1 00:00:01,420 --> 00:00:16,050 George. So I think we have a packed room, completely packed. 2 00:00:16,070 --> 00:00:20,180 I'm glad we don't have to use the other little room because it's never quite as nice. 3 00:00:20,420 --> 00:00:31,310 So welcome to all of you, to the Matamata Institute to see so, so good Christmas lecture, ending a very successful year of public lecture. 4 00:00:31,790 --> 00:00:37,850 And there is no better way to end the year than having Professor Marcus de Soto with us today. 5 00:00:38,030 --> 00:00:44,120 Is the, as many of you know, the University of Sydney chair of Public Understanding. 6 00:00:44,630 --> 00:00:49,010 And it's going to be a real Christmas treat for all of us. Thank you, Phillip. 7 00:00:51,200 --> 00:00:55,250 Do we need any more people sitting? Is there anybody we need? So I think we could actually forget. 8 00:00:56,120 --> 00:01:03,019 Okay, so before we start, actually, we have a word from our sponsor. 9 00:01:03,020 --> 00:01:10,190 For the first time, we have a sponsor for a public lecture. We started very small, but as we are expanding, we were very grateful to have a sponsor. 10 00:01:10,190 --> 00:01:17,870 So I'm always wanted to have a radio show so I can say another word from a sponsor or sponsor today is G Research. 11 00:01:18,560 --> 00:01:23,480 If you're wondering who G research is, I'll tell you G research is a fast growing, 12 00:01:23,690 --> 00:01:28,070 well-established financial research company located in central London. 13 00:01:28,520 --> 00:01:37,790 The quantitative researchers develop ideas to predict return in global markets by finding patterns in large, noisy and rapidly changing data set. 14 00:01:38,270 --> 00:01:45,110 They have opportunities for candidates to apply mathematical concept to real world problems in a relaxed yet dynamic environment. 15 00:01:45,710 --> 00:01:52,430 So I'm particularly grateful because with the support, we are going to be able to expand further our program of public lecture and really 16 00:01:52,430 --> 00:01:56,959 bring the best mathematicians from all around the world to give a public lecture here. 17 00:01:56,960 --> 00:02:02,390 So we have an exciting programme coming. We have talks on the mathematics of crime. 18 00:02:03,080 --> 00:02:07,520 Later next term we'll have talks on mathematics of choice would have another lecture, 19 00:02:07,520 --> 00:02:14,270 I hope, with the coming book of Marcus later on this year and many more coming next year. 20 00:02:14,270 --> 00:02:21,139 So we have an exciting programme of course tonight it's a special night, just not because we have Marcus with us tonight, 21 00:02:21,140 --> 00:02:29,390 but for many of you know, it's also the opening night of Star Wars and that makes it, of course, very special. 22 00:02:29,390 --> 00:02:36,230 And I think there's probably a few of you who came here just to be in a warm environment rather than camping outside the movie theatre with. 23 00:02:36,230 --> 00:02:38,990 And that's perfectly all right. 24 00:02:38,990 --> 00:02:45,710 In the spirit of intergalactic peace, I think we can accommodate you and do a little bit of mathematics in the meantime, 25 00:02:46,280 --> 00:02:49,579 but it really made me think and I said, okay, so that's good. 26 00:02:49,580 --> 00:02:53,240 Star Wars where would Marcus fit in a Star Wars universe? 27 00:02:55,670 --> 00:03:01,139 So first we have, of course, to decide who is the dark side and who is the non dark side, whatever they call it. 28 00:03:01,140 --> 00:03:07,040 To the bright side of the light side where I don't know for you, but for me it's quite clear. 29 00:03:07,040 --> 00:03:07,879 That's an easy one. 30 00:03:07,880 --> 00:03:15,560 You know, if you look at the dark side, guys, they have they have all the the cool gadgets and of course, they have a great sense of fashion. 31 00:03:16,220 --> 00:03:20,390 So for me, clearly the dark side is the applied mathematicians. 32 00:03:22,720 --> 00:03:28,690 Oh, come on. I knew you could imagine a mathematician in a little hurt with the little rope solving equation. 33 00:03:28,930 --> 00:03:32,170 But you wouldn't. You wouldn't have them build the Death Star. 34 00:03:32,190 --> 00:03:35,650 Probably. Right? Right. Okay, so that's the easy one. 35 00:03:35,680 --> 00:03:41,770 No, we have to decide. In that universe, we have the dark side and we call it the pure sides. 36 00:03:42,250 --> 00:03:47,110 Make it easy. So what? 37 00:03:47,500 --> 00:03:53,710 Who would Marcus be? Of course, Marcus is definitely on the pure side is he's a pure mathematician of great renown. 38 00:03:54,070 --> 00:04:01,380 He's made important contribution in group theory, in the theory of prime number is still very active in this field. 39 00:04:01,390 --> 00:04:07,930 He received in 2001. The Berkeley Prize from the London Mathematical Society clearly is very strong. 40 00:04:07,930 --> 00:04:11,560 So which one would he be? Where could he be? Obi-Wan Kenobi. 41 00:04:12,790 --> 00:04:20,320 Yoda? Well, I think he's way too alive to be Obi-Wan and and to articulate for Yoda to be. 42 00:04:22,420 --> 00:04:29,709 I think it would be it would make a great Jar Jar Binks personally because of his ability to entertain people. 43 00:04:29,710 --> 00:04:33,790 But if you think about it, the force is strong in this one. 44 00:04:36,310 --> 00:04:46,570 And in recent years I've heard Marcus talk about algorithm, I've heard about all useful mathematics and how it applies to everyday life. 45 00:04:46,750 --> 00:04:54,010 I even saw him talk about units kilogram metre, something that pure mathematician would never touch. 46 00:04:54,490 --> 00:04:57,880 So I think he's definitely getting corrupted by the dark side. 47 00:04:58,960 --> 00:05:02,410 So that leaves only one main character for me. 48 00:05:02,410 --> 00:05:11,170 Marcus is the Luke Skywalker of mathematics, and maybe he's going to be the one to unite both sides. 49 00:05:11,170 --> 00:05:25,720 Who knows? So but before I ask you to join me, I have to take my laser sabre and tell you about the emergency exit over here and there and there. 50 00:05:25,750 --> 00:05:27,550 We're not going to fight tonight, I hope. 51 00:05:28,120 --> 00:05:36,040 But so if you in case of emergency, you can take any of this exit and you'll end up in open space where nobody will hear you scream. 52 00:05:36,040 --> 00:05:42,189 Of course. So a final word for Marcus. 53 00:05:42,190 --> 00:05:45,250 Please come, Marcus. Come join me. Marcus. 54 00:05:47,050 --> 00:06:06,020 Thank you. Very, very. I actually really enjoy this time of year because there is so much maths hiding everywhere. 55 00:06:06,680 --> 00:06:12,560 But there's something really weird because I'm around my. Carol singers just will not come to my door anymore. 56 00:06:13,040 --> 00:06:18,500 I've wondered why this was in my mind. Children explain to me it's because that every time they come, 57 00:06:18,710 --> 00:06:23,740 you find some mathematical thing in the things that they're singing and they don't come anymore and it's go round it. 58 00:06:24,590 --> 00:06:29,660 So I've got this captive audience now, and so you can sing me carols at the end if you want. 59 00:06:30,140 --> 00:06:35,480 But so what I wanted to do is to give you a little bit of the things that I bought my Carol singers with. 60 00:06:36,080 --> 00:06:40,729 I'm actually I live in East London and so I actually live in quite a Jewish neighbourhood as well. 61 00:06:40,730 --> 00:06:50,930 And we've just finished celebrating. All the Jewish families have had candles in their windows because they just finished celebrating Hanukkah. 62 00:06:51,470 --> 00:06:58,459 Hanukkah is probably the first great sort of mathematical holiday of this time because the candles, 63 00:06:58,460 --> 00:07:07,250 if you know the story of Hanukkah is about the miracle of lights and they light candles because the oil should have only lasted for one day, 64 00:07:07,250 --> 00:07:14,930 actually ended up lasting for eight days. And so they celebrate Hanukkah by lighting candles over the eight days of Hanukkah. 65 00:07:14,930 --> 00:07:17,930 And you get these little boxes with the candles in. 66 00:07:18,440 --> 00:07:25,550 And one of the challenges they set all the kids is can you work out how many candles are in the box? 67 00:07:26,150 --> 00:07:33,530 Because the rules are that on the first day you light one candle but with another candle, which is called the shamash. 68 00:07:33,830 --> 00:07:40,430 So you actually like two candles on the first day of Hanukkah and then the second day you like three candles, 69 00:07:40,440 --> 00:07:47,020 the one that's lighting, and then you like two more. So by the end you you light nine candles in total. 70 00:07:47,030 --> 00:07:51,770 So so that's the challenge you set the kids is, well, how many candles are there in the box? 71 00:07:52,010 --> 00:07:52,999 And actually it's it's quite similar. 72 00:07:53,000 --> 00:07:59,450 It's amazing how, you know, actually all of these holidays are about the fact that it's really dark out there, the Festival of Lights as well. 73 00:07:59,930 --> 00:08:06,920 But Christmas also has a kind of version of this kind of challenge as well, because some of those carols saying is that come round to me, 74 00:08:07,370 --> 00:08:10,310 they're all singing on the first day of Christmas, my true love sent to me. 75 00:08:10,910 --> 00:08:14,959 And so one of the challenges again is, well, how many presents on each day do you get? 76 00:08:14,960 --> 00:08:21,980 And it's a similar sort of problem that you're adding. You're adding up, well, one partridge in a pear tree, two turtle doves. 