The Infinite Monkey Cage - How to Beat the House and Win at Games

Episode Date: January 16, 2017

How to beat the house and win at games. Brian Cox and Robin Ince are joined on stage by mathematicians Hannah Fry and Alex Bellos, psychologist Richard Wiseman and games enthusiast Helen Zaltzman, to get their top tips for winning games and solving puzzles. Do mathematicians make better Poker players, or is psychology the key to the ultimate poker face? Will a knowledge of probability give you the ultimate winning strategy for your next game of Monopoly? (the answer is yes!). How old are the oldest puzzles and why do they involve wolves and cabbages? And how have puzzles involving wolves, cabbages and bridges resulted in the development of whole new branches of mathematics. PRODUCER: Alexandra Feachem.

In our new podcast, Nature Answers, rural stories from a changing planet, we are traveling with you to Uganda and Ghana to meet the people on the front lines of climate change. We will share stories of how they are thriving using lessons learned from nature. And good news, it is working. Learn more by listening to Nature Answers wherever you get your podcast. Hello, I'm Robin Ince. And I'm Brian Cox. And this is the Infinite Mug Cage podcast, which is a longer version than the one you hear broadcast on Radio 4. Let me stop you there, because you have to define what you mean, because it could just be longer because you're moving at

high speed relative to the listener. Oh, yeah, I hadn't really thought of that. Well, I suppose longer in terms of the minute measurement. You see, you're getting into trouble now. Oh, this is really much harder than I thought. You can define it in a particular frame of reference. So you can say in this particular frame of reference where the player is at rest relative to the listener, where the player is at rest relative to the listener, then the recording you may have made off the radio is shorter than the recording on the podcast.

Thursday? Is that a frame of reference, Thursday? It's roughly speaking. It's a starting point, isn't it? Yeah. It's quite imprecise. This is the Infinite Monkey Cage extended version. Hello, I'm Robin Ince. And I'm Brian Cox. Turns out we live in a probabilistic universe. What are the Infinite Monkey Cage, extended version. Hello, I'm Robin Ince. And I'm Brian Cox. Turns out, we live in a probabilistic universe. What are the

chances of that, eh? Actually, never say, what are the chances of that, in a room full of mathematicians and scientists, because they will work it out for you. It's 100%. Anyway, the universe is based on quantum theory, and quantum theory is fundamental, then it must be probabilistic. But is quantum theory fundamental? Probably.

Exactly. Anyway, it's too late for Christmas, but today we are looking at the mathematics in psychology of puzzles. How do you beat the banker? How do you come out of the casino with your shirt still on your back? How have puzzles led to mathematical innovations? Today's show is especially for the Radio 4 puzzle-loving listener, and I do mean the Radio 4 puzzle-loving listener, and I do mean the Radio 4 puzzle-loving listener, not the simple spot the difference is that Radio 2 listeners like.

Oh, look, Jeremy Vine, he's only got one eyebrow in that picture. See, we've spotted three differences already and it's only Tuesday. Nor, indeed, the impenetrable puzzles of the Radio 3 listener. If Klaus has three octaves and Jim has a buzzsaw, what Stockhausen piece are they interpreting at the Gdansk Festival of the Avant Garde? You have three minutes. So, sit down by your baize-covered table,

get your cards out and let the games begin, because tonight's panel of poker faces are... My name is Alex Bellos. I write about maths. My book is Can You Solve My Problems? A Casebook of Ingenious, Perplexing and Totally Satisfying Puzzles. And the game I never lose at is Pooh Sticks.

Which is true. My name is Hannah Fry. I'm a mathematician. My latest book is The Indisputable Existence of Santa Claus. And I very rarely play games because they turn me into a bit of a monster. My name is Helen Zaltzman. I'm a podcaster. I make the podcast Answer Me This and The Allusionist and the game I never lose is Connect Four because no one will play me anymore. I'm Professor Richard Wiseman, psychologist and author of 101 Bets You'll Always Win

and the game I never lose is the three knocks game, where you say to a kid at a party, you have to go under a table and remain there while I knock on the table three times, and you knock twice and walk off. And this is our panel. Well, I have a question for Alex, because I can't leave it there, because how can you arrange never to lose at Pooh Sticks?

Well, because I've only played it once, and I won it. And, in fact, now I'm talking about it, it makes me one ever kind of sporting boast, really. I was the Oxford University Pooh Sticks champion, because it was the Oxford University Pooh Sticks annual event, and I won it. Was it only one round? Yeah, not that many people showed up.

