Thank you, everyone, for coming and thank you Mathematical Institute for inviting me. So can you solve my problems? One rule of life is you get a pun in the title. Always put it in. The subtitle of this talk is going to be a short history of puzzles, with occasional Christmas references and some sparkling mathematics. In other words, you talk a little bit about puzzles, but about the history of puzzles.
We're going to do some puzzles, and I might mention Christmas a few times as a Christmas lecture. Now, it's fantastic for me to be here because it is a return to my alma mater with Marta. The crucial words because I was actually born on this site when it used to be the Radcliffe Infirmary.
This is where I was actually born. So it's quite nice to think that, you know, my life, my wife might say that I was actually born in a mathematical institute, but it's quite nice that the place I was tended to. One also is true. I studied mathematics here. It wasn't math and physics, it was maths and philosophy. And rather wonderfully my old philosophy of maths professor is in the audience, so the pressure is on. It's true. I did a lot of maths when I was at Oxford 30 odd years ago.
I also did lots of journalism. I probably did more journalism than I did maths and I was editor of Cherwell, the university newspaper, and in the last ten years I have combined essentially my two passions. So it wasn't really a passion is writing and mathematics so serious passions and this is what I've tried to do.
My first book was Alex's Adventures in Wonderland, where I went around the world interviewing people as if I was a kind of foreign correspondent in the world of mathematics, interviewing people in the world of maths and sort of explaining what they did and mathematical concepts to the general reader that led to the next one. Alex Through The Looking Glass. Then I wrote with a wonderful mathematician, mathematical artist Edmund Harris, two maths colouring books.
I shouldn't have to mention these because even though they are colouring books, you can look at them and you can colour them in essentially the colouring books because bookshops have a table for colouring books. Really what they are, they are gallery, a gallery of wonderful images from all areas of maths. And even though they look the most kind of junior, so to speak, of all the things that I've done, they probably cover the widest and deepest mathematics.
So two or three years ago, I started to write a puzzle column in The Guardian. And yes, it is a lot of pressure to try and find a good puzzle every two weeks. So if anyone here has a puzzle that they've just invented and they want to throw them my way, please come and save me. And this has resulted in two puzzle books Can't solve my problems and Puzzle Ninja. So Puzzle Ninja, which is not really Christmasy, even though it's just out in time for Christmas, it's about Japan.
I went to Japan at the beginning of the year because the puzzle culture in Japan is unlike anywhere in the world. And I found 200 or so handcrafted puzzles and we presented it in this beautiful kind of Japanese style book, which is lovely to hold. And I promise you, some bloke he might have heard of very, very wise man. Mark is the sort I call this book addictive. And it's that's an understatement.
You will start doing this book and time disappears, which is one of the great joys of doing a puzzle, is that once you get into it, the world disappears and there are whatever distraction there is, you can focus. And it's actually it's such a kind of I found a relaxing thing to do puzzles because all those distractions disappear that I'm going to start off with a puzzle, which is the puzzle on the cover of my book, which for the purposes of this talk I'm calling Santa is belt buckle puzzle.
Santa has got to has laid out five shapes, which could be his buckle and his belt. And it's an odd one out puzzle. So one of these shapes is the odd one out. I'm going to give you a few seconds just to think about it, then we're going to try and solve it together. So it's somewhat together. I'm on this side. We'll start with this one here. Put your hands up. If you think the. Oh, the odd one out is the only small one. One person. Hands up if you think the odd one out is the only blue one.
Somewhat. A couple of about three people. Hands up. The odd one out is the only circle. A hands up if you think the odd one is the only one with no border. Everyone who has put their hands up so far. Just listen. I'm going to repeat what I have just said. This is the only the only small one. The only blue one. The only circle. The only one with no border. What is unique about this one here? Nothing. It's not only anything.
There's nothing. It shares every characteristic, every feature, every property it shares with another one. In other words, the odd one out is the one, which is not an odd one out. So, in fact, as well as a puzzle, this is actually a subversive matter puzzle that can be a kind of camp sort of campaign against odd one out puzzles by kind of ridiculing them.
