1 00:00:11,430 --> 00:00:16,560 Thank you, everyone, for coming and thank you Mathematical Institute for inviting me. 2 00:00:16,890 --> 00:00:22,910 So can you solve my problems? One rule of life is you get a pun in the title. 3 00:00:22,920 --> 00:00:23,730 Always put it in. 4 00:00:25,320 --> 00:00:33,860 The subtitle of this talk is going to be a short history of puzzles, with occasional Christmas references and some sparkling mathematics. 5 00:00:33,870 --> 00:00:37,139 In other words, you talk a little bit about puzzles, but about the history of puzzles. 6 00:00:37,140 --> 00:00:42,870 We're going to do some puzzles, and I might mention Christmas a few times as a Christmas lecture. 7 00:00:43,440 --> 00:00:52,230 Now, it's fantastic for me to be here because it is a return to my alma mater with Marta. 8 00:00:52,680 --> 00:00:58,380 The crucial words because I was actually born on this site when it used to be the Radcliffe Infirmary. 9 00:00:58,590 --> 00:01:03,059 This is where I was actually born. So it's quite nice to think that, you know, my life, 10 00:01:03,060 --> 00:01:09,629 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. 11 00:01:09,630 --> 00:01:13,050 One also is true. I studied mathematics here. 12 00:01:13,050 --> 00:01:15,330 It wasn't math and physics, it was maths and philosophy. 13 00:01:15,570 --> 00:01:21,120 And rather wonderfully my old philosophy of maths professor is in the audience, so the pressure is on. 14 00:01:22,650 --> 00:01:27,540 It's true. I did a lot of maths when I was at Oxford 30 odd years ago. 15 00:01:28,260 --> 00:01:35,190 I also did lots of journalism. I probably did more journalism than I did maths and I was editor of Cherwell, 16 00:01:35,190 --> 00:01:41,070 the university newspaper, and in the last ten years I have combined essentially my two passions. 17 00:01:41,730 --> 00:01:50,280 So it wasn't really a passion is writing and mathematics so serious passions and this is what I've tried to do. 18 00:01:50,310 --> 00:01:52,770 My first book was Alex's Adventures in Wonderland, 19 00:01:52,770 --> 00:01:59,999 where I went around the world interviewing people as if I was a kind of foreign correspondent in the world of mathematics, 20 00:02:00,000 --> 00:02:03,899 interviewing people in the world of maths and sort of explaining what they 21 00:02:03,900 --> 00:02:07,620 did and mathematical concepts to the general reader that led to the next one. 22 00:02:07,980 --> 00:02:17,380 Alex Through The Looking Glass. Then I wrote with a wonderful mathematician, mathematical artist Edmund Harris, two maths colouring books. 23 00:02:17,410 --> 00:02:20,670 I shouldn't have to mention these because even though they are colouring books, 24 00:02:20,820 --> 00:02:27,600 you can look at them and you can colour them in essentially the colouring books because bookshops have a table for colouring books. 25 00:02:28,020 --> 00:02:34,140 Really what they are, they are gallery, a gallery of wonderful images from all areas of maths. 26 00:02:34,410 --> 00:02:39,450 And even though they look the most kind of junior, so to speak, 27 00:02:39,450 --> 00:02:45,180 of all the things that I've done, they probably cover the widest and deepest mathematics. 28 00:02:46,110 --> 00:02:50,819 So two or three years ago, I started to write a puzzle column in The Guardian. 29 00:02:50,820 --> 00:02:54,090 And yes, it is a lot of pressure to try and find a good puzzle every two weeks. 30 00:02:54,210 --> 00:02:59,340 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. 31 00:03:00,420 --> 00:03:06,420 And this has resulted in two puzzle books Can't solve my problems and Puzzle Ninja. 32 00:03:06,780 --> 00:03:14,159 So Puzzle Ninja, which is not really Christmasy, even though it's just out in time for Christmas, it's about Japan. 33 00:03:14,160 --> 00:03:19,110 I went to Japan at the beginning of the year because the puzzle culture in Japan is unlike anywhere in the world. 34 00:03:19,500 --> 00:03:28,740 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. 35 00:03:29,040 --> 00:03:35,519 And I promise you, some bloke he might have heard of very, very wise man. 36 00:03:35,520 --> 00:03:39,540 Mark is the sort I call this book addictive. And it's that's an understatement. 37 00:03:39,720 --> 00:03:46,110 You will start doing this book and time disappears, which is one of the great joys of doing a puzzle, 38 00:03:46,260 --> 00:03:53,190 is that once you get into it, the world disappears and there are whatever distraction there is, you can focus. 39 00:03:53,190 --> 00:03:58,050 And it's actually it's such a kind of I found a relaxing thing to do puzzles because 40 00:03:58,740 --> 00:04:02,370 all those distractions disappear that I'm going to start off with a puzzle, 41 00:04:02,370 --> 00:04:10,530 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. 42 00:04:12,390 --> 00:04:20,100 Santa has got to has laid out five shapes, which could be his buckle and his belt. 43 00:04:20,250 --> 00:04:24,780 And it's an odd one out puzzle. So one of these shapes is the odd one out. 44 00:04:24,780 --> 00:04:28,950 I'm going to give you a few seconds just to think about it, then we're going to try and solve it together. 45 00:04:32,840 --> 00:04:36,260 So it's somewhat together. I'm on this side. We'll start with this one here. 46 00:04:36,620 --> 00:04:39,890 Put your hands up. If you think the. 47 00:04:39,920 --> 00:04:44,990 Oh, the odd one out is the only small one. 48 00:04:48,290 --> 00:04:53,600 One person. Hands up if you think the odd one out is the only blue one. 49 00:04:55,480 --> 00:04:58,630 Somewhat. A couple of about three people. Hands up. 50 00:04:58,970 --> 00:05:09,600 The odd one out is the only circle. A hands up if you think the odd one is the only one with no border. 51 00:05:14,270 --> 00:05:17,420 Everyone who has put their hands up so far. 52 00:05:18,230 --> 00:05:22,040 Just listen. I'm going to repeat what I have just said. 53 00:05:23,180 --> 00:05:26,900 This is the only the only small one. 54 00:05:27,740 --> 00:05:31,910 The only blue one. The only circle. 55 00:05:32,630 --> 00:05:41,970 The only one with no border. What is unique about this one here? 56 00:05:43,260 --> 00:05:46,560 Nothing. It's not only anything. 