77 00:08:22,160 --> 00:08:28,370 And so you've got these kind of the same sort of challenge where you've got to add up the number of presents on each day. 78 00:08:29,120 --> 00:08:35,089 So I always challenge them. Okay, what if Christmas seems to start earlier and earlier every year? 79 00:08:35,090 --> 00:08:38,300 So, you know what if it actually was 100 days of Christmas? 80 00:08:38,690 --> 00:08:41,659 So I always challenge the carols saying I think this is why they never come anymore. 81 00:08:41,660 --> 00:08:49,489 So, okay, so you know what, if they were like 100 iPods, 99 copies of Music of the Primes and all the way down to, 82 00:08:49,490 --> 00:08:55,340 you know, could you tell me how many presents that you're going to get on the hundredth day of Christmas? 83 00:08:55,820 --> 00:09:00,740 And of course, that's a nice way to to view this, because you already saw this kind of triangle building up. 84 00:09:00,740 --> 00:09:07,520 So if you stack all of the presents up, so, for example, you've got one down here and then you've got two turtle. 85 00:09:07,790 --> 00:09:12,290 This is a partridge in there and two turtle doves, three French hens and four watts. 86 00:09:13,770 --> 00:09:18,440 Yeah, that's what you say. So these are actually examples of things called the triangular numbers. 87 00:09:18,440 --> 00:09:26,540 So the triangular numbers apply both to counting the number of candles in Hanukkah and also the number of presents you get on each day of Christmas. 88 00:09:26,810 --> 00:09:31,580 And there's a nice way to actually quickly calculate how many presents there are in here, 89 00:09:31,970 --> 00:09:37,310 because what you do is you take two of these sort of sets of presence and you slot them together. 90 00:09:37,880 --> 00:09:42,230 So if you do that, you get a rectangle and it's very easy to count things in a rectangle. 91 00:09:42,230 --> 00:09:48,170 So if I had 100, the hundred Days of Christmas, I stacked all of those presents up and I took another copy of that. 92 00:09:48,170 --> 00:09:55,910 I put them together. I now have a rectangle which is has 100 boxes down one side, but 101 down the other side. 93 00:09:56,420 --> 00:09:58,610 So that's two copies of what I'm trying to count. 94 00:09:59,000 --> 00:10:06,710 So if I do 100 times 101, I divide that by two, I'll find out how many there are in that triangular set of boxes. 95 00:10:07,010 --> 00:10:13,490 So that gives you a nice sort of formula actually to calculate the number of presents you'll get on each day of Christmas. 96 00:10:13,970 --> 00:10:16,970 But kind of she has a little trick to it, of course, because. 97 00:10:17,690 --> 00:10:23,510 So you think. Oh, right. So it's the ninth triangular number is the number of candles that'll be in that box. 98 00:10:23,900 --> 00:10:27,110 But of course, a little trick you have to remember is that there's no first day. 99 00:10:27,380 --> 00:10:31,880 So you have to take one of that number because you start with two plus three plus four. 100 00:10:31,880 --> 00:10:37,459 So it isn't quite the ninth triangular number is the ninth triangular minus the first candle which never gets used. 101 00:10:37,460 --> 00:10:41,840 So so that's always a trick to see how clever the kids in a kind of carol. 102 00:10:41,840 --> 00:10:47,120 So it's actually you've got 44 candles in that box, but of course, actually. 103 00:10:47,330 --> 00:10:51,740 So that's a kind of nice problem. It's sort of a two dimensional triangle you've got there. 104 00:10:51,950 --> 00:10:57,680 But actually I rather like Christmas because there's a little bit more of a challenge inside that problem. 105 00:10:57,760 --> 00:11:01,710 Because yeah, on each day of Christmas you've got a triangular number of presents. 106 00:11:01,720 --> 00:11:05,380 But what about over the whole of the 12 days of Christmas? 107 00:11:05,560 --> 00:11:09,670 How many presents do I get from my true love over the 12 days? 108 00:11:09,670 --> 00:11:16,239 Because then actually you start the problem starts to become a little bit more interesting because it's not just two dimensional. 109 00:11:16,240 --> 00:11:20,979 So we've seen the triangular numbers give you the number of presents on each particular day. 110 00:11:20,980 --> 00:11:24,790 But what about over the whole of the 12 days of Christmas? 111 00:11:25,000 --> 00:11:28,120 Well, the interesting thing is that this is a three dimensional problem. 112 00:11:28,120 --> 00:11:31,390 So Hanukkah, the Jewish holiday, is a sort of two dimensional holiday. 113 00:11:32,050 --> 00:11:35,980 Christmas goes one dimension up. We've got a three dimensional holiday for Christmas. 114 00:11:36,670 --> 00:11:40,389 So the point is, you know, you get a partridge in a pear tree on the first day, 115 00:11:40,390 --> 00:11:45,880 but on the second day, you also get another partridge in a pear tree and the two turtle doves. 116 00:11:46,360 --> 00:11:53,739 So already you've got this kind of little stack coming up. And then the Partridge in the Pantry, two title ups and the three French hens. 117 00:11:53,740 --> 00:11:59,620 So already you've got some kind of got three partridges down there so you can see what happens. 118 00:11:59,860 --> 00:12:06,099 You actually are building up these triangles until you get some kind of like actually a sort of pyramid effect. 119 00:12:06,100 --> 00:12:11,679 So if it's 12 days of Christmas, actually, what you've got to do is to count things in this kind of pyramid. 120 00:12:11,680 --> 00:12:14,829 So this is a triangular based pyramid. 121 00:12:14,830 --> 00:12:19,180 So we call these the tetrahedral numbers. So how many? 122 00:12:19,270 --> 00:12:23,679 So you've got 12 layers of this. Is there some clever way to work out how many? 123 00:12:23,680 --> 00:12:26,979 You know, I could count them all up, but I'm rubbish at mental arithmetic and I made some mistake. 124 00:12:26,980 --> 00:12:30,310 And so is there some clever way to find out how many there are? 125 00:12:30,550 --> 00:12:31,540 Because I remember what we did. 126 00:12:31,540 --> 00:12:37,930 We took with a triangle, so we took two triangles and put them together and we can make a rectangle and then that's easy to count. 127 00:12:38,230 --> 00:12:44,920 Well, the wonderful thing is you can do the same thing with these tetrahedron, so this time you need six tetrahedron. 128 00:12:45,250 --> 00:12:48,670 So this is what we're trying to add up. Those are the all the triangular numbers. 129 00:12:48,910 --> 00:12:59,559 But you can put six of these kind of pyramids together into a box, which then has 12 one on one side, 13 along the other on 14 up the top. 130 00:12:59,560 --> 00:13:05,260 So actually, this gives you another clever way, a three dimensional way to see how many boxes you will need in total. 131 00:13:05,500 --> 00:13:09,250 So we take 12 times, 13 times 14 divided by six. 132 00:13:09,800 --> 00:13:14,860 The rather remarkable thing is you've got a present for every day of the year except for Christmas. 133 00:13:15,610 --> 00:13:19,059 So 364 presents inside there. 134 00:13:19,060 --> 00:13:23,650 So, so actually. So we've got you know, it's quite a nice little trick for that. 135 00:13:24,220 --> 00:13:31,790 So a three dimensional holiday. That's great. And here's how far I have to go in promoting science across the universe. 136 00:13:31,900 --> 00:13:39,490 So The Daily Telegraph got me to explain this problem, and the editor brought along a Santa outfit, which I had to dress up in the snow. 137 00:13:40,510 --> 00:13:43,060 So there's me trying to calculate how many presents there are. 138 00:13:43,210 --> 00:13:47,650 But I thought, you know, let's get really geeky and let's have our own sort of a holiday. 139 00:13:48,130 --> 00:13:52,360 So I thought, you know, so you've got a two dimensional Jewish holiday, three dimensional. 140 00:13:52,450 --> 00:13:59,140 Why don't we have a sort of four dimensional holiday? So I sort of started to think about a4d festive celebration of science. 141 00:13:59,740 --> 00:14:03,520 So if you think about it, actually, the triangles is a kind of single summation. 142 00:14:03,520 --> 00:14:07,150 You're adding up a sum of things, but the trunk as a pyramid numbers, 143 00:14:07,300 --> 00:14:11,770 it's a sum of sums because we're adding up the triangles which are themselves sums. 144 00:14:12,010 --> 00:14:16,909 So actually to make a four dimensional holiday, what we've got to do, I mean, we can do a sort of song for this. 145 00:14:16,910 --> 00:14:20,680 So I was trying to think on so I've called it's science maths. 146 00:14:20,950 --> 00:14:26,469 So on the first day of science and I asked my geek friend sent to me what I thought Bose on in the LHC seemed quite good. 147 00:14:26,470 --> 00:14:27,790 So that's fine. 148 00:14:27,790 --> 00:14:34,960 But now we're going to go to the second day of science, maths, and I have to do a cut of some of his, some of his, some to get this to be a4d holiday. 149 00:14:35,530 --> 00:14:39,249 So so on the second day of science nice my friend sent to me. 150 00:14:39,250 --> 00:14:43,450 So I've gone for two twin primes and it goes on in the LHC. 151 00:14:43,450 --> 00:14:46,809 But in order to make it for D, I have to add that on again. 