And in fact, that was the reason why it was invented by a Russian-American puzzle expert, Tanya Herrmann, over who wanted it, who doesn't like odd one out puzzles. Wants to use this to show why. Now, I like this because a good puzzle is also a good story. You you take it, it has a beginning. You set the problem and you work it out. You kind of. That there's a path that you need to go.
There might be some false terms. Then you have the kind of the insights and then things are neatly tied up at the end. So I think that's another reason why I like this puzzle, is that it takes us on a journey. It's just that the journey at the end is also really like a joke. Now I feel I should mention something about Christmas now, because it's been about 3 minutes and I haven't. And this is a Christmas lecture. So I want to go back in time to Christmas Day, the year 800.
And if I was in the History Institute, everyone would know exactly what important event happened on Christmas Day, eight hundreds, thankfully. I mean, the Mass Institute, it was when Charlemagne was crowned, was made Emperor of Rome, and he was king of the Franks, but he was in Rome and the Romans made him king of the Roman Empire, the Holy Roman Empire. So he became empire of pretty much all of Western Europe.
And Charlemagne is interesting, not just because his for his sort of political aspect of because he was the head of an intellectual resurgence, the towards the end of the Middle Ages in Europe and his mentor, his educator, his teacher is probably the most important early figure in the history of puzzles. And this person was Alcuin of York. So Alcuin of York from York ended up working for Charlemagne, starting a sort of oboe.
He was teacher at the Palace School in Aachen, where Charlemagne was lived, and he also set up a kind of network of European seminaries. He also invented joined up writing the rules of amazing things. And in 799, he wrote a letter to Charlemagne in which he says, I hear I enclose some arithmetical curiosities to amuse you. The arithmetical curiosities were discovered about 100 years later and attributed to Alcuin.
The propositions and facts on the Classics Institute because you do me from my pronunciation of Latin. The propositions at Aquinas Giovanni's the problems to sharpen the young, which is really the beginning of puzzle culture. So the proposition is also interesting. It's the first piece, the first text we have written in Latin with original mathematical ideas.
So even though it's towards the end of the Roman era, if not really beyond it, and all through the Roman era, they didn't invent come up with new mathematical ideas. Obviously they did maths to do all their engineering, but not in the way the kind of coming in, the ideas, the way that the Greeks did. So what were in this document? There are about 60 puzzles, several totally new types of puzzle. And the most famous one is probably the most famous riddle of all time.
And it's the one where you have a traveller who is travelling with a wolf, a goat and a bunch of cabbages. He gets to a river. He needs to cross the river. And he has a at his disposal a boat, and he can only take one item at a time. He can't leave the wolf with the goat because wolf the goats nor the goat with the cabbages because goats he cabbages. How does he get everything across in the shortest number of trips? Now, I'm not going to go through this puzzle in much detail.
I just want to bring up what makes this a puzzle is the first time someone had used whimsy and fun to sweeten the pill to make a magical mathematical puzzle, or is really a logic puzzle entertaining to do. So you're told this and you think, Well, this is a fun situation. It's quite comic. You also want to know how it's done in order to solve it. You don't actually need any technical expertise. And that's the thing about what what's difference between a puzzle and a problem?
I think all puzzles are problems. But not all problems are puzzles. Something to be a puzzle. It's got to be require the minimum amount of technical expertise that sort of anyone can do it. And. All you need to know is simple logic. You need to work out, obviously, what is the thing? That there's only one thing that you can do if you can't leave items with B or B with C, you have to leave A with C and then you work out how it progresses.
Then the other thing, which is the mark of a good puzzle, is that there is some kind of surprise. There is something that you're learning, something either about mathematics or about the world or about your own thought process.
And what is interesting about this puzzle, it's so simple, is that you get this counterintuitive realisation that you have, that you work out that in order to get everything across, you need to take one thing across and then back and then across again, which feels completely counterintuitive. Aren't you kind of going backwards to do that? So that's something that's sort of really nice about this puzzle and this puzzle you talk about things are going viral.