57 00:05:46,590 --> 00:05:52,919 There's nothing. It shares every characteristic, every feature, every property it shares with another one. 58 00:05:52,920 --> 00:05:59,340 In other words, the odd one out is the one, which is not an odd one out. 59 00:06:00,300 --> 00:06:03,000 So, in fact, as well as a puzzle, 60 00:06:03,240 --> 00:06:16,060 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. 61 00:06:16,110 --> 00:06:23,610 And in fact, that was the reason why it was invented by a Russian-American puzzle expert, 62 00:06:23,610 --> 00:06:27,060 Tanya Herrmann, over who wanted it, who doesn't like odd one out puzzles. 63 00:06:27,300 --> 00:06:32,760 Wants to use this to show why. Now, I like this because a good puzzle is also a good story. 64 00:06:33,570 --> 00:06:37,049 You you take it, it has a beginning. 65 00:06:37,050 --> 00:06:44,470 You set the problem and you work it out. You kind of. That there's a path that you need to go. 66 00:06:44,500 --> 00:06:49,480 There might be some false terms. Then you have the kind of the insights and then things are neatly tied up at the end. 67 00:06:49,780 --> 00:06:53,890 So I think that's another reason why I like this puzzle, is that it takes us on a journey. 68 00:06:53,900 --> 00:06:57,250 It's just that the journey at the end is also really like a joke. 69 00:06:58,100 --> 00:07:03,469 Now I feel I should mention something about Christmas now, because it's been about 3 minutes and I haven't. 70 00:07:03,470 --> 00:07:11,000 And this is a Christmas lecture. So I want to go back in time to Christmas Day, the year 800. 71 00:07:12,410 --> 00:07:20,899 And if I was in the History Institute, everyone would know exactly what important event happened on Christmas Day, eight hundreds, thankfully. 72 00:07:20,900 --> 00:07:32,120 I mean, the Mass Institute, it was when Charlemagne was crowned, was made Emperor of Rome, and he was king of the Franks, 73 00:07:32,570 --> 00:07:37,430 but he was in Rome and the Romans made him king of the Roman Empire, the Holy Roman Empire. 74 00:07:37,610 --> 00:07:40,430 So he became empire of pretty much all of Western Europe. 75 00:07:40,820 --> 00:07:48,800 And Charlemagne is interesting, not just because his for his sort of political aspect of because he was the head of an intellectual resurgence, 76 00:07:49,560 --> 00:07:55,910 the towards the end of the Middle Ages in Europe and his mentor, his educator, 77 00:07:55,910 --> 00:08:02,810 his teacher is probably the most important early figure in the history of puzzles. 78 00:08:03,140 --> 00:08:13,530 And this person was Alcuin of York. So Alcuin of York from York ended up working for Charlemagne, starting a sort of oboe. 79 00:08:13,580 --> 00:08:24,110 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. 80 00:08:24,260 --> 00:08:27,500 He also invented joined up writing the rules of amazing things. 81 00:08:27,710 --> 00:08:36,620 And in 799, he wrote a letter to Charlemagne in which he says, I hear I enclose some arithmetical curiosities to amuse you. 82 00:08:38,740 --> 00:08:43,750 The arithmetical curiosities were discovered about 100 years later and attributed to Alcuin. 83 00:08:44,050 --> 00:08:49,990 The propositions and facts on the Classics Institute because you do me from my pronunciation of Latin. 84 00:08:50,230 --> 00:08:59,140 The propositions at Aquinas Giovanni's the problems to sharpen the young, which is really the beginning of puzzle culture. 85 00:08:59,620 --> 00:09:09,550 So the proposition is also interesting. It's the first piece, the first text we have written in Latin with original mathematical ideas. 86 00:09:09,940 --> 00:09:13,750 So even though it's towards the end of the Roman era, if not really beyond it, 87 00:09:15,040 --> 00:09:19,080 and all through the Roman era, they didn't invent come up with new mathematical ideas. 88 00:09:19,090 --> 00:09:25,870 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. 89 00:09:25,900 --> 00:09:29,120 So what were in this document? 90 00:09:29,140 --> 00:09:34,030 There are about 60 puzzles, several totally new types of puzzle. 91 00:09:34,450 --> 00:09:38,020 And the most famous one is probably the most famous riddle of all time. 92 00:09:38,320 --> 00:09:47,110 And it's the one where you have a traveller who is travelling with a wolf, a goat and a bunch of cabbages. 93 00:09:47,110 --> 00:09:50,260 He gets to a river. He needs to cross the river. 94 00:09:51,220 --> 00:09:56,410 And he has a at his disposal a boat, and he can only take one item at a time. 95 00:09:56,830 --> 00:10:04,300 He can't leave the wolf with the goat because wolf the goats nor the goat with the cabbages because goats he cabbages. 96 00:10:04,510 --> 00:10:10,780 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. 97 00:10:11,920 --> 00:10:17,260 I just want to bring up what makes this a puzzle is the first time someone had 98 00:10:17,260 --> 00:10:25,629 used whimsy and fun to sweeten the pill to make a magical mathematical puzzle, 99 00:10:25,630 --> 00:10:30,550 or is really a logic puzzle entertaining to do. So you're told this and you think, Well, this is a fun situation. 100 00:10:30,670 --> 00:10:36,700 It's quite comic. You also want to know how it's done in order to solve it. 101 00:10:37,860 --> 00:10:43,810 You don't actually need any technical expertise. And that's the thing about what what's difference between a puzzle and a problem? 102 00:10:44,310 --> 00:10:49,300 I think all puzzles are problems. But not all problems are puzzles. 103 00:10:49,930 --> 00:10:56,410 Something to be a puzzle. It's got to be require the minimum amount of technical expertise that sort of anyone can do it. 104 00:10:56,980 --> 00:11:02,260 And. All you need to know is simple logic. 105 00:11:02,300 --> 00:11:09,470 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, 106 00:11:09,650 --> 00:11:15,470 you have to leave A with C and then you work out how it progresses. 107 00:11:15,710 --> 00:11:21,140 Then the other thing, which is the mark of a good puzzle, is that there is some kind of surprise. 108 00:11:21,440 --> 00:11:26,840 There is something that you're learning, something either about mathematics or about the world or about your own thought process. 109 00:11:27,290 --> 00:11:32,750 And what is interesting about this puzzle, it's so simple, is that you get this counterintuitive realisation that you have, 110 00:11:33,050 --> 00:11:35,840 that you work out that in order to get everything across, 111 00:11:36,140 --> 00:11:42,770 you need to take one thing across and then back and then across again, which feels completely counterintuitive. 