152 00:14:46,810 --> 00:14:51,639 So we had another repeat of a Bose on in the LHC and this way we do this, 153 00:14:51,640 --> 00:14:57,290 we'll build up a thing where the number of scientific things will get well, will be you have to be a foot. 154 00:14:57,310 --> 00:15:02,110 We'll have four dimensional boxes that we'll be putting together. So there's a challenge for you I want to work out. 155 00:15:02,470 --> 00:15:05,709 So I decided how many days should Science Match have? 156 00:15:05,710 --> 00:15:10,540 Well, it should be a prime number because I'm obsessed with primes, so I've said it's going to have 13 days. 157 00:15:10,540 --> 00:15:19,120 So I actually sent out a sort of request across Twitter for people to come up with things for all of the 13 days of science, maths, so, 158 00:15:19,300 --> 00:15:24,810 so hey, so we had the Higgs Bose on which there we would think there's one, there might be more than one, of course, actually we're not sure. 159 00:15:24,820 --> 00:15:31,030 Twin Primes three laws of Motions for pairing basis five Platonic Solids six quarks is spinning. 160 00:15:31,240 --> 00:15:38,590 Seven base units measuring eight bits are pointing nine making the numbers prime generating ten row shock inkblots, 161 00:15:39,190 --> 00:15:46,390 diagnosing 11 dimensions, stringing 12 astronauts moonwalking and 13 Neptune moons and orbiting. 162 00:15:46,810 --> 00:15:48,040 So then that's a challenge for you. 163 00:15:48,040 --> 00:15:53,950 You now have to work out how many of these things you get over the 13 days of Christmas using four dimensional pyramids. 164 00:15:54,280 --> 00:15:57,370 So there's a time as well you can send me if you think there is a better. 165 00:15:57,500 --> 00:16:02,030 Choices for your ones. I'm quite happy to get them through Twitter if you think some of those are a bit rubbish. 166 00:16:03,330 --> 00:16:08,180 So you can see, although there are lots of good maths already in some of the songs and the festivities with singing. 167 00:16:08,420 --> 00:16:13,910 But of course this is the time of year when it's really party time and party time. 168 00:16:14,270 --> 00:16:16,430 There's also some great maths in politics. 169 00:16:17,150 --> 00:16:21,260 So one of the things about parties is that especially at Christmas, everyone seems to get off with each other. 170 00:16:21,920 --> 00:16:26,000 And so the challenge is, okay, they're all going to get off with each other. 171 00:16:26,300 --> 00:16:34,520 Can we get the pairings to work? So actually there isn't some dreadful breakdown in the office after everyone sort of 172 00:16:34,520 --> 00:16:37,340 paired up and people are unhappy because they wanted to go off with somebody else. 173 00:16:37,790 --> 00:16:42,640 So actually maths has come up with a great algorithm in order to you know, 174 00:16:42,650 --> 00:16:50,180 if you've got a party and you want to pair people up so everybody will be there, all of the pairings will be stable. 175 00:16:50,750 --> 00:16:56,090 So I thought I'd show you how this works. It's just wonderful algorithm. So I'm going to need some volunteers for this. 176 00:16:56,090 --> 00:17:02,030 So we're going to have eight people at my party. So I need four boys and four girls, basically. 177 00:17:02,030 --> 00:17:07,159 So can I have four volunteers? And we're going to try and show you how this mathematical algorithm works up. 178 00:17:07,160 --> 00:17:10,370 So so everything is kind of safe, so. Yeah. Okay, that's good. 179 00:17:10,420 --> 00:17:13,820 You have one here? Yeah, you can come up. Excellent. Yeah. Good, right. 180 00:17:14,000 --> 00:17:17,720 Yeah. Back there. So I think that yes. 181 00:17:18,350 --> 00:17:23,419 Hand up their great you want to call it excellence. Good. So let's, let's bring you up. 182 00:17:23,420 --> 00:17:31,159 And so the boys, you take a king and you go over there and the girls, you take a queen and you come over here. 183 00:17:31,160 --> 00:17:34,370 So if you want to come up, great, we'll see how we're doing. 184 00:17:34,730 --> 00:17:39,530 So you're going to take you take the king of diamonds. So we've got this, right? 185 00:17:39,530 --> 00:17:42,919 So we go to my boys. Okay, so two, two more girls on each. Yeah, great. 186 00:17:42,920 --> 00:17:46,430 Excellent. And they're right. Good. Absolutely. 187 00:17:47,150 --> 00:17:51,470 Great. So let's see. So we got them. I'm going to put them in order. 188 00:17:51,680 --> 00:17:55,129 I play a lot of poker. So. So king of spades. 189 00:17:55,130 --> 00:17:58,760 I'm going to put you here. And king of diamonds. I'm going to put your hair. 190 00:17:59,210 --> 00:18:02,600 Excellent. So what have you in the order. And then queen of spades. 191 00:18:02,600 --> 00:18:07,219 Right the end. So that's good. You can swap round. Poker order goes like that. 192 00:18:07,220 --> 00:18:10,640 Excellent. Right. So what I got is on each of the sides. 193 00:18:10,640 --> 00:18:19,010 So. So they basically ordered such that they give me a little order here about who they fancy. 194 00:18:19,520 --> 00:18:24,350 So here we go. So let's see the king of spades. So let's see your first choice. 195 00:18:24,560 --> 00:18:28,580 You really fancy. Oh, yeah. Let's make sure we get this the right way around. This the trouble with symmetry. 196 00:18:28,970 --> 00:18:35,120 So, yeah. Make sure your cards are. Yeah, exactly. So the and your ones are down this side, so. 197 00:18:35,120 --> 00:18:38,509 Yeah. So you want to turn yours that way. Excellent. Right. 198 00:18:38,510 --> 00:18:41,510 So, so you really fancy the queen of diamonds? 199 00:18:41,660 --> 00:18:44,780 Okay, that one there. Yeah. Okay. 200 00:18:44,960 --> 00:18:48,260 And which one? You really don't want to end up with? 201 00:18:48,260 --> 00:18:52,219 The Queen of spades. Yeah. Okay, so. So that's the kind of order we got here. 202 00:18:52,220 --> 00:18:57,530 Well, let's see what you, you know, what's what's your kind of wish? So the queen of spades here on course. 203 00:18:57,530 --> 00:19:01,099 Your first choice is the king of spades. Really wants to do well. 204 00:19:01,100 --> 00:19:06,979 Ain't going to happen, is it? So then you've got the king of diamonds, king of hearts, and the king of clubs. 205 00:19:06,980 --> 00:19:11,090 Right down there is the one you really don't want to end up with. So the challenge here is, 206 00:19:11,090 --> 00:19:17,149 can we find a way with all of these preferences to pair everyone up so that nobody 207 00:19:17,150 --> 00:19:20,510 will run off with somebody else because they there's a better option for them? 208 00:19:20,690 --> 00:19:23,780 You see what would happen if we so let's pair up. 209 00:19:24,260 --> 00:19:27,560 So we're going to do the king of spades and the Queen of spades. So if you'd like to come to the front here, 210 00:19:27,770 --> 00:19:32,989 let's suppose we so you're going to pair up there and then we'll have the king of hearts and the Queen of hearts. 211 00:19:32,990 --> 00:19:39,410 So king of hearts, if you come down here, let's see if you were going to suppose we paid them up like this, why wouldn't this work? 212 00:19:39,560 --> 00:19:42,270 Well, the trouble is that the you know, 213 00:19:42,320 --> 00:19:47,240 we've already seen the king of space really doesn't want to be with the queen of spades, but he wants to run off. 214 00:19:47,240 --> 00:19:51,380 He'd be quite happy. He'd be quite happy. The second choice on his is the queen of Hearts. 215 00:19:51,590 --> 00:19:55,669 But what about the queen of hearts? I mean, queen of hearts may not like the king of spades, actually. 216 00:19:55,670 --> 00:19:59,000 You know, you've been paired with the king of hearts as your bottom choice. 217 00:19:59,240 --> 00:20:04,070 King of Spain. So king of hearts, queen of spades is the next one up. So actually you'd be quite happy to run off. 218 00:20:04,520 --> 00:20:07,790 So both of you, if we try pair this off, we'd have a disaster. 219 00:20:07,790 --> 00:20:10,909 They'd run away with each other. So that would just just wouldn't work. 220 00:20:10,910 --> 00:20:12,500 Okay, so you want to get back to your places. 221 00:20:12,530 --> 00:20:19,939 So here's the challenge is, is there a way to pair these up such that nobody will run off with somebody else in these things? 222 00:20:19,940 --> 00:20:23,300 Maybe there isn't the way to do this. Maybe it's always some instability. 223 00:20:23,510 --> 00:20:29,080 If you think about rock, paper, scissors, you know, always there's something which beats something else and you can never make it work. 224 00:20:29,090 --> 00:20:32,930 So something is sort of stable when there is a way to do this. 225 00:20:32,930 --> 00:20:36,860 And so the way to do this is to get so let's put those back. 226 00:20:37,850 --> 00:20:45,319 So the algorithm that was that was come up with is that the queens all propose to your first choice. 227 00:20:45,320 --> 00:20:48,860 KING So look at all the ones on the top of your list. 228 00:20:49,070 --> 00:20:53,000 What I want you to do is go and stand in front of the one that you would really like. 229 00:20:53,060 --> 00:20:56,240 Okay? So if you go, let's see how many, but who's who's the. 230 00:20:56,340 --> 00:20:59,920 Dorsey of this kind of outfit, I think. Oh, yeah, they all want to. 231 00:20:59,970 --> 00:21:03,290 Yeah. Most of them want to be. Yeah. 232 00:21:03,300 --> 00:21:06,450 Look at him. Isn't he cute? Wow. Yeah, it's the hat. 233 00:21:06,450 --> 00:21:10,319 I think we started it, so. Look at them. They're buzzing around. These two go. 234 00:21:10,320 --> 00:21:15,149 Poor guys are like, No, nobody wants us, but. Oh, well, at least you. So you got somebody proposed to use. 235 00:21:15,150 --> 00:21:20,770 That's one. Okay, but what do we do about this lot? Okay, well, basically, you get to choose. 236 00:21:20,790 --> 00:21:24,209 So. So who's your top choice? Queen of diamonds. 