This is probably the most viral puzzle of all time, even though obviously it was only spreading the speed of horseback because there is probably not one society civilisation in the world apart from maybe a few in sort of that Bike Australia and the Amazon that hasn't incorporated some version of this puzzle into kind of into folk stories. And there are actually books written about all the different ways this puzzle has spread around the world. And this is Quinn who at first appears an owl.
QUINN Another type of puzzle that Alcuin invented that I thought was quite fun we could all do together is the genre is called the kinship riddle is riddles about weird families. Okay so this is from this propositions and. If two men marry each other's mothers, what is the relationship between their sons? Again, this is funny. You can't think about that without I mean, it's funny because you think, what kind of family is that?
That's never going to happen. But also, you know, it can't be that difficult to solve. You don't know any mass, really. You only need to know what the relationship of people is within a family. What happens when you marry? What happens when you have a child? You try to do that on your in your head. It's really difficult. You just have twists in and out and it starts to be funny because they why can't they solve this simple thing? And so it's it's kind of it's all teases you a bit.
So I thought, let's solve this. I think personally my favourite answer to this question is. Tense, but it's actually quite easy to solve when you start to write things down. And again, for most puzzles, you really need a pencil. I have an eraser and the back of an envelope. So let's say, okay, two men marry each other's mothers. Let's call them A and B, Albert and Bernard. So we'll draw the family tree.
Albert has married Bernard's mum of three. Bernard's mum is the mum of Bernard with Bernard's dad, who is invisible in this testing. The puzzle obviously must have existed and Bernard marries Albert's mum. That's essentially what we get. Two men marry each other's mothers. We know that they have sons. So let's choose some nice medieval names. Steve and Trevor. That's essentially once we put it like that, it's quite easy to see the answer. Stephen Trevor We could have done it another way round.
They are both step uncle or step nephew, which we just say uncle, nephew, the uncle or nephew to each other. So once you write it down, it's not that difficult, but it's quite fun and the process of working it out is quite enjoyable. Now, once we have this interesting family, it's quite fun just to see, Well, how far could we take it? Like, what else is going on here? So obviously, Albert. Mum should be up there also because Albert's mum is the mother of Albert.
And then you see that Bernard has quite a peculiar relationship to Albert's mum because as well as being married to her, she is also his grandmother. So that makes him grandfather or step grandfather to himself. And I thought, well, that's kind of crazy. Surely there's never ever been any time in the history of the world since Alcoa this has ever been the case.
And you would be wrong. And people of my age anyway will remember the Rolling Stones and Bill Wyman actually was an it was his own grandfather for a short while because he went out with a much younger girl and it turns out that his son then married. I think they're about to marry. And there's a big scandal. Not sure if they actually did or not. Mandy Smith's mother. So truth is stranger than fiction. Okay, now we need to do some more Christmas and stuff.
This next question, I think, would be a really good question. If there was such a thing as the Elf Academy, because obviously elves, I think they that they ride the reindeers around to try and deliver all the presents. So they got to have a pretty good knowledge of geography. So I know this isn't the geographical institute, but why this is related to mathematics will become clear. Which is the furthest west of the following cities.
Okay. Might be going to do it by hands again. Might it be Edinburgh? No good class. How about Glasgow? Possibly it's not good enough with hands up or not. Liverpool. Manchester. Or the capital of the West Country Plymouth. Well, you are all wrong. And if I had asked this at the Mathematical Institute at the University of St Andrews, everyone would have got it right. Because Glasgow is the furthest west. So why is this interesting?
Everyone thinks that Plymouth should be left because everyone thinks that intuitively we see the British Isles as kind of north south. Why do we do that? What's the mathematics behind that? Obviously, the surface of the earth is three dimensional. It's a sphere. If we understood. And we were always looking at this fear. We would know that England, the British Isles is basically a diagonal poking west.
But when you turn something from a three dimensional space and you project it to make a two dimensional representation, the map here, you lose certain things and. One of the things that you lose is the sort of intuitive idea about which about how it all fits together. And I grew up in after being born on, you know, right here on the stage. I moved to Scotland.
