# Magic and the Mathematics of Shuffling Cards Astonishes Math Professor

I’m going to show you a video of magic trick, then we’ll talk about the mathematics of shuffling cards. This is my friend Shiloh Ahlstrand. He’s is a magician member of the world’s most prestigious magic club, the famous Hollywood Magic Castle. He has entertained for all kinds of notable people – for Hollywood’s elite, national and state dignitaries, and even royalty. In this video, he accomplishes a really difficult feat with playing cards that according to mathematics, should be impossible.>>Now I’d like you to help me shuffle these cards up. Take about half the deck. Okay. Just shuffle them up several times. You don’t have to do anything fancy. I’m not.>>Okay>>If you want, you can cut the
cards a few times We’ll shuffle them up a little bit more, but not a lot more, because shuffling is boring>>Alright. Last time.>>Okay. I’ve learned from a friend of mine, who’s a mathematician, that every time one shuffles up
a deck of playing cards, it’s likely that one has come up
with an ordering of cards that is unique in human history. Could I have you come over here? I’m going to use the table. Now, you’ve shuffled your cards, and I’ve shuffled mine.>>Yes.>>Let’s turn over our top cards. Again, the next. And the next. And so on together. Remember, we’ve each shuffled our cards. So, mathematically, this is increasingly unlikely. Yet, with magic, nothing is impossible. Or as a magician, as I like to say, Impossible is nothing!>>Are you kidding me?>>I have goosebumps. That’s weird.>>Wow, that was amazing. We’ll talk about the magic later. But first, Let’s talk about the mathematics of this claim. Is it really true that if you
properly shuffle a deck of cards, it’s likely that you have come up with a unique ordering of cards, that has never been seen before in all of human history? The answer is yes. I like this question because the answer feels so non-intuitive. Surely, somebody somewhere must have shuffled cards into that same order? But the mathematics says “no.” It’s very unlikely anyone ever has. In order to calculate the probability, we need to know two things. One, how many possible orderings
of cards there are. And two: how many times cards have ever been shuffled in human history. Then we divide to find the probability. First, how many different orderings are there? Let’s look at just four cards. If you play around a bit with these cards, you can count and see that there are 24 different ways to put
these four cards in order. Here’s the full list. But that’s tedious. We can use mathematics to count the different orderings like this. Think of each place as an empty location. Into this first spot, we have
four different possibilities. I’ll pick this card. For the next spot, we have three possibilities. I’ll choose this one. Next, there are only two possibilities, and there’s only one possibility
for this last spot. To determine how many
total possibilities there were, just multiply these together: four times three times two times one, which equals 24, as expected. This is called four factorial. If we do this with all 52 cards,
we get 52 factorial, which is this huge number with 68 digits, or about 80 unvigintillion. Every time you shuffle a deck of cards, you’re putting them into one of those orderings. Okay, so let’s compare that to how many times decks of cards has ever been shuffled,
in all of human history. Now, of course, that’s impossible to know, but we can calculate a maximum by overestimating. Standard French-suited playing cards have existed for less than 700 years. If we count 7 billion people shuffling a deck of cards once per second, day and night for the past 700 years, that would be a huge over estimation. But still, that number is less than 155 quintillion, which is a big number. But it’s still way, way smaller than 80 unvigintillion. So, what are the chances that any deck of cards has ever been shuffled into this exact order? Just divide. It’s less than this percent, less than a one in 500 quattuordecillion chance. That means that yes, this is the first time in human history any deck of cards has ever been
in this exact order. So, what was happening Shilo’s video? How in the world is it possible
that all those cards matched up? Well, I don’t know, but doing some similar mathematical calculations tells us it could not have happened randomly. Here’s something you can try yourself: grab a deck of cards, shuffle them up good, and then look at them, and you’ll be the first person
in all of human history to ever see cards in that exact order. And I think that’s pretty magical. Okay, subscribe to see more,
and thanks for watching.