It wasn't a very popular fresher, but I showed up, I threw my stick in and got there first. And you've intentionally never played it again, even with your children or anything like that? You won't play it because you do not want to have that. The opportunity has not arisen, but I quite like the fact that I've got 100% record. Mathematically, I'm beginning to lose faith in your statistics

in some of your books, if you're prepared to go, just the once, but nevertheless. But the question was, what do I never lose at? And I do never lose at that. I just don't trust you at all. Richard, I do trust you for... You are very good at confounding people with obscure puzzles. Now, please don't put Brian Cox on the table

and say you're going to knock three times, but have you got a challenge for Brian, a puzzle challenge? Strangely enough, I have. I have a puzzle challenge. Here it is. Here's my bet. I bet I can say 50 words in 60 seconds and not one of them will involve the letter A. Not one of them contains the letter A.

50 words, 60 seconds, sounds impossible, but you can do it. Brian, you're very clever. You're a professor, apparently. So, um... LAUGHTER 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 100, 200,

300, 400, 500, 600, 700, 800, 900, 1,000. 101, 102, 300, 400, 500, 600, 700, 800, 900, 1,000, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, yeah. Well, that's a solution very close to the one I had, which is simply to count from 1 to 50. There's no letter A in any of the words from 1 to 50. Ah. Yeah, I'd worked it out for 10, so I thought, I'll give up there and just go 101, 102, 103. So once he said 1,000, though, so that's...

The lowest number is 101. Oh, I said 1,000. Yeah, you said 1,000. So you threw to round two, but, Brian, I'm afraid you're just going to have to go home. I was close, though. It wasn't bad. I was being very giving. I realised he'd made a mistake, and I thought, you know what, we'll just carry on and pretend that you'd solved the puzzle. There's no way in this show we edit out when he makes a mistake.

Desperate bid to make the boy human. So I should find out, because we'll start off on poker. Hannah, you've written books and done lectures which have shown many different ways, quite inscrutable ways, of using mathematics to be a victor. In terms of using mathematics to be a victor. In terms of using mathematics, what would be the system that you believe would improve for those people listening now who are poker players?

Well, so ultimately, poker is just a game of logic, really. You're playing the probability of you winning against the probability of somebody else winning. So in the very simplest way, what you need to do is work out, based on what cards you have at that moment, how many different cards there are left in the pack that could potentially give you a winning hand.

It's quite easy to count up those cards. It's called counting outs. Also, based on what's on the table at that time, you can work out what the other players around the table are likely to have in their hands that could potentially beat your hand. So putting those two things together, you can work out your probability of winning. And that's kind of fairly simple to do. That's not too hard to do. The sort of next level to that, if you get really good at poker, is that you then need to compare your chances of winning at that moment with how much money you stand to win in the pot

and how much it will cost you to get there, how much you have to bet to leave yourself in the game, and kind of comparing those two odds effectively, the odds of the pot and the odds of you winning at that moment. I mean, then there's sort of so many more levels to that because that's just if you're playing perfectly rationally and you're just applying the rules of probability and nothing else but you also want to bluff a little bit you want to make yourself unpredictable for other players you want to make yourself unexploitable and that

kind of takes you into whole different areas of game theory of psychology as well making sure that you're sort of masking what you're actually doing. So, Alex, would you say that psychology is more important for the poker player than once you've got the basic mathematical rules, that whole kind of poker face stuff? I mean, I wonder, for instance, is it better to have a poker face or is it better to be constantly gurning and making just never-ending ridiculous faces every time you get, like, kind Like Keith Moon drumming or whatever,

constant kind of change of faces, so that people just go, well, and also fear for their own life, I suppose. I think that you need the basis of understanding what the numbers are, but after that, it really is, by the top level, more of a game of psychology. I mean, it's not been the case that all the top professors of mathematics

have gone on to be poker players at all, although there are lots of great poker players who were great maths students. Richard, can you hide your... Is it possible to just, as Robin said, make such random facial expressions

that you can hide everything that happens to you? So you look at your hand, it's a winning hand. Can you always read someone? No, we're very bad at reading one another. So part of the problem is that we think we're good lie detectors and telling that someone else is lying. And in fact, we're pretty terrible. And during a poker game, it's really bad because you're going to want to believe one thing or another. You're going to want to believe your opponent's got a good or a bad hand. And so that's going to mess up your judgment and also they are going to be quite excited either way so if they're bluffing they've got a bad hand or they've got a good hand they're going to be

sweating and breathing heavily and all those sorts of things that people are excited do so there's not much you can do if you do want to cover your lies you can do what you're sort of talking about there which is what's called sort of squishing an emotion where you show a micro expression for just a few moments of happiness or sadness and then squish it with a smile or a silly facial expression it's not such a bad thing to be doing but you're not going to have many friends by the end of the poker game helen you or someone i remember uh years ago i'm sure it was spelling games that you were kind of you were more a linguistic uh games person than a numbers games person. Yeah, well, some letters games like Scrabble are built on mathematical principles, which is why it's not fun.