237 00:21:24,210 --> 00:21:28,320 Yeah. Okay. So I'm afraid the other two get rejected. So if you come home, I'd say yeah. 238 00:21:28,320 --> 00:21:31,350 So you could get the queen at times for the moment. So the other two. Come here. 239 00:21:31,620 --> 00:21:34,949 You got to find something else to do. So. Right. Okay, so. 240 00:21:34,950 --> 00:21:39,210 So that's the first round of this. So we seen the first round, but the two of them got rejected. 241 00:21:39,870 --> 00:21:44,549 So these are provisional pairings. But now, okay, so first choice you're not going to get. 242 00:21:44,550 --> 00:21:51,360 So let's go for your second choice. Okay. So if you want to go and propose to your second choice, so let's see where they go. 243 00:21:51,360 --> 00:21:58,110 I think we get now. So. So it may work. 244 00:22:00,480 --> 00:22:04,230 Great. Oh, poor king of hearts is kind of like, shucks, I'm not got anybody. 245 00:22:04,380 --> 00:22:08,820 But now you got a choice now. That's right. So. So you got queen of hearts and queen of plump. 246 00:22:08,850 --> 00:22:12,030 So which one do you like? Oh, that one. Well, I'm sorry. 247 00:22:12,030 --> 00:22:15,320 You got to come away then. So how is that? That didn't last it. 248 00:22:15,330 --> 00:22:20,940 Gosh. Well, okay, so you come over here, so you've just got rejected, so let's reject you. 249 00:22:21,120 --> 00:22:24,359 Okay, so now so you king of clubs is your top choice. 250 00:22:24,360 --> 00:22:28,170 So you got the king of diamonds so you could go and propose to the king of diamonds. 251 00:22:28,860 --> 00:22:33,930 Okay, let's see what the king of the why now, King of Dimes is gonna go and see the king of diamonds. 252 00:22:34,000 --> 00:22:37,049 Yeah, he's. There he is. Okay, so now let's see. 253 00:22:37,050 --> 00:22:40,110 Which one do you prefer? So your top choice is queen of Queen of hearts. 254 00:22:40,110 --> 00:22:43,410 You got your top choice. So, yeah, you're going to reject this one here. 255 00:22:43,740 --> 00:22:47,570 Okay. Oh, death. It's terrible, isn't it? Yeah. So. 256 00:22:47,760 --> 00:22:50,940 So we got this one now, so. All right, so you just got rejected again. 257 00:22:51,240 --> 00:22:54,780 Christ, God, that's two rejection. So we're right down here. 258 00:22:54,780 --> 00:22:58,200 Okay, King of hearts. Okay, so now you propose a way to the King of hearts. 259 00:22:58,680 --> 00:23:01,740 Finally, the king of hearts gets lucks out. So. 260 00:23:02,550 --> 00:23:07,830 And now everyone is paired up with a single person. And this algorithm, if you carry this algorithm on, 261 00:23:08,250 --> 00:23:15,690 it means that you can prove that this will lead to a stable pairing such that now if anybody looks and sees, 262 00:23:15,690 --> 00:23:20,370 well, I actually would prefer somebody else, but they have got somebody who's better than the one who's proposing. 263 00:23:20,370 --> 00:23:25,680 So so we find now that nobody is going to run off with anyone else, even though they might not have got their top choice, 264 00:23:25,920 --> 00:23:30,450 they can't offer somebody else who would say, well, I'm already with somebody I prefer than you. 265 00:23:30,450 --> 00:23:34,739 So. So this algorithm works. Now, the intriguing thing is we could have done this the other way round. 266 00:23:34,740 --> 00:23:37,680 We could have started with the Kings proposing all to the Queens. 267 00:23:38,790 --> 00:23:43,020 Does anyone think do you think that would give a different answer to the way these were paired? 268 00:23:43,110 --> 00:23:47,120 If you put your hand up, if you think it would lead to the same answer. Yeah. 269 00:23:47,180 --> 00:23:54,180 Proximity could do things that actually might lead to a different answer. It's kind of an even split that actually does lead to a different answer. 270 00:23:54,190 --> 00:24:01,360 Not always, but very often. So it matters which order is it the women proposing to the men or the men proposing to the women? 271 00:24:01,810 --> 00:24:09,530 I think in this way the men lock out. They get the best that they could possibly get and the women get the worse that they could possibly get it. 272 00:24:09,550 --> 00:24:10,690 But we did it the other way around. 273 00:24:10,690 --> 00:24:15,490 It was swap over so the women would get the best that they could possibly get out of this and the men would not do so well. 274 00:24:15,790 --> 00:24:20,110 Okay. Let's give them a round of applause. Thank you very much. Put your cards on the table. 275 00:24:25,860 --> 00:24:31,290 So this is actually an algorithm that was come up with for for trying to solve this problem. 276 00:24:31,290 --> 00:24:38,240 It's called the stable marriage problem. And it's one of the only algorithms ever to win a Nobel Prize. 277 00:24:38,250 --> 00:24:47,170 So Galen Shapely came up with this girl who'd already died, but Shapely won the 2012 Nobel Prize in Economics for coming up with this algorithm. 278 00:24:47,170 --> 00:24:51,510 And it's a very powerful algorithm that can be used in a lot of different circumstances. 279 00:24:51,780 --> 00:25:00,720 One of the first was sort of pairing up not a lot at a party like this, but students applying to placements at university. 280 00:25:01,350 --> 00:25:06,479 And interestingly, the way they applied the algorithm, of course, the universities knew what they were doing. 281 00:25:06,480 --> 00:25:10,379 They made the students propose to the university. 282 00:25:10,380 --> 00:25:14,850 So the universities got the best that they could and the students got the worse that they could do. 283 00:25:14,860 --> 00:25:18,210 The students suddenly work this out after a while, and so they say This isn't fair. 284 00:25:18,390 --> 00:25:22,980 So now they flipped it over. So the students, the universities now propose to the students. 285 00:25:22,980 --> 00:25:26,280 So the students get the best possible result out of this some. 286 00:25:27,030 --> 00:25:34,830 So but another interesting place is used in the NHS by and America as well kidney transplants. 287 00:25:35,130 --> 00:25:39,510 Very often you might have a partner who is quite happy to give their kidney, but they don't match. 288 00:25:40,050 --> 00:25:45,330 And then the problem is you've got to find somebody who does match, but why would they give you the kidney? 289 00:25:46,020 --> 00:25:48,389 And so if you've got just to it's very simple. 290 00:25:48,390 --> 00:25:55,410 If you could find two and they'd be quite happy to swap over and donate a kidney to somebody else if their partner will donate a kidney, 291 00:25:55,410 --> 00:25:58,860 which will save their partner. So that's is quite simple. 292 00:25:58,860 --> 00:26:04,530 But this matching algorithm, if you look at the number of people who are prepared to give kidneys across the country or in America, 293 00:26:04,530 --> 00:26:07,680 for example, it can get quite complicated trying to match these up. 294 00:26:07,890 --> 00:26:17,400 And this a similar sort of algorithm has being used and I think the longest one was this particular train of people here who managed to be matched up. 295 00:26:18,360 --> 00:26:23,700 So this algorithm was actually saving lives in order to be able to find the right match for somebody, 296 00:26:23,700 --> 00:26:29,009 which I think is quite amazing what algorithms of course all over the place and 297 00:26:29,010 --> 00:26:35,969 algorithms are at the heart of how I find my Christmas presents to get people actually, 298 00:26:35,970 --> 00:26:38,190 because, you know, basically I don't know what to do. 299 00:26:38,190 --> 00:26:43,170 So I, I type into a search engine and say, what's the best Christmas present you could give to somebody? 300 00:26:43,710 --> 00:26:50,070 And this is the amazing thing. I mean, Google is, I think, one of the most extraordinary algorithms and it's like almost like magic. 301 00:26:50,070 --> 00:26:53,880 I mean, yeah, but of course, it's not match, it's not elves. They're sort of saying, Oh, 302 00:26:55,560 --> 00:26:59,910 it's just a really amazing piece of mathematics that is at the heart of finding 303 00:26:59,910 --> 00:27:04,980 out what the best website is for for the present that you want to give. 304 00:27:05,460 --> 00:27:09,330 So I thought I'd do a little demonstration to explain how this algorithm works. 305 00:27:10,050 --> 00:27:11,970 So let's suppose we got, say, 306 00:27:11,970 --> 00:27:22,470 three websites that's they're offering presents for Christmas and we're going to try and order and see which one is the most popular, 307 00:27:22,470 --> 00:27:30,930 which one is the most likely one that I should go to for this. So you might know that this Google algorithm works on the basis of the fact that 308 00:27:30,930 --> 00:27:36,120 it's the links between websites which actually rank the order of the websites. 309 00:27:36,120 --> 00:27:40,589 But how does this work? So suppose that I had this particular way of linking. 310 00:27:40,590 --> 00:27:46,530 So suppose that website A or also recommends both B and website C. 311 00:27:46,530 --> 00:27:50,130 So that link from website A to website B and website C, 312 00:27:50,490 --> 00:27:58,140 website B only links to website website B only links to website C and website C only links to website A. 313 00:27:58,860 --> 00:28:05,999 Now with this particular set up, which do you think Google is going to put right at the top as the most popular? 314 00:28:06,000 --> 00:28:09,450 We're going to do a little vote. Okay. So you can you can put your hand up in a second. 315 00:28:09,450 --> 00:28:19,049 So who thinks that website is going to end up at the top of the pile with the Google ranking with this particular way that network is connected? 