112 00:11:42,860 --> 00:11:44,690 Aren't you kind of going backwards to do that? 113 00:11:45,170 --> 00:11:50,000 So that's something that's sort of really nice about this puzzle and this puzzle you talk about things are going viral. 114 00:11:50,120 --> 00:11:52,400 This is probably the most viral puzzle of all time, 115 00:11:52,580 --> 00:12:05,690 even though obviously it was only spreading the speed of horseback because there is probably not one society civilisation in the world apart from 116 00:12:05,690 --> 00:12:14,300 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. 117 00:12:14,570 --> 00:12:21,110 And there are actually books written about all the different ways this puzzle has spread around the world. 118 00:12:21,320 --> 00:12:24,709 And this is Quinn who at first appears an owl. 119 00:12:24,710 --> 00:12:29,990 QUINN Another type of puzzle that Alcuin invented that I thought was quite fun we could all 120 00:12:29,990 --> 00:12:38,990 do together is the genre is called the kinship riddle is riddles about weird families. 121 00:12:39,110 --> 00:12:43,340 Okay so this is from this propositions and. 122 00:12:44,540 --> 00:12:49,040 If two men marry each other's mothers, what is the relationship between their sons? 123 00:12:49,280 --> 00:12:56,269 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? 124 00:12:56,270 --> 00:13:00,890 That's never going to happen. But also, you know, it can't be that difficult to solve. 125 00:13:00,920 --> 00:13:04,879 You don't know any mass, really. You only need to know what the relationship of people is within a family. 126 00:13:04,880 --> 00:13:08,990 What happens when you marry? What happens when you have a child? You try to do that on your in your head. 127 00:13:09,080 --> 00:13:15,500 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? 128 00:13:17,150 --> 00:13:20,540 And so it's it's kind of it's all teases you a bit. 129 00:13:21,500 --> 00:13:30,290 So I thought, let's solve this. I think personally my favourite answer to this question is. 130 00:13:31,700 --> 00:13:37,849 Tense, but it's actually quite easy to solve when you start to write things down. 131 00:13:37,850 --> 00:13:42,980 And again, for most puzzles, you really need a pencil. 132 00:13:43,280 --> 00:13:49,729 I have an eraser and the back of an envelope. So let's say, okay, two men marry each other's mothers. 133 00:13:49,730 --> 00:13:54,140 Let's call them A and B, Albert and Bernard. So we'll draw the family tree. 134 00:13:54,170 --> 00:14:02,420 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. 135 00:14:02,420 --> 00:14:06,560 The puzzle obviously must have existed and Bernard marries Albert's mum. 136 00:14:06,770 --> 00:14:11,300 That's essentially what we get. Two men marry each other's mothers. We know that they have sons. 137 00:14:12,020 --> 00:14:15,050 So let's choose some nice medieval names. 138 00:14:15,440 --> 00:14:25,140 Steve and Trevor. That's essentially once we put it like that, it's quite easy to see the answer. 139 00:14:25,740 --> 00:14:27,960 Stephen Trevor We could have done it another way round. 140 00:14:27,990 --> 00:14:34,560 They are both step uncle or step nephew, which we just say uncle, nephew, the uncle or nephew to each other. 141 00:14:35,160 --> 00:14:40,890 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. 142 00:14:41,310 --> 00:14:47,880 Now, once we have this interesting family, it's quite fun just to see, Well, how far could we take it? 143 00:14:47,910 --> 00:14:51,180 Like, what else is going on here? So obviously, Albert. 144 00:14:52,360 --> 00:14:55,900 Mum should be up there also because Albert's mum is the mother of Albert. 145 00:14:56,290 --> 00:15:07,000 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. 146 00:15:08,590 --> 00:15:12,370 So that makes him grandfather or step grandfather to himself. 147 00:15:12,840 --> 00:15:14,280 And I thought, well, that's kind of crazy. 148 00:15:14,290 --> 00:15:20,710 Surely there's never ever been any time in the history of the world since Alcoa this has ever been the case. 149 00:15:21,400 --> 00:15:34,510 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 150 00:15:34,810 --> 00:15:41,740 for a short while because he went out with a much younger girl and it turns out that his son then married. 151 00:15:42,280 --> 00:15:45,370 I think they're about to marry. And there's a big scandal. 152 00:15:45,390 --> 00:15:48,610 Not sure if they actually did or not. Mandy Smith's mother. 153 00:15:49,390 --> 00:15:55,180 So truth is stranger than fiction. Okay, now we need to do some more Christmas and stuff. 154 00:15:56,110 --> 00:16:02,079 This next question, I think, would be a really good question. 155 00:16:02,080 --> 00:16:06,219 If there was such a thing as the Elf Academy, because obviously elves, 156 00:16:06,220 --> 00:16:11,320 I think they that they ride the reindeers around to try and deliver all the presents. 157 00:16:11,500 --> 00:16:14,320 So they got to have a pretty good knowledge of geography. 158 00:16:14,800 --> 00:16:20,260 So I know this isn't the geographical institute, but why this is related to mathematics will become clear. 159 00:16:22,160 --> 00:16:25,460 Which is the furthest west of the following cities. 160 00:16:27,020 --> 00:16:31,130 Okay. Might be going to do it by hands again. Might it be Edinburgh? 161 00:16:32,660 --> 00:16:36,260 No good class. How about Glasgow? 162 00:16:39,480 --> 00:16:42,600 Possibly it's not good enough with hands up or not. Liverpool. 163 00:16:45,820 --> 00:16:52,330 Manchester. Or the capital of the West Country Plymouth. 164 00:16:56,480 --> 00:17:08,600 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. 165 00:17:08,810 --> 00:17:13,490 Because Glasgow is the furthest west. So why is this interesting? 166 00:17:13,490 --> 00:17:23,270 Everyone thinks that Plymouth should be left because everyone thinks that intuitively we see the British Isles as kind of north south. 167 00:17:24,620 --> 00:17:30,050 Why do we do that? What's the mathematics behind that? Obviously, the surface of the earth is three dimensional. 