No offence. But I used to play a lot of Boggle because that rewards people that like words and stack them up. But there's not really the same stakes as poker. But if they started introducing that to Vegas casinos, I'd be quite interested, I think. Your Radio 4 listeners, come on, international boggle for money. I am imagining this has got to happen.

The scoring system for Scrabble is based on a sound mathematical principle, isn't it? It is, yeah. It's not random, it's very carefully worked out. It is, yeah. It's not random, it's very carefully worked out. It is, yeah. So it's based on some analysis of how letters were used back in the 1930s when it was invented. So I think the inventor, who's called Alfred Moshe Butz, incidentally, one of the best names ever,

he went through the New York Times and sort of counted up how different letters were used and then based the point system on that. And it ends up being something very similar to the Shannon entropy, which is used in sort of encoding messages. So it's based totally on the probability of a letter coming out. But actually very recently, because of the huge data set that you have with sort of Google Books and so on,

a couple of different computer scientists redid the analysis of the letter distribution have with sort of Google Books and so on, a couple of different computer scientists redid the analysis of the letter distribution, the sort of frequency of different letters, and found that it's changed quite a lot, actually, since the 1930s. So Z, for example, is used a lot more now than it was before. So there was a suggestion, actually, some of the scoring on the tiles of Scrabble was slightly off

and they should be readjusted to sort of account for how languages have changed. But the Scrabble community looked at all the data and the analysis by the computer scientists and were like, actually, no, leave our game alone. What's the Shannon's entropy? So Shannon entropy, it's a measure of how much uncertainty or how much information is associated with a particular message. So for example, a letter like Z, which is still quite rare, so I think it comes up about 0.07% of the time, if you have a three-letter word that begins with Z, there's not very much uncertainty in what that word could be.

There's only a few that it could be. So there's a lot of information that's encoded in that letter. Whereas if you have a three-letter word that begins with an H, for example, a much more common letter, there's a lot uh there's a lot more uncertainty a lot less information so therefore it's worth a lower Scrabble score. Helen can I just check as a regular Boggle player if you've got a z in the uh the letter you've got there what are you imagining is going to be the best word you can get out of that? In Boggle it doesn doesn't matter. It's just the longest word that is the best. But Scrabble rewards bad words,

because you just have to remember every two-letter word in the language and then you're golden. What's the three-letter word combination with the lowest Shannon entropy? What letter starts three-letter words with the lowest Shannon entropy? And they said Scrabble couldn't work on the radio. What would it be? X. It would start with an X, presumably. X, yeah. X, yeah.

No, lowest. So, lowest and highest, you want really common letters. So, it would be T, E... Oh, the highest, yeah, yeah. Or the blank. Or the blank. Blank. Is there a three-letter word beginning with X?

That was like listening to a John Cage concert. That was fantastic. When you are playing... Going back to the card games, getting away from the spelling for a while, we will come back, don't worry. When you... In terms of you're playing a game, you seem to be on a losing streak, what is the best thing? Again, mathematically, what is the best thing for someone to do in that situation?

I think it would be easy just to say a game which is just rolling a dice. It's just rolling a dice. There's no point really changing your strategy and I think this because rolling a dice has no memory of what came before. If the game is you roll a dice, someone else roll a dice and basically it's who gets the highest. You've just got to keep on going and if you are losing three times don't think oh my god I'm going to lose again. The chance of you losing that fourth time is exactly the same chance as in the beginning. And the idea that you think that the past might somehow predict

the future in a game like that is called the gambler's fallacy. And it's something that we all do, but normally we would do it the other way around, which is we've won a few times, we think, well, hey, I'm really good at this, or actually you're not really good at it at all. It's complete randomness. And that, again, is the gambler's fallacy. Is that, Richard, again, the understanding of psychology, is that really what makes most people gamble,

is that they don't realise that they get caught up in all of these kind of human interpretations of what the situation will lead to. They're on a winning street, they're on a losing street, whatever it might be. It's in that same way we have sometimes people who have certain lucky things they do before they play baseball or whatever it might be uh well i think you know they're after an easy win but you have to remember if you go to vegas you know got those massive casinos all these big chandeliers and you know

that that wasn't built on on the uh the proceeds from the winners um so uh that primarily you you you're going to lose your your shirt That's how it's all stacked up against you. And so we don't like the idea of being in uncertain situations, particularly when they really matter to us. And so we start carrying all these weird superstitious rituals in order to try and affect chance. And so we'll carry our lucky rabbits for it wasn't quite so lucky for the rabbit and all of that. So yeah, I think there's a whole psychology there. And part of it is that you want to get your money back, and so if you've lost your money, you keep on going,

you keep on investing more and more, because at some point you're not going to have any more money, and that's the end of that. So, yeah, I think there is a whole psychology there that keeps people at those gaming tables, unfortunately. Plus, they sometimes spray testosterone on the tables in order to make it more attractive for people. So that's why there's all those paintings that people have

of dogs playing poker. They've been drawn in the same way it's been drawn towards trousers. That's correct, yes. I should say that thing about testosterone. I'm not certain if that's true. But if it is, that would be quite good. Aren't they just kept at the tables by the ebbing away of any other purpose in life.