316 00:28:19,050 --> 00:28:24,510 So put your hand up if you think website A should be the one which will be at the top of the list. 317 00:28:25,050 --> 00:28:28,140 So we've got some votes for a few votes, right? Okay. 318 00:28:28,290 --> 00:28:32,280 Who thinks the website B is the one that's going to go top of the pile? 319 00:28:33,540 --> 00:28:36,870 Very much for you in there. Okay. Losing weight. I see. 320 00:28:37,050 --> 00:28:42,000 It's a loss. Are you going to see? That's interesting. Okay, so how does Google work this out? 321 00:28:42,570 --> 00:28:51,660 Well, basically the idea is that, yeah, Google is very clever because it's quite hard to kind of boost your own website's ranking in this. 322 00:28:51,670 --> 00:28:57,660 There's not much you can do. You've got to wait for other people to link to you. And that's the importance of how the Google algorithm works. 323 00:28:58,080 --> 00:29:02,040 So I need three volunteers are going to come up and be my websites for me and we're going to do a little. 324 00:29:02,310 --> 00:29:06,510 It's a great you can come up. And who else? It's it's very mechanical. 325 00:29:06,510 --> 00:29:09,750 You don't have to do much maths. You just have to. Great. Excellent. We have one there. 326 00:29:10,440 --> 00:29:15,569 One more. Yes, one. You come. Great. So. So let me show you how this algorithm works. 327 00:29:15,570 --> 00:29:22,110 Okay. So I'm going to give we're going to give each website a kind of equal amount of value right at the beginning. 328 00:29:22,440 --> 00:29:26,820 Okay. So do you want to be website a? Oh, you've got a ruby red for. 329 00:29:27,420 --> 00:29:30,420 Excellent. All right. Yeah. And I'm ready. Have you read the recent Ruby Red? 330 00:29:31,680 --> 00:29:37,400 I do the codes for Lauren Child's Ruby Red for books. And I'm really proud of my code for this one. 331 00:29:37,410 --> 00:29:40,979 So each book is about a different sense. 332 00:29:40,980 --> 00:29:49,530 And so we did one about twice this time. So I did this amazing four dimensional code for my one, and it was like, Yeah, sorry, I'm getting it right. 333 00:29:49,530 --> 00:29:53,550 So you, me, which I be and we, we get sizes, so we're going to put website B over here. 334 00:29:54,390 --> 00:29:59,820 Okay. So I'm giving them equal value at the moment. So how does the Google algorithm work to see which one is the best? 335 00:30:00,030 --> 00:30:06,900 So I'm going to put these pots down here. So you a so essentially what you do is you share your value. 336 00:30:07,050 --> 00:30:11,790 If you link to a website, several websites, you have to share your value between those websites. 337 00:30:12,330 --> 00:30:15,810 So for example, you've got each of you got eight balls inside here. 338 00:30:16,230 --> 00:30:24,840 So you link to both of these websites. So you're going to share your balls between four balls, two website B and four balls website C. 339 00:30:24,840 --> 00:30:32,220 Okay. But you, you only link to one website, so you will link to website C, so you're going to give all your value to website C. 340 00:30:32,250 --> 00:30:36,510 Okay. And the same for you. You link to website, are you giving all your balls to website? 341 00:30:36,870 --> 00:30:42,689 So what basically what we're trying to do is to run this and see how the value gets distributed among the websites. 342 00:30:42,690 --> 00:30:50,180 So let's offer you go you. That's the pots there. So you're going to go to the pot and fill up the other person's pot with the value. 343 00:30:50,190 --> 00:30:51,810 So let's open these up. 344 00:30:52,200 --> 00:31:02,760 So all of them so you're giving four balls each, but all of your balls are going into that one and all of yours are going into them half off. 345 00:31:03,300 --> 00:31:10,770 So in the first round we distributed the ball so, so half ago on the here from the website, but they swapped all of those over. 346 00:31:10,950 --> 00:31:16,420 So now the way the value is being distributed and you can see that's a website but not know many of you went 347 00:31:16,430 --> 00:31:21,360 to website B and that's kind of right because it's not getting much value on this round at the moment. 348 00:31:21,360 --> 00:31:28,290 Website. See, you've got ten balls inside there, so you're being highly valued at the moment, but that's not good. 349 00:31:28,290 --> 00:31:32,790 Not one round isn't good enough, so we need to run it again. So let's do exactly the same thing again. 350 00:31:32,790 --> 00:31:38,999 So take the balls. So put the empty ones down. Take the ones that you got filled up and you do the same thing again. 351 00:31:39,000 --> 00:31:42,389 So you're equally distributing your balls and you're distributing them. 352 00:31:42,390 --> 00:31:47,310 It's all to one website. So if you go, let's open these up. 353 00:31:48,210 --> 00:31:51,240 So what happens now is website C, you still going to stay ahead? 354 00:31:51,750 --> 00:31:58,950 Well, after we it on this one website B it's got a little bit is still pretty low but what's my CS going down now and the 355 00:31:58,950 --> 00:32:06,350 website is going up so now it's not clear whether you know which one is it website B you website C so a website. 356 00:32:06,420 --> 00:32:10,140 So it's a website. See your website. So let's do it one more time. 357 00:32:10,170 --> 00:32:15,149 Let's distribute. So, so, so basically this algorithm first of all, is just keeps on distributing. 358 00:32:15,150 --> 00:32:18,270 So if you go again, let's redistribute them. Where do they go? 359 00:32:21,030 --> 00:32:25,859 Okay. So now, yes, you put half there and make sites. 360 00:32:25,860 --> 00:32:29,909 And so again, it's after that round website B is gone up a bit actually. 361 00:32:29,910 --> 00:32:34,139 So so is this thing ever going to stabilise or is it just sort of pinpointing all around? 362 00:32:34,140 --> 00:32:37,560 And this is the kind of challenge of this. So I think, am I doing one more round? 363 00:32:37,860 --> 00:32:44,099 Yeah, let's do one more round, see what happens. So this is the kind of what the algorithm is doing. 364 00:32:44,100 --> 00:32:48,780 So if you go share them around. But what we want to do is does it stabilise after a while. 365 00:32:48,960 --> 00:32:50,010 If we kept on doing this, 366 00:32:50,010 --> 00:33:00,870 would one website would the websites kind of stabilised one of the moments that's we seem to get equal value for website and website C and that's the 367 00:33:00,870 --> 00:33:07,500 interesting thing is how can you work out mathematically without going through this thing over and over again until you see some sort of stability? 368 00:33:08,070 --> 00:33:13,440 Well, this is actually a really powerful piece of mathematics, because essentially each time I'm doing this, 369 00:33:13,440 --> 00:33:17,690 I've got a little so this is the hardest part of maths I'm going to do, but there's a little matrix. 370 00:33:18,090 --> 00:33:24,930 So the number of balls that are at a website, A, B and C, when we do this round and we redistribute, 371 00:33:25,170 --> 00:33:31,440 basically it's the way this matrix acts on this little column vector gives us the new distribution of balls. 372 00:33:31,710 --> 00:33:36,160 So I want to know, is there a. Play at distributing the ball such that the thing is actually stable. 373 00:33:36,160 --> 00:33:43,750 So it doesn't keep on changing each time. And this is actually a really powerful piece of maths called it's finding the eigenvalue of a matrix. 374 00:33:43,930 --> 00:33:49,749 So actually and this is at the heart of many things like quantum physics, it's about eigenvalues. 375 00:33:49,750 --> 00:33:55,770 So we want to know, is there a way of distributing those balls such that when we redistribute, they don't change ever. 376 00:33:56,470 --> 00:34:00,640 And in this case, actually, we distributed them with 2/5 to website, 377 00:34:00,640 --> 00:34:07,330 a 2/5 to website C and a fifth to website B, then this distribution would actually stabilise. 378 00:34:07,330 --> 00:34:09,399 And so this is actually how Google works. 379 00:34:09,400 --> 00:34:16,150 It looks for the way that the websites are linked together and then what's the waiting to give to each website? 380 00:34:16,150 --> 00:34:21,340 So should when I do this redistributing, I don't actually get any change. 381 00:34:21,350 --> 00:34:27,040 So this is amazing thing. Eigenvalues of matrices basically are the heart of how Google works. 382 00:34:27,370 --> 00:34:32,139 So if we change that and I put a 0.4, 0.4, so let's change that. 383 00:34:32,140 --> 00:34:35,680 So yeah, so we've got a distribution of 2/5 to face and one fifth, 384 00:34:36,130 --> 00:34:42,250 if we ran the websites again and did the little algorithm we did here, we'd see that the thing would stabilise. 385 00:34:42,250 --> 00:34:46,450 And those are the values, the ranks that are given to each particular website. 386 00:34:47,080 --> 00:34:52,590 Okay, just give our websites a round of applause. So. Thank you. 387 00:34:55,270 --> 00:35:01,030 Oh, you can leave that. So the interesting thing is it's your intuition that to see was actually the most 388 00:35:01,030 --> 00:35:05,260 powerful because it seems to be being linked to by both the other websites. 389 00:35:05,710 --> 00:35:12,610 But actually both C and A are equally valued with this particular ranking. 