168 00:17:30,970 --> 00:17:36,170 It's a sphere. If we understood. And we were always looking at this fear. 169 00:17:36,200 --> 00:17:42,710 We would know that England, the British Isles is basically a diagonal poking west. 170 00:17:43,910 --> 00:17:49,600 But when you turn something from a three dimensional space and you project it to make a two dimensional representation, 171 00:17:50,000 --> 00:17:53,990 the map here, you lose certain things and. 172 00:17:55,510 --> 00:18:02,230 One of the things that you lose is the sort of intuitive idea about which about how it all fits together. 173 00:18:02,680 --> 00:18:07,540 And I grew up in after being born on, you know, right here on the stage. 174 00:18:07,720 --> 00:18:09,190 I moved to Scotland. 175 00:18:09,430 --> 00:18:16,120 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, 176 00:18:16,720 --> 00:18:19,870 which is round well round about there. 177 00:18:21,580 --> 00:18:27,430 So in Scotland, this is this is everyone knows this. So basically that's why there are so many Scottish elves. 178 00:18:29,200 --> 00:18:37,090 So another Christmas question. Who knows what happened on December the 25th, 1642? 179 00:18:38,020 --> 00:18:44,190 It's a slight trick question. Yes. I like your thinking. 180 00:18:46,210 --> 00:18:51,820 Newton. Correct. Newton was born, but the reason was a bit of a trick question. 181 00:18:52,090 --> 00:18:59,530 He was born on the 25th of December, but 100 or 100 or so years later, 182 00:18:59,620 --> 00:19:04,060 because that was in the Julian calendar, it was revised and then we have the Gregorian calendar. 183 00:19:04,070 --> 00:19:07,510 So now he was born in the we say his born on the 4th of January. 184 00:19:07,570 --> 00:19:09,850 It's nice that he was born on Christmas Day, 185 00:19:10,030 --> 00:19:19,390 and that's a nice link to talking about the next person who I have chosen to be important in the history of puzzles, 186 00:19:19,660 --> 00:19:28,030 which is this man, William Weston. So William Whiston, little known mathematician, but he was good friends with Newton, 187 00:19:28,030 --> 00:19:35,170 and Newton was Lucasian, the first Lucasian professor of math at Cambridge, and Whiston was his successor. 188 00:19:35,920 --> 00:19:43,600 Whiston was also controversial. He only lasted eight years because he was expelled from Cambridge for heresy. 189 00:19:44,200 --> 00:19:49,540 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 190 00:19:49,540 --> 00:19:59,710 things and how you can't really ring fence into one subject so that it somehow its influence is felt in other spheres and. 191 00:20:00,840 --> 00:20:04,709 When Newton came up with his laws of Motion and Come the Clockwork Universe, 192 00:20:04,710 --> 00:20:09,090 how it all fits together had a knock on effect on people's understanding of religions. 193 00:20:09,090 --> 00:20:16,799 And so there was lots of kind of. At the time if people sort of reconsidering their Christianity, holding onto Christianity, 194 00:20:16,800 --> 00:20:20,640 but having kind of different, interesting and some say heretical views. 195 00:20:21,030 --> 00:20:27,360 And William Winston was part of a kind of small kind of Christian sect who believed that the Holy Trinity was wrong. 196 00:20:27,630 --> 00:20:32,880 It should, because Christ didn't have the same kind of value as the other two. 197 00:20:33,120 --> 00:20:35,910 And for not believing the Trinity, he was expelled. 198 00:20:36,150 --> 00:20:43,950 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. 199 00:20:44,550 --> 00:20:47,070 If you were to articulate the Wikipedia page for him, 200 00:20:48,660 --> 00:20:55,410 what he is tends to be most known for is that he really campaigned hard for the Board of Longitude offer, 201 00:20:56,040 --> 00:21:06,930 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. 202 00:21:07,440 --> 00:21:13,740 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, 203 00:21:13,860 --> 00:21:19,889 which he never did. But what is ironic and nice about that is that he is now remembered by people like me 204 00:21:19,890 --> 00:21:27,120 for coming up with probably the most famous maths puzzle about navigating the globe. 205 00:21:27,620 --> 00:21:35,820 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. 206 00:21:36,150 --> 00:21:42,370 Very simple but interesting mathematical sort of curiosity wrapped within this puzzle. 207 00:21:42,390 --> 00:21:46,470 A man walks around the circumference of the world. How much further does his head travel than his feet? 208 00:21:47,070 --> 00:21:49,320 Again, I love this type of puzzle because you look at it, 209 00:21:49,320 --> 00:21:54,240 you think that's a funny thing to say to think about, and you want to know the answer and your intuition. 210 00:21:55,170 --> 00:21:59,160 I have is probably quite different to what the answer is. So you get this nice surprise on the way. 211 00:21:59,580 --> 00:22:02,460 Well, the first thing I need to do because of this is a Christmas lecture. 212 00:22:02,640 --> 00:22:06,870 What type of crazy man is going to want to walk around the circumference of the world? 213 00:22:08,780 --> 00:22:16,579 Thank you. Santa walks, runs, comes to the worlds because just say there's norovirus is going. 214 00:22:16,580 --> 00:22:20,060 Randel is reindeer, and he has the walk to deliver all the presents. 215 00:22:20,390 --> 00:22:24,590 Okay. How much this has had travel in his feet. Let's. 216 00:22:27,280 --> 00:22:32,000 Think about it and let's draw a sketch on the back of the envelope up here. 217 00:22:32,020 --> 00:22:37,930 Essentially what I'm asking is this. We can assume this is like a math lecture. 218 00:22:38,330 --> 00:22:40,870 You can assume that someone can actually walk around the world. 219 00:22:41,230 --> 00:22:52,280 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. 220 00:22:52,600 --> 00:22:56,560 Right. So you're walking around, this is Santa here. 221 00:22:56,590 --> 00:23:03,340 He's going to walk all the way around the distance that the head travels more than the feet 222 00:23:04,240 --> 00:23:11,469 is the circumference of that that basically the dotted line minus this comes from the earth, 223 00:23:11,470 --> 00:23:17,560 the dotted line minus the circumference of the shaded circle. 224 00:23:18,100 --> 00:23:22,210 Now, you would have thought that this is 40,000 kilometres that he's walking around. 225 00:23:22,870 --> 00:23:34,920 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. 226 00:23:35,920 --> 00:23:44,290 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. 227 00:23:44,560 --> 00:23:51,010 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. 