It's a strong glue. Well, some games are quite social. Craps, for example, they appear to have quite a good time, but everyone else appears to me sort of deeply depressed, just sitting there. I went to Vegas, I've been to Vegas quite a lot, but I never replay. I went to a roulette table once,

and it seemed to me that what happens is you put your money down, something happens, they take your money. That seems to be the... Fun game. And that didn't quite do it for me. Does anyone want to explain the statistics of the roulette wheel? There are even numbers, odd numbers and then the... Yeah, the zero is the

thing that really messes you up. That's the thing that's really in the house's favour because, I mean, I think crazy people bet on individual numbers but um sort of more sane people bet on sort of black red odd even um but because zero is neither that's the thing that skews the probability in in the house but otherwise it would just be like rolling dice just betting on a simple roll of the dice so when you say some people will just go on an individual number, are those people likely to keep going back and playing that way? Is it that thing of the unlucky gambler

who keeps going down to the better shop, that keeps putting money on the horse that doesn't? But they believe that somewhere in their narrative, almost the more that they fail, one day that number is going to be the one that makes them... And they can't give up that addiction. The National Lottery, people do that the whole time. I think that for some people, just the fun

of the thought that you might win something is actually worth the quite small amount of money that you might pay. I think with the lottery, that's interesting because if you choose the same numbers week on week, I think you then become terrified that the one week you don't is the week those numbers are going to come up. So you keep on going. So I think people should

always change their numbers every week. Terrible thing to do. Do not choose the same numbers each week. Does anyone here, again on this panel, as it's mathematician and psychologist, do any of you play the lottery? I only bought one lottery ticket. We were doing a project on psychology of luck and we had a thousand of the luckiest people in the UK.

They said they were lucky. And they all sent in their lottery numbers one week. And so we got this database andiest people in the UK, they said they were lucky, and they all sent in their lottery numbers one week. And so we got this database and we looked at the numbers that they were choosing most frequently and we thought, surely, scientifically, that should be the winning set of numbers. And we put on a pound and bought the ticket and not one single number came up. Devastated.

Helen, having now watched Psychologists, Two Mathematicians, if you are placed in a situation of having to have a card game either with psychologists or mathematicians, on the evidence you've seen so far, and presuming that Richard hasn't chosen the table so you know it's clean, what do you... Which group would you go for? I think I'd go for mathematicians

because I'd be worried that psychologists knew more about me than I know. They know nothing. They know nothing. They know nothing. Well, this is very disillusioning. It's true. I mean, the only thing, if you do psychoanalysis and so on, is that when you know nothing about the person at all, you just simply say, how do you feel about that?

I can't cope with that. I don't want to self-analyse because then it will all come out. And how do you feel about that? Don't know. Oh, no. self-analyse because then it will all come out. And how do you feel about that? Don't know. I can't. No, I've got... I want to talk about monopoly because I know, Hannah, you've been talking about monopoly.

And actually, today, I was wandering around seeing people talking about what this was about, and when I said monopoly, the statistics, the mathematics of monopoly, everybody went, tell me about that. So tell me, because Monopoly seems at first sight like the most random of games.

How can you have a strategy in Monopoly? Well, the trick to winning at Monopoly is noticing that not every square is created equal. Some squares are much more likely to be landed on than others. And the reason for that is because Monopoly has a few curveballs. So throwing three doubles in a row, sending you to jail. The go to jail square. The chance and community chess cards

that kind of send you off around different places. And what that does is tweak the probability of landing in different places across the board. So once you know that you can work out your chances of landing in different places and add into that the value of every property to work out what the best possible squares are.

So it is a non-random game because of things like the jail square. Exactly. So the next question then for everyone, I should say we're recording this approaching Christmas. It will be broadcast in January when everyone will be sick of Monopoly. So give us a clue then. So given that the jail square is the most likely to land on, which is worthless, how should you use that piece of information?