And one of the first things they teach you with incredible pride in Scotland is that the westernmost point of the British Isles is the peninsula, which is round well round about there. So in Scotland, this is this is everyone knows this. So basically that's why there are so many Scottish elves. So another Christmas question. Who knows what happened on December the 25th, 1642? It's a slight trick question. Yes. I like your thinking.
Newton. Correct. Newton was born, but the reason was a bit of a trick question. He was born on the 25th of December, but 100 or 100 or so years later, because that was in the Julian calendar, it was revised and then we have the Gregorian calendar. So now he was born in the we say his born on the 4th of January. It's nice that he was born on Christmas Day, and that's a nice link to talking about the next person who I have chosen to be important in the history of puzzles,
which is this man, William Weston. So William Whiston, little known mathematician, but he was good friends with Newton, and Newton was Lucasian, the first Lucasian professor of math at Cambridge, and Whiston was his successor. Whiston was also controversial. He only lasted eight years because he was expelled from Cambridge for heresy.
And so one of the things I like about talking about maths is that you get to talk about how maths affects lots of different things and how you can't really ring fence into one subject so that it somehow its influence is felt in other spheres and. When Newton came up with his laws of Motion and Come the Clockwork Universe, how it all fits together had a knock on effect on people's understanding of religions.
And so there was lots of kind of. At the time if people sort of reconsidering their Christianity, holding onto Christianity, but having kind of different, interesting and some say heretical views. And William Winston was part of a kind of small kind of Christian sect who believed that the Holy Trinity was wrong. It should, because Christ didn't have the same kind of value as the other two. And for not believing the Trinity, he was expelled.
And he spent most of the rest of his life only living, giving mass lectures in the coffeehouses of London and kind of arguing for his religious views. If you were to articulate the Wikipedia page for him, what he is tends to be most known for is that he really campaigned hard for the Board of Longitude offer, which was set up by the government to offer a prize for the folks who could invent a machine or implement that could work out longitude while at sea.
And the reason why he was so keen for them to set this up in this prize money was because he thought he would be able to solve it and win the money, which he never did. But what is ironic and nice about that is that he is now remembered by people like me for coming up with probably the most famous maths puzzle about navigating the globe. Okay. And this also fits in with our Christmas theme question is this is the fuse the first person to realise this was really interesting.
Very simple but interesting mathematical sort of curiosity wrapped within this puzzle. A man walks around the circumference of the world. How much further does his head travel than his feet? Again, I love this type of puzzle because you look at it, you think that's a funny thing to say to think about, and you want to know the answer and your intuition. I have is probably quite different to what the answer is. So you get this nice surprise on the way.
Well, the first thing I need to do because of this is a Christmas lecture. What type of crazy man is going to want to walk around the circumference of the world? Thank you. Santa walks, runs, comes to the worlds because just say there's norovirus is going. Randel is reindeer, and he has the walk to deliver all the presents. Okay. How much this has had travel in his feet. Let's. Think about it and let's draw a sketch on the back of the envelope up here.
Essentially what I'm asking is this. We can assume this is like a math lecture. You can assume that someone can actually walk around the world. We are assuming that the world is a perfect sphere, so the circumference is like a grand circle and we're going to say at exactly 40,000 kilometres. Right. So you're walking around, this is Santa here.
He's going to walk all the way around the distance that the head travels more than the feet is the circumference of that that basically the dotted line minus this comes from the earth, the dotted line minus the circumference of the shaded circle. Now, you would have thought that this is 40,000 kilometres that he's walking around. The head is always, you know, the feet rules. I'm going to say that Santa has the average height of a British adult male 1.8 metres.
His head is always 1.8 metres from the ground. So this up there is 1.8 is going all the way around and at the bottom it's also 1.8 metres from it. You're going to think it's going to be, you know, in the hundreds of miles, probably it's going to be like a long the head is travelling a lot longer. It feels like a lot longer. Anyway, let's work this out simply.
We do this in a little bit of technical knowledge, but I'm thinking that everyone we should we should all know this, but just in case revision, the circumference of a circle is two pi are okay. But we are. So we've all got that. Well, what is the distance travelled by the feet to power? What is the distance? Travelled by the head. It's too high and the radius is going to be H plus R to pi plus h. That's two way up was super to pi h. So the disk that the.