228 00:23:51,010 --> 00:23:53,890 It feels like a lot longer. Anyway, let's work this out simply. 229 00:23:54,550 --> 00:24:01,000 We do this in a little bit of technical knowledge, but I'm thinking that everyone we should we should all know this, 230 00:24:01,000 --> 00:24:05,079 but just in case revision, the circumference of a circle is two pi are okay. 231 00:24:05,080 --> 00:24:08,120 But we are. So we've all got that. 232 00:24:11,440 --> 00:24:14,740 Well, what is the distance travelled by the feet to power? 233 00:24:15,250 --> 00:24:23,200 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. 234 00:24:23,510 --> 00:24:27,130 That's two way up was super to pi h. So the disk that the. 235 00:24:28,070 --> 00:24:33,080 The, um, the difference is going to be the bottom one minus the top one. 236 00:24:34,360 --> 00:24:41,860 Which is two pi h. Okay, so let's work that out as two times two, 3.1, four or thereabouts. 237 00:24:42,070 --> 00:24:45,480 So I was 1.8 which is 11 metres case. 238 00:24:45,580 --> 00:24:54,250 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, 239 00:24:54,250 --> 00:24:59,560 Euclid's elements and every new mathematician that came along did a new version with that kind of annotated notes and thoughts. 240 00:24:59,800 --> 00:25:02,890 And William Wisdom came out with one when he was a professor of maths. 241 00:25:03,190 --> 00:25:11,260 And in it he said, Isn't this interesting? Because when a man walks around the earth, this is so certainly 11 metres or. 242 00:25:12,370 --> 00:25:20,490 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. 243 00:25:21,180 --> 00:25:27,060 Nowhere does the answer include ah, the radius of the earth. 244 00:25:27,450 --> 00:25:28,440 In other words, 245 00:25:28,800 --> 00:25:39,790 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. 246 00:25:40,710 --> 00:25:47,180 Okay. Trivially if you're walking around dot. The head, the feet don't walk anything at all. 247 00:25:47,300 --> 00:25:54,380 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, 248 00:25:54,560 --> 00:25:59,720 you're walking around the largest sphere that it's possible to get in the universe still. 249 00:26:01,750 --> 00:26:10,720 The head travels only 11 metres more than the feet, which is something which is surprising and interesting. 250 00:26:11,290 --> 00:26:15,810 So. I prefer the puzzle set about a man walking around the earth because it's a red. 251 00:26:15,870 --> 00:26:19,780 That's how it was originally spotted by Western theory and 50 years ago, 252 00:26:20,140 --> 00:26:28,660 but also because the way that it is normally stated this problem, this problem, this puzzle is as the rope around the earth puzzle. 253 00:26:30,560 --> 00:26:33,620 The wrap around the puzzle. It might give you a bigger wow, 254 00:26:34,220 --> 00:26:46,010 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. 255 00:26:46,550 --> 00:26:50,750 Then someone extends this rope by one metre. 256 00:26:51,870 --> 00:26:59,490 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. 257 00:26:59,670 --> 00:27:03,600 So it's the same height all around the earth. And this is just where it gets complicated. 258 00:27:03,600 --> 00:27:10,170 Must be like, Why do you want to do that? So sometimes they say it's the iron bar around the earth. 259 00:27:10,320 --> 00:27:14,840 Do you have an iron bar around the earth? And you extend the iron bar by one metre and then you pull it up. 260 00:27:14,850 --> 00:27:17,850 So the iron bar is above the circumference at exactly the same. 261 00:27:18,060 --> 00:27:27,990 The question is by extending something from 40,000 kilometres to 40,000.001 kilometres, 262 00:27:27,990 --> 00:27:34,560 I mean, it's a tiny fraction of what animal can get underneath. 263 00:27:36,290 --> 00:27:40,400 And at first you would think, Oh, my God. Just like nothing I think would end. 264 00:27:40,820 --> 00:27:44,120 But actually, it's a small dog. 265 00:27:44,450 --> 00:27:45,590 That small dog could get under. 266 00:27:46,190 --> 00:27:54,710 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, 267 00:27:54,740 --> 00:28:00,200 that is that is surprising because it would be also the case that if you had a rope around Jupiter, 268 00:28:00,230 --> 00:28:06,140 around the biggest possible thing as a sphere and the entire universe, the exact same thing would happen. 269 00:28:06,290 --> 00:28:09,950 Extend that by a metre. Lift it up. You can get this dog underneath. 270 00:28:11,290 --> 00:28:18,730 So if I was asking that question or I was being told that question, I would say, hang on a second. 271 00:28:19,720 --> 00:28:24,940 You've got all the work to get this rope going all the way around the earth. 272 00:28:26,100 --> 00:28:32,280 You're extending it by one metre with a purpose of trying to get animals to go underneath it. 273 00:28:33,120 --> 00:28:40,350 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? 274 00:28:41,040 --> 00:28:47,040 So that's also a really interesting question. What animal now can get underneath it? 275 00:28:49,550 --> 00:28:59,210 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. 276 00:28:59,570 --> 00:29:05,840 But it turns out that if you pull if you have this rope around the earth extended by metre and you pull it. 277 00:29:07,440 --> 00:29:10,440 Because we're going to talk about festive animals. 278 00:29:10,710 --> 00:29:19,270 You can get about. A pyramid of a hundred reindeer will be able to go underneath it. 279 00:29:19,540 --> 00:29:23,140 It's basically a 122 metres high, the height of Centrepoint in London. 280 00:29:23,950 --> 00:29:30,700 Which is, again, kind of counterintuitive. You're extending it only by a tiny amount and you get so much slack. 281 00:29:31,870 --> 00:29:35,679 Now quite often when I give my talks, I, I about the philosophy. 282 00:29:35,680 --> 00:29:43,059 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? 283 00:29:43,060 --> 00:29:46,150 Why what? How does that affect real life? 284 00:29:46,510 --> 00:29:53,200 Well, the fact that we've been talking about circles but also works in straight lines if you give something and small. 285 00:29:54,120 --> 00:29:59,940 Extend something a tiny amount, you get so much more slack than you would intuitively think. 286 00:30:00,210 --> 00:30:03,270 And this is why you get things like this happening. 