So because the jail square is the most likely to be landed on, that means that there are the most people leaving the jail square than any other square on the board and because you're rolling two dice you're most likely to get a six seven or eight if you roll two dice not every set of numbers that you can roll between two and twelve is equally likely and so that sort of six seven eight throws which very likely, takes you then to the orange set and then round the corner to the red set. And then when you throw again, you end up sort of in red and yellow. So actually the most likely square, the property that's most likely to be landed on in the board, taking all of those things into account, is actually Trafalgar Square. Because there's also, I think it's a community chess card which sends you directly there.

because there's also, I think it's a community chess card which sends you directly there. But the orange set is very well visited as well. And you should explain why six, seven and eight are the most likely numbers for a two dice throw. Yeah, so when it comes to throwing two dice, if you think of throwing a two, there's only one possible way that you can throw a two, which is throwing a one on each dice.

Whereas if you're throwing a seven, there are several different ways that you can get there. You can have a three and a four, a five and a two, which is throwing a one on each dice. Whereas if you're throwing a seven, there are several different ways that you can get there. You can have a three and a four, a five and a two, a six and a one, and so on. And so just the different combinations of how the dice can be thrown changes the chances of each different total dice roll coming up. So it's six, seven...

So I can see how you can generate quite a complex pattern going round and round a board. So there's a periodicity to it., 7, 8. So I can see how you can generate quite a complex pattern going round and round a board. So there's a periodicity to it. There is, yeah. But if dice rolls were the only thing, then after you'd been round the board a few times, after the game really got going, actually those probabilities would end up evening out

so that every square would be equally likely to be landed on. But it's those curveballs that kind of tweak the probabilities. The thing is, though, it's not just the chances of landing on each square that matters, it's also how much money you can make from those squares. Ultimately, monopoly is about sort of, you know, forcing your friends and family into poverty while you get all the money. So once you take the money into account, then actually it depends on how many opponents you have as to what the best possible sets are. So if you only have one or two opponents then it's light blue and orange that are the best possible sets.

If you have two or three then it's orange and red and if you have three or more actually it's the green set that becomes the best possible one to have. But basically forget about Mayfair and Park Lane because the probabilities are just too low and they're too expensive. I think I read that you said that isn't Park Lane the lowest probability square? So how does that work?

Because I couldn't see that immediately. It's just kind of where it's sitting on the board, really. It doesn't have any squares around it that are sort of those curveballs that you're being sent to. So the go-to-jail square, for instance, which is just a few before Park Lane, nobody ever ends up landing on there, so that ends up taking up quite a lot of people away

from ending up in the next throw landing on Park Lane. So only 2.1% of throws end up taking you to Park Lane on the board. Monopoly was invented by a woman who was fiercely against capitalism and thought that people would play Monopoly and then become against anti-capitalist ideas because it was such a horrible system. So it's worked out well, as you can tell. But invented at the turn of the last century

in order to stop people becoming capitalists. And that was... Didn't you tell me that she was a Quaker? I believe she was a Quaker. Ah, right. So when you say, I believe she was a Quaker, we place that in the testosterone tablecloth arena. You know, Pluto was invented to stop people killing other people with tiny bits of lead piping. Actually, the mathematics of Monopoly.

Now, when someone lists that, they might go, why do mathematicians waste time looking at the mathematics of a board game? But I presume behind all of this is something which we can broaden out in terms of looking at life. I think it's a problem to ask why mathematicians do anything, because you've just got to ask why mathematicians do anything, because you've just got to let the mathematicians do what they do, and even if they do it because it's fun,

because they want to be their friends at Monopoly or whatever, because the amount of times that something that a mathematician has discovered that much later has become of fundamental importance to some amazing scientific discovery. So I think that the role of puzzles and games is kind of crucial to inspiring mathematicians to kind of play around with things. And probably the best example of a puzzle or a game that created a whole new branch of mathematics

would be the Bridges of Königsberg puzzle, which was a puzzle in the 18th century um in koenigsberg which has got seven bridges and you had to work out is there a route across all seven bridges back to the start without going over a bridge twice and there isn't and leonard euler who's the kind of the top mathematician of his day one of the greatest mathematicians ever, in order to prove that you couldn't do it, invented graph theory, really. He sort of made an abstraction of what was going on, you know, so for each side of a bridge, that was a node, and then there was like a line going from node to node across the bridges,

and he made what looks now like a kind of a network, and he, with some theorems about how networks work could prove categorically that you couldn't solve the bridges of konigsberg problem and that has become you know an extraordinarily rich and vibrant important field in mathematics i mean all the kind of network stuff that we're talking about now wouldn't be around if it wasn't for leonard euler because he was playing some fun game so so what is graph theory used for today? Tons of stuff. So you can look at how people are connected, sort of social connections. You can look at how

diseases spread through populations using it. You can look at the street network, sort of visualising it through a series of sort of nodes and links rather than actual physical roads. And then you can use that to look at how transport flows, to look at how crime patterns

emerge in a city i mean just the absolutely endless if it is used to do transport flows because that's exactly what the bridges of curlewsburg problem was in the first place so is it is that similar to there was the one when you're a kid i think it's one of the first i can't remember this one it's something like you've got a crocodile and a scorpion and a guinea pig and the guinea pig you know, someone has to get travelled over also with a bag of grain or something. I can't remember. I'm not so hot on the ESOP. You mean the wolf, the goat and the bunch of cabbages?