The, um, the difference is going to be the bottom one minus the top one. Which is two pi h. Okay, so let's work that out as two times two, 3.1, four or thereabouts. So I was 1.8 which is 11 metres case. Really. Not a lot. It's kind of wow. I mean, that's why and William Whiston Well, there's basically one math textbook for 2000 years, Euclid's elements and every new mathematician that came along did a new version with that kind of annotated notes and thoughts.
And William Wisdom came out with one when he was a professor of maths. And in it he said, Isn't this interesting? Because when a man walks around the earth, this is so certainly 11 metres or. It's only two page. The most interesting isn't necessarily it isn't really that 11 metres is if you look at the answer to PI. Nowhere does the answer include ah, the radius of the earth.
In other words, it does not matter what sphere the man centre walks around the head is always walk and always travelling only 11 metres more than the fate. Okay. Trivially if you're walking around dot. The head, the feet don't walk anything at all. And the head walks 11 metres. That's to player. But also just say you are walking around the moon, you're walking around Jupiter, you're walking around the largest sphere that it's possible to get in the universe still.
The head travels only 11 metres more than the feet, which is something which is surprising and interesting. So. I prefer the puzzle set about a man walking around the earth because it's a red. That's how it was originally spotted by Western theory and 50 years ago, but also because the way that it is normally stated this problem, this problem, this puzzle is as the rope around the earth puzzle.
The wrap around the puzzle. It might give you a bigger wow, but it's a lot more weird and bizarre and complicated to explain because you need to say there is a rope around the scum of the earth and it's taut. Then someone extends this rope by one metre. Extend the rope by one metre. Then what the person does is you need to pull up the rope above as it comes of the earth. So it's the same height all around the earth. And this is just where it gets complicated.
Must be like, Why do you want to do that? So sometimes they say it's the iron bar around the earth. Do you have an iron bar around the earth? And you extend the iron bar by one metre and then you pull it up. So the iron bar is above the circumference at exactly the same. The question is by extending something from 40,000 kilometres to 40,000.001 kilometres, I mean, it's a tiny fraction of what animal can get underneath.
And at first you would think, Oh, my God. Just like nothing I think would end. But actually, it's a small dog. That small dog could get under. And I think by the time if you've managed to follow all the stuff about this rope and then pulling it up, you would you would think, wow, that is that is surprising because it would be also the case that if you had a rope around Jupiter, around the biggest possible thing as a sphere and the entire universe, the exact same thing would happen.
Extend that by a metre. Lift it up. You can get this dog underneath. So if I was asking that question or I was being told that question, I would say, hang on a second. You've got all the work to get this rope going all the way around the earth. You're extending it by one metre with a purpose of trying to get animals to go underneath it. Why are you going to try and levitate it above that? Why don't you just pick it up at one point and pull that rope as high as you can?
So that's also a really interesting question. What animal now can get underneath it? And this is something that it does depend on the size of the sphere. And you actually need trigonometry level, if not beyond trigonometry to work out. But it turns out that if you pull if you have this rope around the earth extended by metre and you pull it. Because we're going to talk about festive animals. You can get about. A pyramid of a hundred reindeer will be able to go underneath it.
It's basically a 122 metres high, the height of Centrepoint in London. Which is, again, kind of counterintuitive. You're extending it only by a tiny amount and you get so much slack. Now quite often when I give my talks, I, I about the philosophy. So I'm not that interested in applications. I'm interested in just like it's interesting like baclofen and say, well, why is that useful? Why what? How does that affect real life?
Well, the fact that we've been talking about circles but also works in straight lines if you give something and small. Extend something a tiny amount, you get so much more slack than you would intuitively think. And this is why you get things like this happening. So when in the heat, if you have a rail track, you only need to extend that rail track by a really small amount. And the slack that you get actually creates really counterintuitively large bumps.