390 00:35:12,850 --> 00:35:19,690 And it's quite I get asked actually quite a lot by advertisers now in advertising companies used 391 00:35:19,690 --> 00:35:24,130 to be overrun by people doing English literature because they wanted good copy and sort of. 392 00:35:24,490 --> 00:35:28,899 But now advertising firms are full of mathematicians who are trying to reverse engineer the Google 393 00:35:28,900 --> 00:35:34,360 algorithm because they basically want to put their product right at the top of the ranking. 394 00:35:34,900 --> 00:35:39,190 And so, you know, this is pretty public, but actually the real mechanics of Google, 395 00:35:39,190 --> 00:35:42,819 there's a lot of hidden things because they don't want people kind of reverse 396 00:35:42,820 --> 00:35:47,050 engineering it and finding a way to get their website at the top one way. 397 00:35:47,080 --> 00:35:51,760 The other thing I get a lot of approaches is will you link your website? 398 00:35:51,760 --> 00:36:00,610 Here are the Maths Institute to my website because actually some of the places with the highest page rank across the world are universities. 399 00:36:00,880 --> 00:36:04,400 So Oxford University has one of the highest PageRank because a lot of you know, 400 00:36:04,420 --> 00:36:08,470 we have a lot of value, a lot of people are linking to us and it boosts our page rank. 401 00:36:08,680 --> 00:36:13,809 So if I then link to somebody else, I'm giving them a lot of these boards and a lot of value. 402 00:36:13,810 --> 00:36:22,270 So so it's interesting that, you know, of course, the more I do that, the more I'll I'll sort of water down the particular value. 403 00:36:22,270 --> 00:36:29,860 But it's interesting that people are coming to me because I have powerful page rank because of my place at this university, 404 00:36:29,980 --> 00:36:32,410 and that will actually boost their page rank as well. 405 00:36:33,010 --> 00:36:41,530 So this, of course, was discovered by these two geeky mathematicians, Page and Brin, in their in their garage in the West Coast of America. 406 00:36:41,530 --> 00:36:50,229 And just using this simple tool of eigenvalues, of matrices, they've been able to make their their billions, basically. 407 00:36:50,230 --> 00:36:54,970 So I think it's a wonderful example of pure maths. 408 00:36:55,120 --> 00:37:01,170 Eigenvalues were considered rather pure. And I think the the idea of the dark side and light side, you know, this is really disappearing. 409 00:37:01,180 --> 00:37:06,280 The amount of pure mathematics that are now popping up in sort of an applied area 410 00:37:06,340 --> 00:37:09,850 kind of area like this really shows there's really no difference between us. 411 00:37:10,330 --> 00:37:18,040 So the hope is, of course, that when you put in, you know, best birthday present of what will pop up is maybe Amazon recommending a copy of my book. 412 00:37:18,080 --> 00:37:20,500 In fact, this is this is what happened to me. 413 00:37:21,100 --> 00:37:30,420 So sometimes these algorithms don't really work because I got this letter a couple of days ago saying, Dr. de Sautoy, maybe you'd like these books. 414 00:37:31,810 --> 00:37:36,790 Yeah, yeah, yeah, I do like those books. And so they recommended three copies of my books. 415 00:37:36,970 --> 00:37:41,260 And one of the trouble is, some of the times these algorithms really just because they're not, 416 00:37:41,750 --> 00:37:45,340 they, you know, you could have done this to make sure I didn't get a copy. 417 00:37:45,340 --> 00:37:51,940 I don't mind being recommended copies of my books, but, um, but sometimes on some kinds of these algorithms can do some weird things. 418 00:37:52,810 --> 00:38:02,650 So it's an interesting example of a book that wasn't very popular book, but when this person put it in, he was interested in this book for his PhD. 419 00:38:03,370 --> 00:38:08,230 He put this book in The Making of a Fly, The Genetics of Animal Design. 420 00:38:08,680 --> 00:38:14,210 So, you know, they're second hand books on Amazon and basically, you know, this prize. 421 00:38:14,530 --> 00:38:18,580 So some of these second hand bookshops, they use algorithms to generate the price of these books. 422 00:38:18,910 --> 00:38:27,070 Now, when he looks at this book, he got a big surprise because the book was on one website for $1,000,000, 423 00:38:27,430 --> 00:38:33,550 730,000, and on another website over $2 million, he said, This is really weird. 424 00:38:33,760 --> 00:38:38,850 He wrote to Peter Lorre and said, you know, you know, your book is selling, you know, why is it selling for so much? 425 00:38:39,010 --> 00:38:44,770 You didn't know either. So this guy came back a few days later to say, you know, maybe it'd sort itself out. 426 00:38:45,350 --> 00:38:52,389 But two days later, the thing had gone up. So one one bookseller was offering it for 18 million. 427 00:38:52,390 --> 00:38:55,780 Wow. This is amazing book. 428 00:38:55,780 --> 00:39:00,760 This must be that. And the other one, 23 me. I wish somebody buy my book for $23 million. 429 00:39:01,900 --> 00:39:11,620 So, so what was happening here? What was going wrong that somehow, weirdly, you know, why were they competing with each other to to sell this book? 430 00:39:13,480 --> 00:39:23,350 Well, actually, the guys started to analyse what was happening over each of the days, and he saw basically there was a multiplier going on. 431 00:39:23,350 --> 00:39:27,999 So so each website was looking at each other and they were using the price of 432 00:39:28,000 --> 00:39:33,610 the other website in order to determine the price that they put their book on. 433 00:39:34,510 --> 00:39:39,639 So Prof. Nath, for example, is always trying to undercut the other one. 434 00:39:39,640 --> 00:39:42,700 They're basically, well, if you want to buy there, but we got it a little cheaper. 435 00:39:43,450 --> 00:39:47,890 And we the beauty books was slightly over pricing their book. 436 00:39:48,940 --> 00:39:53,740 And of course, the way they'd weighted it meant that the cumulative effect. 437 00:39:53,810 --> 00:39:58,190 Was that actually the book was being pushed up and up and nobody was wanting to buy this book. 438 00:39:58,470 --> 00:40:02,330 I was reading it so nobody had noticed. And this thing was just over time. 439 00:40:02,990 --> 00:40:07,460 I mean, it's an exponential growth. Doesn't take too long to blow this book up to the price here. 440 00:40:08,300 --> 00:40:15,060 Now, here's a question for you. You can understand why pro-Nazi were wanting to undercut, you know, so. 441 00:40:15,080 --> 00:40:18,110 Okay, well, we'll make it a little cheaper than the other bookshops, so you can understand that. 442 00:40:18,290 --> 00:40:23,210 But what's going on with the other books? Why are they actually got their algorithm? 443 00:40:24,230 --> 00:40:28,520 Making the book a little bit more expensive than the other website. Anyone got any ideas? 444 00:40:28,520 --> 00:40:32,590 Why? Yeah. Exactly. 445 00:40:32,620 --> 00:40:35,680 That could be one reason you think, oh, go with a really cheap one. 446 00:40:36,010 --> 00:40:39,100 And maybe the other one's got a higher quality, you know, it's got everyone's ranking it. 447 00:40:39,280 --> 00:40:43,060 So maybe you'll go, Oh no, I'd rather trust the person and pay a little bit more. 448 00:40:43,240 --> 00:40:45,639 It was only a little bit of a factor, so that's one idea. 449 00:40:45,640 --> 00:40:52,710 But actually it turned out that probably wasn't the reason you got an idea that already broke, listed it fast and the other cost. 450 00:40:55,240 --> 00:41:01,360 Yeah. But then why bawdy books? It's actually got their algorithm always looking at the other website and multiplying it a little bit more. 451 00:41:01,360 --> 00:41:06,710 So it's over time. That's right. So yeah. Yeah. 452 00:41:06,950 --> 00:41:11,680 Maybe they just wanted to mess with everyone. Yeah. Yeah, well, they certainly succeeded. 453 00:41:11,720 --> 00:41:17,510 Exactly. Yeah, yeah. But, you know, they're not I'm nobody's going to say pay 23 million for that book. 454 00:41:18,160 --> 00:41:25,210 I my thought was that 40 books didn't have a copy of Making the Fly. 455 00:41:26,840 --> 00:41:31,600 They didn't have a copy. So they just kept on looking. And the other one said, okay, if anyone orders from us, we buy their book. 456 00:41:33,290 --> 00:41:39,800 But if we buy their book, we need to pay a little bit more so we can pay for it to be delivered and then we send it on. 457 00:41:39,980 --> 00:41:44,120 So we need to crank it up a little bit. And so I think that's what was happening. 458 00:41:44,120 --> 00:41:47,120 They didn't have that book at all and they needed to have a little multiplying factor. 459 00:41:47,300 --> 00:41:51,060 But it ended up, you know, with it costing 23 million, that is. 460 00:41:51,570 --> 00:41:57,500 And that's the trouble with these algorithms. Sometimes they just aren't sensitive to kind of with these sort of anomalies happening. 461 00:41:57,710 --> 00:42:01,910 And this happens in the stock market. So there was this amazing thing which happened. 462 00:42:02,420 --> 00:42:07,010 And so there were a lot of algorithms at work in the stock market which essentially, 463 00:42:07,010 --> 00:42:12,709 you know, you want something which acting much faster than a human brain can react to. 464 00:42:12,710 --> 00:42:18,710 And so when things change, it automatically does something. These algorithms that are in the stock market, 465 00:42:19,520 --> 00:42:28,010 we think that they were probably responsible for a rather remarkable flash crash that happened on the 6th of May 2010, 466 00:42:28,130 --> 00:42:30,560 when in about an a couple of minutes, 467 00:42:31,130 --> 00:42:39,320 the value of the stock market just absolutely crashed because these algorithms are working faster than anybody could react to. 468 00:42:39,320 --> 00:42:41,750 And they just sort of basically the same thing happened. 