end: N!

This is shorthand for N*(N-1)*(N-2)*… and so on… *3*2*1

So 3! = 3*2*1 and 6! = 6*5*4*3*2*1

Now for this card trick, we have 52 cards. If we ask how many different ways there are to order 52 cards the answer is simple: it’s 52! (That is, 52*51*50*…*3*2*1)

### Put another way, there are 52! possible outcomes from shuffling a 52 card deck.

Still have that randomly shuffled deck in your hands? Awesome. Hold on to it for just a little while longer.

I’m trying to show you there’s nothing up my sleeve.

How many times has a deck of 52 cards been shuffled in the history of the world?

Clearly, we have no idea.

But what we can do is come up with an upper bound. A value so big that whatever the true answer is, it’s definitely smaller than our upper bound.

How long have cards been around? Hundreds of years? Thousands? Screw it. Let’s pretend they’ve been around since the beginning of time itself—a cool 15 billion years or so ago.

Next thing to estimate: How many decks are there in the world are there being shuffled at any given moment? Think of all the casinos and random card games among friends being played at any moment. Thousands? Millions? Well, if the human population of Earth is 7 billion or so people, let’s round up and say 10 billion.

### Let’s pretend all 10 billion of those people are playing cards simultaneously.

Each has their own deck.

Okay, now let’s imagine that these 10 billion folks have been doing nothing but shuffling their own deck of cards every single second for as long as cards have existed. Which, we are assuming, to be conservative, is the age of the universe.

There’s a cool website where you can find the age of the universe in seconds.  According to this site, our universe is 436,117,076,900,000,000 seconds old.

That’s a lot of seconds. Now let’s multiply this value by 10 billion. Why? This gives us our estimate for the number of times a deck of cards has been shuffled in the history of the universe, which we are pretending is 10 billion decks a second since the beginning of time.

### So, what’s the actual value for our estimate?

4,361,170,769,000,000,000,000,000

This… is a big number. But it’s also an upper estimate. The true number of times a deck of 52 cards has been shuffled is much, much less.

Here’s the thing, as big as that number appears to be, we’ve already—within this same post—encountered a number that is much, much, much larger.

It’s 52!

### The number of possible ways that a deck of 52 cards could be ordered.

How big is 52! ?

52! =

.

.

.

Wait for it…

.

.

.

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

### This… is a much, much bigger number. It has 68 digits.

By comparison, the number we calculated above for the number of times a deck of cards has been shuffled in the history of the universe has a measly 25 digits. This number is 43 digits bigger!!

Still have that deck?

Good. I’m finally ready to show you the trick.

The deck you are holding in your hands contains a particular arrangement of 52 cards that has—in all likelihood—never before existed in the history of the universe.

It’s never existed in Vegas or Casablanca or in the many bridge games played by grandmother’s throughout history.

### If you need a moment for this to sink in, I totally get it.

If it seems like an insane claim, I get it. 52 cards. How could it be possible to arrive at an ordering of 52 cards that has never been realized at any time in the past?

Because 52! is such an unfathomably large number. Even shuffling 10 billion decks a second for the entirety of the universe’s history would only check off a tiny—like, seriously, minuscule—fraction of all those 52! possible orderings.

### Here’s another way of thinking about it.

By some estimates, there are something like 1025 stars in the known universe. Recall that 52! is a number like 1067. Again, absurdly bigger than 1025. This means that—at least by this estimate—there are vastly more ways of arranging a deck of 52 cards than there are stars in the entire universe.

So, in this admittedly very narrow sense, one could argue that each time you play a game of Hold ‘Em, you are holding an entire universe (of possibilities) in your hand.

I cannot be the only person who thinks this is mind-blowing. Can I?

Jesse Stone

Jesse B. Stone loves science and writing. Apologies if you were looking for the "Jesse Stone" played by Tom Selleck in the CBS movies.