Right. Moving on to the next. Century or maybe two centuries after that in the Victorian Times. And it's all about Lewis Carroll. So. A great boom time for puzzles was the Victorian era, especially the late Victorian era. And one of the reasons for that was the growth in media and magazines and newspapers. Actually people realised that people wanted to read and do puzzles for pleasure.
And Lewis Carroll, obviously, he's an Oxford don, much more famous for Alice in Wonderland and Through The Looking Glass. But he also wrote several puzzle books, none of them particularly successful because his wasn't good at writing a good puzzle. They were a lot more complicated and a lot more too too difficult, really, for them to be of general appeal. But he did invent a type of logic puzzle which has become a hugely popular type of logic puzzle and actually.
Really. Sort of useful and good applications and pretty much any computer science course that you do, you will start to do you will play around the puzzle like this and it's puzzles where you have some people who tell the truth all the time and some people who lie all the time. So puzzles involving truth tellers and liars. And Lewis Carroll was the first person and then it was quite late in his life in the 1890s, he scribbled around and.
He worked this precise puzzle, which he just did A, B and C, but because this is a Christmas lecture, I've done it with different names. And this is his original puzzle dance. She says, the dancer tells lies. Dancer sells it, Prancer tells lies. Prancer says that both Dasher and Dancer tells lies who's telling the truth. And in order to solve this, we're going to act this out. So I need three volunteers. Three young volunteers? Yeah, you three. Brilliant. Can you come up here?
Yeah. One, two, three. Okay. Well, what have you had? The hand that was you with you? Well, whoever. So one is going one of you is going to be Dasher, one is Dancer, one is Prancer. Brilliance if you come and stand here three in a row. Yes. If you look at sort of entrance exams and things to do, computer science studies that's full of questions about she tells us we're going to get you, all three of you in a row like this. So you. What's your name? Kathy. Kathy is Dasha.
Chi Chi is Dancer and Alexandra is Prancer. So I forgot the original names. I'm just telling you with Dasher, Dancer and Prancer. Okay, so how do you solve something like this? Well, what you need to do first. Well, this is one way to solve it, is that you need to assume a truth value. I assume that someone is telling the truth and then just, like, work through and see what happens. So let's assume that Dasher is telling the truth if you're telling the truth.
Then you're a liar, okay? You're a liar so that everyone knows you're a liar. This is the international symbol of being a liar. Opinion is deeper than you are now, not a lawyer. That's right. I that you are now a liar. You say that Prancer tells lies, but you're a liar. So actually you must tell the truth. Okay, so you don't know. Now's for you if you're telling the truth. If you say that, both of you tell lies. But you're not telling the truth.
That's incorrect. So we have a system that is inconsistent, doesn't work. So this is not a solution to the puzzle. So we can eliminate the possibility that you are telling the truth. Unfortunately, Dasha, I think you might be a liar. So if you're a liar, could you put on the symbol of lying? So Dasha is not lying. But you say that dancer tells lies. But if you're a liar, that means that you tell the truth. Where does system hear the either a liar or truth teller?
There's nothing in between. So if you're telling the truth, you say that Prancer is a liar. So if you're a liar, we need to get one of these noses for you. You're a liar. You say that Dasher danced, tell lies. Both, which is not true. So you're actually. It's a consistent system. You're lying because that's not the case. You're saying that's true, but it's not the case. So you're lying.
So we have here a system that works. The truth values make system consistent, which means who is telling the truth? Only one of you is. Dancer. Thank you very much. Radicals are trained there. Thank you. Okay. Since we are on Canuck noses and lying and Pinocchio is always on the Christmas time. Tenuous link. I thought we would have some fun here about Pinocchio's nose. So this is what we're all going to do this together. Pinocchio's nose is five centimetres long.
Each time he tells a line, his nose doubles in length. After so nine lines, his nose will be roughly the same length as who thinks it's a domino. Think good a tennis racket. A snooker table. Tennis court. A football pitch. Okay. Well, let's just have a look. Have a look. Well, let's go through the math. They say so nine lines doubling each time. That's two to the nine times five. Sets me to the original length.