So there's a traveller, he's been travelling all day, and he's been travelling with his wolf, his goat and his bunch of cabbages, and he gets to a river. He's got to get to the other side. Fantastic. And then he sees there's a boat there, but the boat only has space for himself and one of those items. We're taking a bunch of cabbages as an item. So how does he do it?

Because he cannot leave the goat with the cabbages because the goat will eat the cabbages, and he cannot leave the wolf with the goat because the wolf will eat the goat. How does he do it? And this is a puzzle which was first written down in the 8th century and i would say is probably the most viral in the sense that it's probably spread to more people in the history of civilization than any anyone else but by the 13th century

there's a text that says that every five-year-old in the world can solve this problem this puzzle and now it's you know it's a logic puzzle it's one of the very very first logic puzzles but it's studied in kind of anthropology and kind of culture because it's become a kind of a folk fable in pretty much every civilized every culture in the world so that is something which again we're talking about why do mathematicians play around with puzzles this is something which, again, we're talking about why should mathematicians play around with puzzles? This is something which has brought so many people to enjoy the process of kind of logical thinking and the enjoyment of actually using your brain,

and that's something which I think should be celebrated. Richard, is logical thinking difficult for human beings? Is it sort of an alien response to the world? Something that has to be learnt, taught? Well, I think it is. I mean, I don't think it comes naturally to us. I mean, in this instance, you think, as a psychologist, a man arrives with a wolf, a goat and some cabbages. I'd be thinking, what's going on?

I wouldn't be thinking, how does he get those across? I'd think, what kind of man is this? He's travelling with a wolf and a wolf and some cabbages. Yeah, but he might have been travelling with his wife originally and then he kind of screwed up the going across the river the first time when the wolf's eaten my wife. So it could be that it was a much bigger group initially and this is what he's left with after going, left the wolf there again. See, I think that's a more interesting scenario.

I'd like to explore that as a psychologist. Wouldn't they all have eaten each other before they got to the river anyway? Now we're getting somewhere, you see. But to get back to your question, yes, I don't think we naturally think in these kind of ways. We think sort of emotionally and intuitively. And what's nice about some of these puzzles

is they're counterintuitive. The answer is, oh, my goodness, they're in front of you all the time or whatever. And that kind of makes it kind of appealing. It's like getting a joke. You suddenly realise you can rearrange the world in your head. So I think sometimes the difference perhaps between psychologists and mathematicians is a psychologist eventually makes up the answer

and mathematicians are determined to get it right. There's an element of rigour in mathematics. Unbelievable. He's got a wolf, he's got a goat, he's got some cabbages and no-one's asking the why question. It's a kind of a cliche that mathematics almost

should be, I'm going to say should be useless, but you know what I mean? In a very powerful sense, there's certainly pure mathematicians. There doesn't have to be a contact with the real world in terms of usefulness, although, as you've said time and again, these puzzles lead to useful mathematics.

Is that a fair characterisation for many mathematicians? It's just in this abstract world of puzzles. I think so. Traditionally, maths is divided into pure maths, which is that totally pointless, just playing around with shapes and patterns, and the applied maths, which is trying to solve problems. But actually, you can't really divide the two, I think, at all. And all mathematicians will be a little bit interested in how what they're doing might apply,

but also want to just have fun. I mean, math is kind of just like doodling in your brain with ideas just to see where it goes, because you're all the time kind of creating new languages, and it is kind of playful and creative, and I think that that's something which is great about puzzles, is that it isolates that aspect of mathematics, and I think that you...

I mean, I'm not a professional mathematician, but from what professional mathematicians are telling me, the reasons why they do it and they carry on doing it is that there is always that kind of childlike playfulness that the subject always contains. Yeah, I mean, I think maths is sort of the ultimate playground, really, the ultimate logical playground. Is there a character type, Richard, that goes into mathematics?

Is there a particular...? I mean, where would you even come across a wolf? LAUGHTER I mean, where would you even come across a wolf? Just leave it. Could be a wild boar. A wild boar's not going to eat a goat. It totally changes the plot. They're vicious, wild boars. It's not going to eat a goat. Crocodile.

It's more likely to eat the man, isn't it? You're making it worse. You're saying it's ridiculous he's got a wolf. Why do you think he's got a crocodile? They should have just travelled separately and this would be fine. You see? So I think what's interesting about games, like Monopoly and so on, what I find amazing psychologically is people are so happy when they've won,

they kind of go, oh, yes, look at me, I've won. And you think, hold on a second, a lot of this had to do with the roll of the dice, or the way you shuffle cards, or whatever the game is, chance. You won because of a chance event, but instead of putting anything out there like that, you kind of go, look at me! I've won!