469 00:42:41,750 --> 00:42:46,549 But instead of escalating the price of that Amazon book, it basically just crash the whole thing. 470 00:42:46,550 --> 00:42:52,160 And, you know, people were just going crazy. You know, this is one person's reaction to seeing what the [INAUDIBLE] is happening. 471 00:42:52,580 --> 00:42:54,950 But we then very quickly, the whole thing, it picked up again. 472 00:42:55,640 --> 00:43:01,250 And it seems now that there was somebody sitting in West London in with his parents, actually with Alison, 473 00:43:01,250 --> 00:43:07,970 and he created these algorithms to basically just sort of manipulate the market a little bit. 474 00:43:08,240 --> 00:43:11,600 But these algorithms eventually led to this flash crash. 475 00:43:11,600 --> 00:43:18,229 And he's, I think, still waiting trial for actually manipulating the markets in this way with these algorithms, 476 00:43:18,230 --> 00:43:21,920 although it's not clear whether he knew the effect that they would have. 477 00:43:22,100 --> 00:43:26,419 So very interesting questions coming up now with these algorithms whether actually, you know, 478 00:43:26,420 --> 00:43:33,350 whose if you write an algorithm, are you responsible for the effects of the algorithm, which of course. 479 00:43:33,680 --> 00:43:37,459 I mean, you still are. I think so, just to warn you. 480 00:43:37,460 --> 00:43:46,250 Yeah. Okay. So the the challenge I actually set in my my the title of my talk was this travelling centre problem. 481 00:43:46,250 --> 00:43:48,469 So this is, I think one of my favourites is, you know, 482 00:43:48,470 --> 00:43:56,270 a centre has this great challenge on Christmas Eve of having to find his way across the whole of the planet, 483 00:43:56,270 --> 00:43:59,959 delivering all of these presents down chimneys. 484 00:43:59,960 --> 00:44:04,070 And and so, you know, how how does he manage to do this? 485 00:44:04,070 --> 00:44:07,850 He has to find the most efficient path to deliver all of these presents. 486 00:44:08,270 --> 00:44:14,719 This is one of his things. Now, have you gone on Google on Christmas Eve? And you can see where where Santa is on any particular moment. 487 00:44:14,720 --> 00:44:18,710 So here he is. He caught him on the top of Mount Everest delivering. 488 00:44:19,520 --> 00:44:22,549 So, you know, the challenge is, you know, you're going across the planet. 489 00:44:22,550 --> 00:44:29,090 Is there an efficient way to find the shortest path so he can get round all of the chimneys? 490 00:44:29,480 --> 00:44:37,080 So I tell you, a little challenge. When you came in, you got the Santa's grotto there and various cities that they have to visit. 491 00:44:37,370 --> 00:44:39,320 So I don't know, will you be having a go at this, 492 00:44:39,710 --> 00:44:47,720 but the idea is you start at Santa Grotto and you go to visit each of the chimneys in turn and then come back to Santa's grotto here. 493 00:44:48,380 --> 00:44:51,680 And the challenge is, can you find the shortest path around this? 494 00:44:52,040 --> 00:44:59,390 Now, I've been telling you about these wonderful algorithms to match people up, wonderful algorithms to find you presents on Google. 495 00:44:59,990 --> 00:45:07,520 But this presents a real challenge to mathematicians because we cannot find an efficient algorithm other than TRON an error. 496 00:45:07,640 --> 00:45:12,110 I mean, basically, I've got a lot of you here. Have you all tried trying to find a path around here? 497 00:45:12,110 --> 00:45:19,879 We probably would find one, which was minimal, but basically there's no kind of efficient way to see how to find the shortest route here. 498 00:45:19,880 --> 00:45:25,820 I mean, you can have a go at this, as it were, to see whether you can find the shortest path. 499 00:45:25,870 --> 00:45:27,620 We'll see whether anybody comes up with it. 500 00:45:28,160 --> 00:45:34,040 The trouble is, that's the only thing we really know how to do, is to try one path after another and just see, 501 00:45:34,070 --> 00:45:37,070 try all the different paths and see which one comes out the smallest. 502 00:45:37,370 --> 00:45:40,549 But that is a very inefficient way to do it. 503 00:45:40,550 --> 00:45:46,490 And the more and more cities you have to visit, the more different possibilities there are around this network. 504 00:45:47,150 --> 00:45:52,160 And so Point Sans has got a lot more than this to to visit. So this is an example of a problem. 505 00:45:52,590 --> 00:45:57,770 In fact, it's one of these millennial problems, so called the NP versus P problem. 506 00:45:58,010 --> 00:46:03,409 And these problems have the particular quality that once you kind of find a solution. 507 00:46:03,410 --> 00:46:12,070 So suppose you want to find a path around there. Which has a sort of which is less than, say, 240 miles or something once you've found one. 508 00:46:12,250 --> 00:46:14,470 You know, you've you've you've found a path. 509 00:46:14,620 --> 00:46:22,060 But trying to find a path, which is to say less than 240 miles, requires just searching for the needle in the haystack. 510 00:46:22,390 --> 00:46:29,080 So there are a lot of these problems out there which have this quality that to in order to find the most efficient solution, 511 00:46:29,080 --> 00:46:32,450 you just got to try sort of one after another after another. 512 00:46:32,470 --> 00:46:37,030 But when you find it, you can prove very quickly that it is the fastest solution. 513 00:46:37,300 --> 00:46:42,310 So, in fact, we don't know whether there's a very efficient algorithm to find this path. 514 00:46:42,520 --> 00:46:46,780 And this is one of the challenges. So we have these seven millennial problems. 515 00:46:46,780 --> 00:46:53,319 So now businessmen in America, Land and Clay offered $1,000,000 to anybody who could solve one of these seven problems. 516 00:46:53,320 --> 00:46:56,650 So one has already been solved. It's called the Poincaré conjecture. 517 00:46:57,550 --> 00:47:01,150 This is the my third book was about some of these problems. 518 00:47:01,150 --> 00:47:04,180 So the other one is the Riemann hypothesis about primes. 519 00:47:04,990 --> 00:47:12,160 But this problem many regard as perhaps the most important because it has so many applications to the real world around us. 520 00:47:12,160 --> 00:47:20,020 So the NP versus P problem is basically can you find an efficient algorithm to find the shortest path around centres 521 00:47:20,020 --> 00:47:26,380 network or can you prove that there isn't one other than by trying all of them out and seeing which is the smallest? 522 00:47:27,010 --> 00:47:30,650 The intriguing thing is that there are many problems which have this quality to them. 523 00:47:30,670 --> 00:47:34,150 So one of my favourite is the the premiership problem. 524 00:47:34,990 --> 00:47:40,180 So you might detect here that I'm an Arsenal supporter because I took this little freeze 525 00:47:40,180 --> 00:47:44,649 frame before Leicester beat Chelsea and went back on the top of the premiership. 526 00:47:44,650 --> 00:47:48,550 So this is a small moment a couple of days ago when we were still top. 527 00:47:49,540 --> 00:47:54,100 But the challenge here is, well, can you put pressure on Aston Villa, who we beat at the weekend? 528 00:47:55,270 --> 00:48:01,990 A Right down there at the bottom. But is it still technically possible for Aston Villa to win the premiership? 529 00:48:04,180 --> 00:48:09,190 So what is the challenge there? It means, okay, well, Aston Villa are basically going to win all their games. 530 00:48:09,190 --> 00:48:17,499 Okay. But it's not enough because there are other teams who will be winning games or drawing and maybe it's a bit like the stable marriage problem. 531 00:48:17,500 --> 00:48:23,709 Is there a way to distribute the wins and losses and draws such as Aston Villa will stay 532 00:48:23,710 --> 00:48:28,990 top with all of their wins and we can make sure that no other team beats them weirdly. 533 00:48:29,050 --> 00:48:36,640 So many years ago, when I was young, you used to get two points for a win and one point each when you drew. 534 00:48:37,300 --> 00:48:40,990 This means that actually the end of the season, you know, 535 00:48:40,990 --> 00:48:47,950 the total number of points in the division because it didn't really matter whether it was a win or a draw or a loss. 536 00:48:48,700 --> 00:48:53,410 It was basically, you know, everyone would be those two points would be distributed around. 537 00:48:53,530 --> 00:48:59,379 But then something changed. The premiership decided, okay, now we want to incentivise people to win and they changed it. 538 00:48:59,380 --> 00:49:04,060 So you get three points for a win and only one each if you draw. 539 00:49:04,270 --> 00:49:08,140 So now at the end of the season, we don't know how many of the total points will be. 540 00:49:08,260 --> 00:49:11,499 If everyone drew all their games, it would be as small as possible. 541 00:49:11,500 --> 00:49:18,399 If there were no draws, it would be much bigger. The weird thing is that before we did this change, 542 00:49:18,400 --> 00:49:25,810 there was an efficient algorithm to work out whether Aston Villa would actually have a possibility to win the Premiership. 