Okay. You might be really good and be able to do two for like no one can remember that if you run one thing today, it's that two to the ten is about a thousand as a really, really useful thing just to know just for sort of making good estimates to two tens, about thousand, it's actually 1024. So two to the nine is going to be half. That's about 505 or ten, 12, so 5 to 12 times five centimetres, which obviously half a metre and a five metres or times five.
25 metres, so most of you were wrong. It is a tennis court and it's an interesting. I'm a journalist. Or maybe an ex journalist. I didn't do much journalism at the moment, but journalists always trying to describe how big things are and also is is the size of several football pitches when it gets bigger, say, the size of several times the size of Wales. But the reason why is the football pitch is football pitches.
You think? Well, the stadiums, they've always got a 100 metre racetrack right next to them. So the rule of thumb. Football pitcher stated, You know, there's always going to be years that 100 metres will be 100 metres. So something which is a quarter the size of a football pitch is going to be a tennis court. Now that is the level one question, the level two question. Neurones on the ready.
If Pinocchio's nose is literally one inch, that's 2.54 centimetres long and has a weight of six grams and is attached to a 4.18 kilogram wooden head. After how many fibs with a break and how long would it be? What sorts of rights is a good question. Mahogany? No, I don't know, because the reason why I bring this up, the answer is apparently 1315 children, eight metres because of this is the only thing in the entire election day which is actually the subject of a proper academic paper.
This is what you study at the University of Leicester. Okay. At the end of the 19th 19th century, there were two huge figures in puzzled them Sam Lloyd in America and Henry Dudney in the UK. And these two people in a sense didn't. He was a much more interesting mathematician, a fascinating man who was completely self-taught, and yet quite a lot of the maths that he found in his puzzles went on to be stuff that proper mathematicians still study to this day.
Sam Lloyd. Leaders in America became a lot more famous and a lot richer. He was very kind of American and entrepreneurial. And I want to use one of Sam Lloyd's puzzles, because it's actually actually easier to do as a group. And it's called the Canals of Moses from about 100 years ago. What we have we have this is a picture of Mars. What? It's a start of the letter T. And then you need to get your way through all of the canals and come back to t.
Saying a sentence in English. There is some sentence in English that you can say. This is quite a famous puzzle and had a lot of attention at the time. And Sam Lloyd said that when the puzzle originally appeared in a magazine, more than 50,000 readers reported this. No possible way. Yeah. It's a very simple puzzle. When the puzzle originally appeared in the magazine, more than 50,000 readers reported There is no possible way. Yet it is a very simple puzzle. Have you got the answer?
Don't tell us what it is. There is no possible way. It is amazing how many people when will not the penny will not drop when you are literally saying to them again and again there is no possible way. And what I like about this puzzle is that it really brings out the fact that sometimes when you start a problem, if you're looking in one direction, it's impossible to kind of go back from that direction and start in a new direction that if you are totally focussed on one way,
even if someone is telling you the answer. It's hard to see it when it's right there in front of you. So as well as read the question you need to. Sometimes kind of forget everything you know, and then start again afresh. And often this is why this puzzles are quite similar to magic tricks that what magicians are sometimes trying to do is. Do something here but make you look over there. Right. Sort of sleight of hand and puzzles often.
That is exactly the same thing. You want to lead the solver in one direction when the solution might be really obvious, but it's much more fun to try and strike them, take them, go in a different and different way. And actually, there's a famous puzzle, which is about the light bulb and the three switches. And this is a classic puzzle because it's set as a maths puzzle, but actually the answer is a kind of physics answer, cause it's all about that the bulb is going to heat up.
But if you start trying to solve that puzzle. Thinking about mathematical combinations. You just not going to get it. He's just not going to get it because you're stuck in this way of thinking for the math puzzle. So sometimes you need to kind of go back. And once I was at a book festival and I was in the green room and I was chatting to Jo Nesbo or, you know, Nesbo, the Norwegian sort of bestselling crime writer. And he's knowing, you know, puzzles Guy, Tell Me a puzzle.