Disgusting. I was thinking about a lot of people say this about the stock market. I don't know if it's true, maybe someone, a lot of people say this about the stock market. I don't think it's true, maybe someone can comment, but there's this famous analysis that you could throw darts at the FTSE 100 and just invest in that and you would do statistically as well over time as someone who thinks that they're reading patterns in the market.

Oh, we didn't experiment years... But that's a character type, isn't it? Yes, that's right, we did an experiment years ago where we had three, we had a professional investor, we had a financial astrologer that looked at when companies were formed and on the basis of that predicted when to invest in them. For the radio listeners, that was a very surprised look. And a five-year-old child. And we gave them all £5,000 to invest and then we tracked them, I think, for ten days and the kid outperformed the other

two. And we said to the astrologer, you know, are you surprised at the outcome? And she said no because the child is Pisces and they're traditionally very lucky. Helen, are games, do you think, in the end the actual, the agony for the loser,

the frequent frustration, again as we're saying, this will be going out a few weeks after Christmas where, you think, in the end, the agony for the loser, the frequent frustration, again, as we're saying, this will be going out a few weeks after Christmas, where family rows over pies in trivial pursuit, over Cluedo, Mastermind, whatever it might be, that actually for the benefit of the limited joy of the victor compared to the agony of the losers, says a great deal about humans? Well, in games like monopoly my tactic is usually get into jail as soon as possible sit out the rest of the game in

there because it is not fun that is a game that rewards people that are very into admin and buying a lot of houses and ruining everyone else's lives and that is too close to reality no i think that in in terms of game-playing, I think Helen's exactly right. It should be as much fun as possible. For me, as a psychologist, it's about bonding. I mean, you don't want to sit around at Christmas or whenever and just alienate everyone else around the table

as you go, yes, I won! You think, well, what kind of achievement is that? So these should be about having fun. That just seems to me the key thing. And my mathematician colleagues, much as I respect them, have taken away a lot of that fun for me. I've got a mathematical puzzle, which was ruined for me by another mathematician, actually.

Here's the puzzle. You say to people, give me any three-digit number, and within ten seconds, I'll multiply that number by seven, then 11, and then 13. And that sounds quite good. So what, do I have to give you three...? Any three-digit number, and within ten seconds, I'll multiply it by seven, 11 and 13 in my head.

399. Sorry, 399. The answer is 399, 399. Because seven times 11 times 13 is 1001, and any number times 1001 is itself repeated. I thank you. APPLAUSE I use that, which I go to schools quite a lot. How much do you earn?

I do a version of that. So I was quite disappointed not to hear a wow. When you do that to 400 15-year-olds, you get a wow. You really get a feeling of excitement. And that's kind of nice to think that I'm here talking about mathematics and they're kind of as excited as they would watching a magician or a comedian. Those patterns in numbers are quite fascinating, aren't they? And then you get the really deep questions about patterns

in the prime numbers, for example. Those are three prime numbers, so the prime factors of 1,001 are 7, 11 and 13. But I think the attraction of that puzzle is not the prime numbers, it's the fact that everyone's thinking, I could do that tomorrow. I reckon I could get a tenner for John in the office getting back for that goat puzzle last week. I'm going to just deviate from this script here.

Do you know what? It's the best thing to do because we've run out of time and now you want to get into the deep sides of mathematics. I just wanted to just talk a little bit about prime numbers, just if you had one minute, because we've talked about primes and they are just fantastic, aren't they? And this... The search for patterns in them and all that, give us just a bit of it. I would just say my thing about prime numbers

is that I would say that it is probably the intellectual quest of civilisation that has been going on for the longest and that has involved the most amount of people, which is the search to try and find order in the prime numbers, because it's kind of really where math begins. It's the first interesting pattern that you see in numbers. That's really where math begins.

And the forefront of where math is right now is still to try and find some really, really simple things, things that anyone could understand about patterns of prime numbers. Maybe Hannah can explain some of that. Well, no, I mean, ultimately, they're the sort of fundamental building blocks of numbers. But when you look at where they're distributed in amongst all the other numbers that have divisors,

it's... I mean, they're just... The pattern is just impossible to get a grip on. It seems so kind of haphazard. Yeah, it really does seem so haphazard. And we get very close to things called Mersenne primes, which kind of haphazard. Yeah, it really does seem so haphazard. And we get very close to things called MERS-N primes, which kind of give you a hint as to where some of them might be lurking amongst all the other numbers.