287 00:30:04,140 --> 00:30:13,410 So when in the heat, if you have a rail track, you only need to extend that rail track by a really small amount. 288 00:30:13,620 --> 00:30:20,550 And the slack that you get actually creates really counterintuitively large bumps. 289 00:30:22,580 --> 00:30:30,830 Right. Moving on to the next. Century or maybe two centuries after that in the Victorian Times. 290 00:30:31,040 --> 00:30:39,479 And it's all about Lewis Carroll. So. A great boom time for puzzles was the Victorian era, especially the late Victorian era. 291 00:30:39,480 --> 00:30:44,400 And one of the reasons for that was the growth in media and magazines and newspapers. 292 00:30:44,460 --> 00:30:49,680 Actually people realised that people wanted to read and do puzzles for pleasure. 293 00:30:49,980 --> 00:30:58,350 And Lewis Carroll, obviously, he's an Oxford don, much more famous for Alice in Wonderland and Through The Looking Glass. 294 00:30:58,950 --> 00:31:05,850 But he also wrote several puzzle books, none of them particularly successful because his wasn't good at writing a good puzzle. 295 00:31:06,420 --> 00:31:14,370 They were a lot more complicated and a lot more too too difficult, really, for them to be of general appeal. 296 00:31:14,910 --> 00:31:25,319 But he did invent a type of logic puzzle which has become a hugely popular type of logic puzzle and actually. 297 00:31:25,320 --> 00:31:33,680 Really. Sort of useful and good applications and pretty much any computer science course that you do, 298 00:31:33,830 --> 00:31:37,790 you will start to do you will play around the puzzle like this and it's puzzles where you 299 00:31:37,790 --> 00:31:42,439 have some people who tell the truth all the time and some people who lie all the time. 300 00:31:42,440 --> 00:31:44,960 So puzzles involving truth tellers and liars. 301 00:31:45,290 --> 00:31:54,620 And Lewis Carroll was the first person and then it was quite late in his life in the 1890s, he scribbled around and. 302 00:31:57,590 --> 00:32:04,280 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. 303 00:32:05,210 --> 00:32:09,080 And this is his original puzzle dance. She says, the dancer tells lies. 304 00:32:09,080 --> 00:32:14,660 Dancer sells it, Prancer tells lies. Prancer says that both Dasher and Dancer tells lies who's telling the truth. 305 00:32:14,810 --> 00:32:20,629 And in order to solve this, we're going to act this out. So I need three volunteers. 306 00:32:20,630 --> 00:32:23,870 Three young volunteers? Yeah, you three. Brilliant. Can you come up here? 307 00:32:24,740 --> 00:32:29,379 Yeah. One, two, three. Okay. Well, what have you had? 308 00:32:29,380 --> 00:32:38,740 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. 309 00:32:40,180 --> 00:32:44,260 Brilliance if you come and stand here three in a row. 310 00:32:48,530 --> 00:32:53,410 Yes. If you look at sort of entrance exams and things to do, 311 00:32:53,420 --> 00:32:59,630 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. 312 00:32:59,990 --> 00:33:03,230 So you. What's your name? Kathy. Kathy is Dasha. 313 00:33:03,750 --> 00:33:08,630 Chi Chi is Dancer and Alexandra is Prancer. 314 00:33:09,110 --> 00:33:12,980 So I forgot the original names. I'm just telling you with Dasher, Dancer and Prancer. 315 00:33:13,280 --> 00:33:17,470 Okay, so how do you solve something like this? Well, what you need to do first. 316 00:33:17,480 --> 00:33:22,730 Well, this is one way to solve it, is that you need to assume a truth value. 317 00:33:22,730 --> 00:33:28,220 I assume that someone is telling the truth and then just, like, work through and see what happens. 318 00:33:28,490 --> 00:33:33,800 So let's assume that Dasher is telling the truth if you're telling the truth. 319 00:33:35,020 --> 00:33:39,310 Then you're a liar, okay? You're a liar so that everyone knows you're a liar. 320 00:33:39,340 --> 00:33:46,480 This is the international symbol of being a liar. Opinion is deeper than you are now, not a lawyer. 321 00:33:46,550 --> 00:33:54,460 That's right. I that you are now a liar. You say that Prancer tells lies, but you're a liar. 322 00:33:54,580 --> 00:33:59,140 So actually you must tell the truth. Okay, so you don't know. Now's for you if you're telling the truth. 323 00:33:59,410 --> 00:34:03,490 If you say that, both of you tell lies. But you're not telling the truth. 324 00:34:03,520 --> 00:34:09,670 That's incorrect. So we have a system that is inconsistent, doesn't work. So this is not a solution to the puzzle. 325 00:34:09,850 --> 00:34:13,240 So we can eliminate the possibility that you are telling the truth. 326 00:34:13,270 --> 00:34:19,710 Unfortunately, Dasha, I think you might be a liar. So if you're a liar, could you put on the symbol of lying? 327 00:34:19,810 --> 00:34:24,310 So Dasha is not lying. But you say that dancer tells lies. 328 00:34:24,310 --> 00:34:29,070 But if you're a liar, that means that you tell the truth. Where does system hear the either a liar or truth teller? 329 00:34:29,100 --> 00:34:35,010 There's nothing in between. So if you're telling the truth, you say that Prancer is a liar. 330 00:34:35,340 --> 00:34:39,480 So if you're a liar, we need to get one of these noses for you. You're a liar. 331 00:34:39,510 --> 00:34:44,060 You say that Dasher danced, tell lies. Both, which is not true. 332 00:34:44,070 --> 00:34:48,440 So you're actually. It's a consistent system. 333 00:34:48,440 --> 00:34:51,980 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. 334 00:34:52,130 --> 00:34:58,310 So we have here a system that works. The truth values make system consistent, which means who is telling the truth? 335 00:34:58,340 --> 00:35:01,430 Only one of you is. Dancer. 336 00:35:02,630 --> 00:35:06,320 Thank you very much. Radicals are trained there. 337 00:35:06,320 --> 00:35:15,270 Thank you. Okay. Since we are on Canuck noses and lying and Pinocchio is always on the Christmas time. 338 00:35:15,410 --> 00:35:20,550 Tenuous link. I thought we would have some fun here about Pinocchio's nose. 339 00:35:20,570 --> 00:35:24,350 So this is what we're all going to do this together. Pinocchio's nose is five centimetres long. 340 00:35:24,380 --> 00:35:32,930 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. 341 00:35:34,080 --> 00:35:38,330 Think good a tennis racket. A snooker table. 342 00:35:40,670 --> 00:35:45,170 Tennis court. A football pitch. 