And I told him this puzzle, and he's the only person who ever told that puzzle to who managed to solve it in front of me. And he basically said, okay, you're doing exactly what I do as a crime writer. You know, I'm writing all the characters in the first few pages, and obviously it's blindingly obvious who who did it, who the murderer is. But all I'm trying to do is to make the reader think it's not that person until right at the end.
That's the pleasure of it. And he said, okay, so this puzzle you're telling me it's a mathematical puzzle, but I know that's probably a trick. So what else could it be? What happens to bulbs? And he basically reverse engineered and solved it. And I thought that was very interesting to think that there is a kind of parallel with a great puzzle and also a fiction. So we could do some Christmas Eve time. Very good time.
If there is one mathematical sort of field really that saw of the Christmas Santa Claus has to tackle with every year, surely it is the travelling salesman problem. Okay. Simon says the problem is you have all the places that you need to go get the loop. You need to start where you are. Good visit all of them. Come back to where you started in the quickest or shortest possible way. The first person to start thinking about this was Karl Manga in the 1930s famous mathematician.
But coming it was actually. You see what I did here? Much more famous for something else. The manga sponge. Okay. I don't if anyone is. It doesn't make us money. Well, we some people here, we've got some professors of mathematics. I'll be surprised if they don't know what it is. The manga sponge that we don't know is a really fun, fascinating object that's quite simple to describe and kind of cool to look at, is that let's say we have a cube.
This is a cube. We're going to divide the cube into 27 smaller cubes, just like it's a Rubik's cube, say. So three by three. By three by three. So 27 cubes. And we're going to extract the middle cube in each of the sides, plus the very central cube. Okay, so there's there should be 27. If it's a block, then we take out seven, five, six, one on each side. There are six sides and one in the middle at seven. So we're left with something with 20 sort of cube looks like that.
And then the next process is Let's treat it. It's a fractal object. So we're going to repeat the process, take each cube and do exactly the same thing. So we get this. You see, we've taken each cube and within it we've made 27 small cubes taken at seven centres, and we get this kind of whole, the sort of cube, this growing wholes, and then we carry on one way iteration and we get this. This is a mega cubical, there's a level three mangcu because we've done this region three times.
And one of the reasons why this is a really interesting object is that. Each time we take out holes, we increase the surface area. But we reduced the volume. So. In the limit. We're going to have an object which has no volume, an infinite surface area, which is kind of interesting. Okay. How is this relevant to Christmas? Well, we're going to get there. Sometimes I submit puzzles for Radio four today program now has a puzzle of the day about 650 every morning.
And there was one which I really liked actually. I thought this was a really nice puzzle. I think it was. The University of Leeds set it and it's a match. You've got a cube. Let's forget this for a while. You got a cube and you hold it for a piece of string from one of the vertices that. And then you dip into water. We dip it in halfway. What is the shape on the surface? I think this is really interesting because it's really hard to visualise even though it's a cube.
One of the most simple objects. Really hard to visualise, actually. What you get is a hexagon. I can show that here. It's basically if you slice a cube like this, you get a hexagon. Okay. So you go halfway along each side and then diagonally. So you get to that. That's. Looks like it's why France from here, actually. But if you see the debt, you basically get.
You split as a way of taking a diagonal slice, which you get a hexagon, which is so the both equals equal size, and you also get it with you're dipping the Cubans. The question is. What? Pattern do you get if you slice the manga? Sponge on one of those diagonal slices. Okay. And when I was on this, I had no idea. But I was asked this by a guy called George Hart, who is a well-known American geometrical sculptor and geometry, and is actually also the father of my heart.
He does all these amazing YouTube doodle videos, and he said that this was probably the biggest wow that he knew of in mathematics. And this is a guy who's like knows loads of wows. This is his business, his wives of geometry, because he's they're making these amazing sculptures, which just most people feel, wow. So it's not something that you might be able to work out for yourself. I certainly couldn't. But if you are to slice this level three. Manga sponge with a hexagonal slice.
This is what you get. Which as well as being surprising and wonderful. It's a fantastic image to end on because it's basically like kind of a snowflake or a star at the top, the Christmas tree over in the sky. And it's almost exactly 6:00. So thank you very much.