But we still don't know absolutely for certain exactly where they're supposed to be. But these things, you know, prime numbers, knowing... We know, for instance, that there are an infinite number of them. But knowing the really big ones is just an incredibly valuable quest. I mean, the entirety of sort of internet security

is based on having these enormous, enormous prime numbers, you know, prime numbers that are sort of millions of digits long. And there are sort of computer programmes churning continually all around the world, desperately trying to search for new ones that we can use to sort of keep ahead of internet security. Helen, what for you are the most exciting ideas, again, for kind of luring people into the world of puzzling and mathematics

and problem-solving? Well, you take the goat across first and then the wolf, but then you take the goat back with you, then the cabbages across to where the wolf is, and then the goat is the last trip. Is the right answer. That is... APPLAUSE

I don't accept that. What, you don't accept the system of going across the... No, because wolves eat cabbages. Again, I'm not entirely sure! Wolves eat cabbages, it's well known. Everybody knows that. Google wolves, cabbages, I think we'll know where we'll end up with that one. Right, do skies. Wolves eat cabbages is well known. Wolves eat cabbages. Yeah, yeah, yeah. Google wolves, cabbages. I think we'll know where we'll end up with that one.

Right, do it. So you're always... I'm going to do it. Wolves, cabbages. I was joking about the Google thing, but... No, no, no. Again, as a physicist, we don't really do jokes. Carry on.

This is where David Attenborough is filming all of this. And here we see the physicist using his Google to try and understand wolves. Right, go on. Wolves are merely opportunistic carnivores. When there's a shortage of game, they can survive a short while with a vegetarian diet. Right, there we go. while with a vegetarian diet. Thank you. Right.

There we go. Did it? The ill-posed problem which neglected the possibility that a wolf can be a temporary vegetarian. Right. OK, then, so it turns out your brilliant problem for 1,200 years has been a waste of everyone's time. And it's going to turn out there are eight bloody bridges

at Connersburg as well, isn't it? So... That wouldn't make any difference, would it, going to turn out there are eight bloody bridges at Connersburg as well, isn't it? So, that wouldn't make any difference, would it, if there were eight? It all depends on what rivers they were going across. Ah. If there was a bridge... Oh, you're bringing that up now? The reason why there are bridges is because it's little islands, like, with bridges

across the rivers. It depends where the rivers are going. Oh, yeah, because you had two onto one island. Yes. Right, I think what we've discovered is puzzles really aren't as simple as they might have first appeared. When given a puzzle, ask a lot of questions about the dietary requirement of the animals involved, how many wives the man has, etc.

Was it a Mormon man with a wolf? Was it a motorway bridge or a river bridge? There's a lot of parameters you did not establish at the start, which I think is unfair. What about the psychology of people that, when they can't solve the puzzle, always want to change the question? So, anyway, we asked the audience a question,

and the question was, there's a wolf, there's some cabbages... Puzzles are fun, aren't they? Aren't they fun? So, the audience question was, what game do you think you could beat Brian Cox at and why? So, the answers include... Here's one, the replicant test from Blade Runner, because I'm human. And that's from Robin Ince.

Oh, yeah, I do remember that. Cluedo, because you can always find Professor Cox in the observatory with a telescope. Russian roulette. Oh, my goodness. Russian roulette. Yeah, either way, one of us won't care who won. LAUGHTER Oh, my goodness. Yeah, either way, one of us won't care who won. Don't worry, Brian, Elizabeth will be placed on the list.

So, thank you very much to our guests, who have been Dr Hannah Fry, Professor Richard Wiseman, Alex Bellos and Helen Zaltzman. We're going to now set you some homework. Richard Wiseman, you have a problem set, I believe. I do indeed. You meet a person in the street, you find out that person has two children and one of them is a boy. What are the chances that the other one is a boy?

Well, we will tell you that next week. Brian, have you worked it out yet? Still thinking about cabbages. OK, then. Thank you very much for listening. Goodbye. Turned out nice again. Thank you very much for listening to the extended version of The Infinite Monkey Cage. Now it says here that we're contractually obliged to say that there are other science podcasts available

from the BBC. Oh OK, I'll try and do that. Jim Al-Khalili's Life Scientific. Does that sound upbeat enough? Not really and I don't think it'd have any jokes in it. I don't, you know, I don't think it'd be, I think it's too serious. Yeah, you shouldn't deal with science seriously. It should be flippant

and preposterous. That was the rules of Lord Reith, wasn't it? I think seriously. It should be flippant and preposterous. That was the rules of Lord Reith, wasn't it? I think so. Let's try and channel him. In our new podcast, Nature Answers, rural stories from a changing planet, we are travelling with you to Uganda and Ghana to meet the people on the front lines of climate change. We will share stories of how they are thriving using lessons learned from nature.

And good news, it is working. Learn more by listening to Nature Answers wherever you get your podcasts. you