343 00:35:47,210 --> 00:35:50,930 Okay. Well, let's just have a look. Have a look. Well, let's go through the math. 344 00:35:51,200 --> 00:35:54,940 They say so nine lines doubling each time. 345 00:35:54,950 --> 00:35:58,070 That's two to the nine times five. Sets me to the original length. 346 00:35:59,060 --> 00:36:08,050 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, 347 00:36:08,530 --> 00:36:12,639 it's that two to the ten is about a thousand as a really, 348 00:36:12,640 --> 00:36:20,980 really useful thing just to know just for sort of making good estimates to two tens, about thousand, it's actually 1024. 349 00:36:20,980 --> 00:36:22,360 So two to the nine is going to be half. 350 00:36:22,360 --> 00:36:30,580 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. 351 00:36:31,620 --> 00:36:37,980 25 metres, so most of you were wrong. It is a tennis court and it's an interesting. 352 00:36:39,590 --> 00:36:42,800 I'm a journalist. Or maybe an ex journalist. 353 00:36:43,100 --> 00:36:44,690 I didn't do much journalism at the moment, 354 00:36:44,780 --> 00:36:50,770 but journalists always trying to describe how big things are and also is is the size of several football pitches when it gets bigger, 355 00:36:50,780 --> 00:36:55,250 say, the size of several times the size of Wales. But the reason why is the football pitch is football pitches. 356 00:36:55,400 --> 00:36:59,900 You think? Well, the stadiums, they've always got a 100 metre racetrack right next to them. 357 00:37:00,200 --> 00:37:09,010 So the rule of thumb. Football pitcher stated, You know, there's always going to be years that 100 metres will be 100 metres. 358 00:37:09,160 --> 00:37:13,300 So something which is a quarter the size of a football pitch is going to be a tennis court. 359 00:37:13,630 --> 00:37:18,210 Now that is the level one question, the level two question. Neurones on the ready. 360 00:37:18,310 --> 00:37:28,280 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. 361 00:37:28,300 --> 00:37:35,200 After how many fibs with a break and how long would it be? What sorts of rights is a good question. 362 00:37:36,160 --> 00:37:43,030 Mahogany? No, I don't know, because the reason why I bring this up, the answer is apparently 1315 children, 363 00:37:43,030 --> 00:37:50,980 eight metres because of this is the only thing in the entire election day which is actually the subject of a proper academic paper. 364 00:37:51,160 --> 00:37:55,810 This is what you study at the University of Leicester. 365 00:37:58,110 --> 00:38:12,060 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. 366 00:38:12,340 --> 00:38:20,650 And these two people in a sense didn't. He was a much more interesting mathematician, a fascinating man who was completely self-taught, 367 00:38:20,860 --> 00:38:28,510 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. 368 00:38:30,620 --> 00:38:35,350 Sam Lloyd. Leaders in America became a lot more famous and a lot richer. 369 00:38:35,650 --> 00:38:39,729 He was very kind of American and entrepreneurial. 370 00:38:39,730 --> 00:38:46,360 And I want to use one of Sam Lloyd's puzzles, because it's actually actually easier to do as a group. 371 00:38:46,750 --> 00:38:56,140 And it's called the Canals of Moses from about 100 years ago. What we have we have this is a picture of Mars. 372 00:38:56,410 --> 00:39:04,360 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. 373 00:39:05,880 --> 00:39:13,350 Saying a sentence in English. There is some sentence in English that you can say. 374 00:39:14,100 --> 00:39:16,830 This is quite a famous puzzle and had a lot of attention at the time. 375 00:39:17,190 --> 00:39:22,530 And Sam Lloyd said that when the puzzle originally appeared in a magazine, more than 50,000 readers reported this. 376 00:39:22,530 --> 00:39:24,990 No possible way. Yeah. It's a very simple puzzle. 377 00:39:27,170 --> 00:39:35,510 When the puzzle originally appeared in the magazine, more than 50,000 readers reported There is no possible way. 378 00:39:36,980 --> 00:39:42,150 Yet it is a very simple puzzle. Have you got the answer? 379 00:39:42,970 --> 00:39:46,350 Don't tell us what it is. There is no possible way. 380 00:39:46,830 --> 00:39:54,960 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. 381 00:39:55,470 --> 00:40:05,460 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, 382 00:40:05,970 --> 00:40:13,770 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, 383 00:40:13,890 --> 00:40:18,980 even if someone is telling you the answer. It's hard to see it when it's right there in front of you. 384 00:40:19,190 --> 00:40:23,150 So as well as read the question you need to. 385 00:40:24,920 --> 00:40:29,850 Sometimes kind of forget everything you know, and then start again afresh. 386 00:40:30,440 --> 00:40:36,110 And often this is why this puzzles are quite similar to magic tricks that what magicians are sometimes trying to do is. 387 00:40:37,370 --> 00:40:42,380 Do something here but make you look over there. Right. Sort of sleight of hand and puzzles often. 388 00:40:42,410 --> 00:40:52,370 That is exactly the same thing. You want to lead the solver in one direction when the solution might be really obvious, 389 00:40:52,730 --> 00:40:58,400 but it's much more fun to try and strike them, take them, go in a different and different way. 390 00:40:58,790 --> 00:41:08,060 And actually, there's a famous puzzle, which is about the light bulb and the three switches. 391 00:41:08,510 --> 00:41:12,350 And this is a classic puzzle because it's set as a maths puzzle, 392 00:41:13,010 --> 00:41:17,840 but actually the answer is a kind of physics answer, cause it's all about that the bulb is going to heat up. 393 00:41:18,170 --> 00:41:24,409 But if you start trying to solve that puzzle. Thinking about mathematical combinations. 394 00:41:24,410 --> 00:41:28,400 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. 395 00:41:28,430 --> 00:41:37,759 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, 396 00:41:37,760 --> 00:41:41,930 you know, Nesbo, the Norwegian sort of bestselling crime writer. 397 00:41:41,930 --> 00:41:44,959 And he's knowing, you know, puzzles Guy, Tell Me a puzzle. 398 00:41:44,960 --> 00:41:50,630 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. 399 00:41:51,020 --> 00:41:55,820 And he basically said, okay, you're doing exactly what I do as a crime writer. 400 00:41:56,630 --> 00:42:04,640 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. 