Is every card shuffle unique?

0:00Thanks for downloading the more or less podcast, with a programme that looks at the numbers in the news, in life and in card shuffling. And I'm Tim Haafard. One of the joys of this programme is constantly being surprised by the subtle power of numbers. Often something that sounds utterly simple becomes infinitely astonishing once you just start doing some counting. Or indeed, some shuffling. Famously, if you shuffle a deck of cards, you will end up with an arrangement that's never been shuffled before. Yeah. Because there are just so many ways you can arrange 52 cards. The probability of someone else having done it exactly the same is basically zero. That is Matt Parker, a maths YouTuber and stand-up comedian talking on his podcast, a problem squared. But can that possibly be right? Can just 52 cards be arranged in so many ways that it means every shuffle is unique?

1:03We invited Matt into the studio to show us his workings. Now, I didn't realise how bold a statement I was making at the time, but I'm here to defend it. In fact, I brought in a deck of cards. Okay, good. So I can shuffle them for you and live on radio. This is just like a nice lazy. A nice lazy overhand shuffle. Yeah, I would expect it better from you, Matt, to be honest. Now, okay. Split the deck. If I can get this to work, we'll do a... Oh, riffle shuffle. That's a midair riffle shuffle, by the way, for those of you listening in audio only. So if people are happy, I've done a couple of these. If we're happy to say that's a random arrangement. No one has ever shuffled a deck of cards into this specific order before. World premiere folks. Exactly. You heard it here first. This is through why you are so confident. If you want to work out the number of ways, you can arrange a deck of cards. You're straining into the maths topic of combinatorics in terms of counting combinations.

2:04And specifically, you're using what's called effectorial, which a lot of people will remember as the little shouty exclamation mark on the calculator. Six. Six. Exactly. And that's... It's such a perfect... There's very little maths notation, which is so perfect as using the exclamation mark for something as startling as factorials. Because factorials make exponentials look puny. Yes. They just get so ridiculously big so quickly. And I think what makes them extra startling is they kind of drop out of seemingly such boring, small, contained situations. So let us return to the boring, small, contained situation. How do factorials tell us something about a deck of cards? So if you just wanted to work out how many ways you could arrange one suit, let's say the 13 clubs. There are 13 cards you could put down first. And then there were 12 you could put down next. And there are 11 you can put down after that. And each time you've got one fewer option, because you've just put a card down.

3:07And so to work out the total number of options you're presented with is 13 times 12 times 11 times 10 all the way down. Yes, which is actually multiplied by one. Which is written 13 exclamation mark. 13 exclamation mark. And it's described as 13 factorial. And do you want to have a guess what 13 factorial is? I'll skip you ahead. I'm going to go for it right now. Yeah. Okay, so one times two is two times three is six times four is 24. Go on the whole way up. Times five is 60 times six is 360 times seven is a bit more than 2000 times eight is like 18,000 ish times nine. So like 160,000 times 10 is about is 1.6 million times 11. We're now about 20 million at 12 is about 200 million 13 is about two three billion. Something like that is very, very close. And as you stubborn on the end there, you're pretty quickly going up by what we call

4:08a mass and order of magnitude. Yeah. The number is getting one digit longer every time. So you have millions, 10 millions, 100 millions billions for a bit. And then that's the factorials increase. And then that gets even faster than that. And so it's just over six billion. So I would say you were spot on at that point. We're not arguing the minor details of how many billions. So that's a lot. It's a lot. All right. And then you think you've got to do that for all. I'm not doing that for all 52. No, no. But it would be a similar thing, right? You could estimate it by, oh, I'm going up by 10 or eventually 100. I know. But actually, you say you could estimate it. But I don't think you can. I think you start to see how that is getting very big. Because we went from five was about 60. Yeah. 120. Something like that. And then we were a billion by 13. So yes, you can see how this is accelerated. It gets big fast and it gets bigger faster as you go along. You could always type it into a calculator. Or could you?

5:09Yeah. A lot of calculators will just quit after a while. If you do work it out properly for 52 cards, you're looking at, well, it's eight. And then another 67 digits after that. Right. So we've kind of gone past giving numbers nice names. Yes. And we just say how many digits they've got. Billions, trillion, quadrillions, quintillion, septic. Yeah. You're so far beyond that. Forget it. It convinced me that the number of different possible combinations of a deck of cards is very, very large. It's very big, very, very, very, very large. Still, the claim that there is no chance. Yes. No. Yeah, absolutely right. Now I should say, because just having lots of combinations is one thing. Yeah. But what we actually care about is have two people shuffled the same deck. Yeah. And now we look at every possible pairing of shuffles that may have happened. Okay. So every time someone has shuffled, they've created an arrangement. Yep. And then every subsequent shuffle might match with that or any other historical arrangement.

6:10Correct. And you've got a lot of chances to hit. There's a lot of chances to hit. You still end up with an outrageously big number. So even if you had 10 billion humans shuffling once a second for longer than cards have been invented, and then you run the combinations on what's the odds any two of those matched. You're still looking at a chance of one in four times 10 to the 26. Right. So outrageously big number. However, not strictly zero. And this is why people get upset when I say it definitely hasn't happened. Here's the thing, because people get stuck on the fact it's not exactly zero. Yes. I would say it's indistinguishable from zero. It's or it's effectively zero. They might be a safer way to put it. And there are shades of being basically zero. Because if you say something's one in a million, that might still happen. Yeah. If you say something's one in a billion or one in a trillion, at some point, we don't

7:11have enough opportunities in our universe for it to happen. And because you get so stuck on people saying it's not actually zero, I've tried doing improbable things to give myself a taste of what it's like. I once flipped a coin 10,000 times to see how often it would land on its edge. And I got 14 edges. Yeah. And I was like, oh, actually, it's unlikely, but it's within the realms of things that can happen. 14 edges. It feels like a lot. It was a lot. It was a lot. It tossed a lot of coins. Exactly. But I was there for multiple days flipping a coin over and over. Yeah. I once kept dealing a deck out like a shovel to deck until I picked two cards at random and they matched color and face value. Yeah. I was in a thousand hours. But you realize pretty quickly when you start trying to do improbable things, the difference between a one in a thousand and a one in 10,000 and a one in a hundred thousand and you realize what you could or could not achieve in a year, you know, a lifetime a month.

8:13And so when you realize the moment you're looking at things that start becoming one in a billion, there's no way you can do it as a human. And once we're looking at probabilities with like 20 plus digits, doesn't matter how many humans in on planets across galaxies, you still end up with a ridiculously small effectively zero chance of it happening. Our thanks to friend of the program Matt Parker. Find him on YouTube as stand up maths. If you've seen a number you think we should look at, email us at more or less at bbc.co.uk. Until next time, goodbye.