401 00:42:05,030 --> 00:42:10,140 But all I'm trying to do is to make the reader think it's not that person until right at the end. 402 00:42:10,160 --> 00:42:16,520 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. 403 00:42:16,760 --> 00:42:22,760 So what else could it be? What happens to bulbs? And he basically reverse engineered and solved it. 404 00:42:23,150 --> 00:42:30,080 And I thought that was very interesting to think that there is a kind of parallel with a great puzzle and also a fiction. 405 00:42:31,340 --> 00:42:35,120 So we could do some Christmas Eve time. Very good time. 406 00:42:37,150 --> 00:42:47,320 If there is one mathematical sort of field really that saw of the Christmas Santa Claus has to tackle with every year, 407 00:42:47,680 --> 00:42:51,220 surely it is the travelling salesman problem. 408 00:42:51,520 --> 00:42:58,569 Okay. Simon says the problem is you have all the places that you need to go get the loop. 409 00:42:58,570 --> 00:43:04,960 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. 410 00:43:05,140 --> 00:43:15,820 The first person to start thinking about this was Karl Manga in the 1930s famous mathematician. 411 00:43:16,540 --> 00:43:19,960 But coming it was actually. You see what I did here? 412 00:43:20,230 --> 00:43:23,860 Much more famous for something else. The manga sponge. 413 00:43:24,360 --> 00:43:27,670 Okay. I don't if anyone is. It doesn't make us money. 414 00:43:27,820 --> 00:43:31,240 Well, we some people here, we've got some professors of mathematics. I'll be surprised if they don't know what it is. 415 00:43:31,240 --> 00:43:35,770 The manga sponge that we don't know is a really fun, 416 00:43:35,980 --> 00:43:41,830 fascinating object that's quite simple to describe and kind of cool to look at, is that let's say we have a cube. 417 00:43:42,010 --> 00:43:47,650 This is a cube. We're going to divide the cube into 27 smaller cubes, just like it's a Rubik's cube, say. 418 00:43:47,800 --> 00:43:51,280 So three by three. By three by three. So 27 cubes. 419 00:43:51,610 --> 00:43:58,180 And we're going to extract the middle cube in each of the sides, plus the very central cube. 420 00:43:58,940 --> 00:44:01,990 Okay, so there's there should be 27. 421 00:44:02,290 --> 00:44:06,219 If it's a block, then we take out seven, five, six, one on each side. 422 00:44:06,220 --> 00:44:11,260 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. 423 00:44:11,920 --> 00:44:15,280 And then the next process is Let's treat it. 424 00:44:15,640 --> 00:44:23,740 It's a fractal object. So we're going to repeat the process, take each cube and do exactly the same thing. 425 00:44:24,070 --> 00:44:34,700 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, 426 00:44:35,150 --> 00:44:40,580 the sort of cube, this growing wholes, and then we carry on one way iteration and we get this. 427 00:44:41,180 --> 00:44:47,090 This is a mega cubical, there's a level three mangcu because we've done this region three times. 428 00:44:47,390 --> 00:44:51,440 And one of the reasons why this is a really interesting object is that. 429 00:44:52,870 --> 00:44:56,530 Each time we take out holes, we increase the surface area. 430 00:44:57,590 --> 00:45:01,000 But we reduced the volume. So. 431 00:45:01,990 --> 00:45:09,700 In the limit. We're going to have an object which has no volume, an infinite surface area, which is kind of interesting. 432 00:45:10,330 --> 00:45:15,080 Okay. How is this relevant to Christmas? 433 00:45:15,110 --> 00:45:25,250 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. 434 00:45:25,940 --> 00:45:29,630 And there was one which I really liked actually. I thought this was a really nice puzzle. 435 00:45:30,700 --> 00:45:34,489 I think it was. The University of Leeds set it and it's a match. 436 00:45:34,490 --> 00:45:44,160 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. 437 00:45:44,160 --> 00:45:48,230 And then you dip into water. We dip it in halfway. 438 00:45:48,590 --> 00:45:51,950 What is the shape on the surface? 439 00:45:52,790 --> 00:46:00,270 I think this is really interesting because it's really hard to visualise even though it's a cube. 440 00:46:00,350 --> 00:46:06,350 One of the most simple objects. Really hard to visualise, actually. What you get is a hexagon. 441 00:46:06,810 --> 00:46:12,170 I can show that here. It's basically if you slice a cube like this, you get a hexagon. 442 00:46:12,860 --> 00:46:18,020 Okay. So you go halfway along each side and then diagonally. 443 00:46:18,290 --> 00:46:24,830 So you get to that. That's. Looks like it's why France from here, actually. 444 00:46:25,160 --> 00:46:29,120 But if you see the debt, you basically get. 445 00:46:30,160 --> 00:46:34,360 You split as a way of taking a diagonal slice, which you get a hexagon, 446 00:46:34,540 --> 00:46:39,790 which is so the both equals equal size, and you also get it with you're dipping the Cubans. 447 00:46:39,820 --> 00:46:42,930 The question is. What? 448 00:46:44,110 --> 00:46:48,110 Pattern do you get if you slice the manga? 449 00:46:49,450 --> 00:46:53,260 Sponge on one of those diagonal slices. Okay. 450 00:46:53,280 --> 00:46:59,439 And when I was on this, I had no idea. But I was asked this by a guy called George Hart, 451 00:46:59,440 --> 00:47:05,080 who is a well-known American geometrical sculptor and geometry, and is actually also the father of my heart. 452 00:47:05,260 --> 00:47:14,110 He does all these amazing YouTube doodle videos, and he said that this was probably the biggest wow that he knew of in mathematics. 453 00:47:14,470 --> 00:47:17,530 And this is a guy who's like knows loads of wows. 454 00:47:18,790 --> 00:47:26,140 This is his business, his wives of geometry, because he's they're making these amazing sculptures, which just most people feel, wow. 455 00:47:26,770 --> 00:47:30,940 So it's not something that you might be able to work out for yourself. 456 00:47:30,940 --> 00:47:37,570 I certainly couldn't. But if you are to slice this level three. 457 00:47:38,850 --> 00:47:42,060 Manga sponge with a hexagonal slice. 458 00:47:43,840 --> 00:47:49,580 This is what you get. Which as well as being surprising and wonderful. 459 00:47:49,820 --> 00:47:59,930 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. 460 00:48:00,530 --> 00:48:05,780 And it's almost exactly 6:00. So thank you very much.