543 00:49:25,990 --> 00:49:34,120 But with the change from two points for a win to three points through in suddenly this change and we no longer have an efficient algorithm, in fact, 544 00:49:34,120 --> 00:49:39,759 this has the same quality as a travelling centre problem that's essentially you've just got to try all the different 545 00:49:39,760 --> 00:49:45,490 possibilities of results and see whether there's one which will make sure that Aston Villa will stay top. 546 00:49:46,000 --> 00:49:51,790 And so all of these problems, their various different versions of them say this one about Minesweeper, 547 00:49:52,510 --> 00:49:59,310 there's another one sort of another coming back to the party problem. If you're staging several parties across the Christmas season and you know, 548 00:49:59,350 --> 00:50:03,220 there are people who really don't like each other and you go to invite them to different parties. 549 00:50:03,430 --> 00:50:10,180 So, for example, suppose you cannot invite somebody to the same party if they are linked to the other person to that party. 550 00:50:10,180 --> 00:50:15,160 So Fran and Ian, we can't invite to the same party because they hate each other. 551 00:50:15,730 --> 00:50:19,150 So the challenge is, you know, can we find a way to have as many, you know, 552 00:50:19,150 --> 00:50:23,590 as few parties as possible such that nobody is coming to a party with somebody that they hate? 553 00:50:24,820 --> 00:50:33,460 So this is another version, it seems to be you can only just try different ways, give it your day in doing the invites. 554 00:50:34,040 --> 00:50:39,610 There's another version of this which is kind of interesting, which is so you probably know this thing called the full colour map problem. 555 00:50:40,330 --> 00:50:43,570 Robin Wilson is here. He's written a wonderful book about the full colour problem. 556 00:50:44,260 --> 00:50:47,470 It's this challenge of, you know, if you have any map, 557 00:50:47,620 --> 00:50:54,910 you can always get away with four colours to colour that map such that no countries that have the same border will have the same colour. 558 00:50:55,600 --> 00:50:59,050 But here's the challenge actually. Could you get away with three? 559 00:51:00,070 --> 00:51:05,680 Is three colours enough? So with any particular map, the challenge is well. 560 00:51:06,140 --> 00:51:09,700 Okay. I know thanks to this proof that for will suffice. 561 00:51:09,710 --> 00:51:15,380 But in this particular case can I get away with three? So in this. 562 00:51:15,800 --> 00:51:23,990 So you might. So I think in this case you can't because Luxembourg has three countries around, it is no way to colour those. 563 00:51:24,140 --> 00:51:26,930 But sometimes you can re divide the map such that there are three. 564 00:51:27,170 --> 00:51:35,209 So the challenge of trying to work out whether there are three colours that will suffice is actually absolutely 565 00:51:35,210 --> 00:51:40,640 equivalent to the challenge of trying to work out whether you can do get away with inviting everyone to three parties. 566 00:51:41,090 --> 00:51:44,600 Because if you think about suppose I tried to see whether there were three parties that I 567 00:51:44,600 --> 00:51:49,970 could stage such that nobody here would be invited to a party with somebody they hated. 568 00:51:50,240 --> 00:51:54,620 Well, that's exactly the same sort of problem if I change these now from people to countries. 569 00:51:55,340 --> 00:52:01,040 Actually, it's the same problem as if I was trying to colour the map with three colours as opposed 570 00:52:01,040 --> 00:52:05,510 to four and make sure that no countries with the same border would have the same colour. 571 00:52:06,110 --> 00:52:10,220 And this is the amazing thing about all of these problems, actually, if you can solve one of them. 572 00:52:10,250 --> 00:52:16,460 So if you can solve the travelling side to problem, you end up actually finding an algorithm which will solve all of them. 573 00:52:16,880 --> 00:52:24,050 So we've managed to prove that all of these problems are actually equivalent so that any solution to one of them, 574 00:52:24,050 --> 00:52:29,390 if you do the Minesweeper one or the Premiership one, you can use that algorithm to solve all of the others. 575 00:52:29,390 --> 00:52:34,190 So it's extraordinary. So you only have to analyse one problem in order to be able to work this out. 576 00:52:34,580 --> 00:52:41,569 Now Centre is actually faced not only with the problem of finding an efficient path around the earth, 577 00:52:41,570 --> 00:52:44,000 but there's another version of this which we are going to end with. 578 00:52:44,010 --> 00:52:50,800 So I need to volunteers from this side of the lecture theatre who can great if you can come up and one more volunteer from this line. 579 00:52:50,840 --> 00:52:51,500 Excellent point. 580 00:52:51,620 --> 00:52:59,990 So you're going to come down here and two volunteers from this site who are going to compete against them and a little problem, which is a problem. 581 00:53:00,170 --> 00:53:04,400 So any volunteers from this side? Great. 582 00:53:04,990 --> 00:53:10,410 Yeah. I just need one more. You can bring him. 583 00:53:10,510 --> 00:53:21,450 You will. Okay. So the problem here is I've got basically Santa has to find the most efficient way to pack his sleigh when he's setting off. 584 00:53:21,450 --> 00:53:27,240 And he has to try and get as many people up and presence inside as he can. 585 00:53:27,930 --> 00:53:31,889 So the challenge here is I've got different sorts of sizes of packages now. 586 00:53:31,890 --> 00:53:41,640 The length of each lorry is 150 metres long and you have to find is there a way to choose these presents so you can fill the hole of the lorry? 587 00:53:42,150 --> 00:53:50,430 Okay, so we've got a little time here, so I'm going to give you 30 seconds and see who can find the most efficient way of filling this. 588 00:53:50,430 --> 00:53:53,850 And you mustn't overfill it. You can under fill it, of course, but the sleigh won't move. 589 00:53:54,060 --> 00:53:58,620 So we're going to go with 30 seconds starting now. And you can also kind of have a go there, 590 00:53:58,660 --> 00:54:06,140 see whether you can find is there a way to find those boxes such that you add them up and you can get up to exactly 150? 591 00:54:06,210 --> 00:54:09,750 What's the closest you can get? You mustn't go over 150. Okay. 592 00:54:09,750 --> 00:54:18,809 So that's you don't. 15 seconds left. So basically you've got to find at so many different ways I can stack these up. 593 00:54:18,810 --> 00:54:27,690 So we got seven, five, four, three, two, one and stop there. 594 00:54:27,720 --> 00:54:32,860 Okay, so let's see, let's see what we got. So right now, you know what? 595 00:54:32,910 --> 00:54:36,390 You've got to you've got you you've got to stick them on. 596 00:54:36,420 --> 00:54:41,000 Yeah, yeah. You can't start changing now. Okay, let's see what is always interestingly so. 597 00:54:41,460 --> 00:54:47,880 So this one is a draw here because they got 52, 59 and 27, which fits, but it isn't maximising the thing. 598 00:54:48,180 --> 00:54:52,080 So 30 seconds. Well, because you could have got the whole length, so you got a little bit left here. 599 00:54:52,770 --> 00:54:57,660 Okay. So and this side here, you also went for 52, 59 and 27. 600 00:54:58,290 --> 00:55:01,840 So did anyone manage to get a different way, which got a little bit more efficient? 601 00:55:01,860 --> 00:55:05,580 Yes. 65, 47, 60. 602 00:55:05,850 --> 00:55:09,090 Okay. Let's and that adds up exactly to 150. So well done. 603 00:55:09,780 --> 00:55:15,269 But this is a real cut. But this is the interesting thing, 604 00:55:15,270 --> 00:55:20,610 because here I just asked to two people to do it and they had to try all the different possibilities they didn't get. 605 00:55:21,540 --> 00:55:25,049 You know, interestingly, obviously, you're using the same sort of algorithm. 606 00:55:25,050 --> 00:55:30,240 I mean, one of these algorithms is, you know, well, just put the biggest one in first and then you find, well, that's not really good. 607 00:55:30,240 --> 00:55:33,270 And so you go for the next biggest one down maybe. 608 00:55:33,660 --> 00:55:39,719 But if I ask all of you, basically you're working a bit like everyone trying something different and it was not unexpected. 609 00:55:39,720 --> 00:55:43,830 The maybe with the whole of you here, we would find one person would find the most efficient one. 610 00:55:44,070 --> 00:55:47,400 But again, we don't have an efficient algorithm to find. 611 00:55:47,520 --> 00:55:53,640 It seems such a simple problem. You know, just what's the most efficient way to pack the back of your van or the the sleigh? 612 00:55:53,940 --> 00:55:58,500 And this is another version of this. No problem. So let's give all our volunteers a very grounded ball. 613 00:55:58,510 --> 00:56:07,460 Thank you very much. So I think this season is just jam packed with amazing mass. 614 00:56:07,490 --> 00:56:11,510 I mean, it was all I could do, snowflakes as well and tell you all about that. 615 00:56:12,320 --> 00:56:15,440 But time has run out. So we come to the end of the lecture, they say. 616 00:56:15,770 --> 00:56:21,170 So if you can find a solution to this an efficient way, an algorithm will prove that there isn't an algorithm. 617 00:56:21,380 --> 00:56:27,350 There is $1,000,000 prize. And so the travelling sales centre problem, travelling salesman problem, as it's known. 618 00:56:28,550 --> 00:56:32,900 Did anyone get anything below 240? Yes. 619 00:56:32,910 --> 00:56:36,240 What did you get from the 231 231? 620 00:56:36,720 --> 00:56:41,070 That's amazing. Today, anyway. What are you going to at 2 to 8? 621 00:56:42,300 --> 00:56:45,450 Right. Okay. Yeah. So 2 to 8. 622 00:56:45,450 --> 00:56:51,240 Wow. You see, even I got it wrong, you see. So I thought 2 to 8 is even less than I said. 623 00:56:51,630 --> 00:56:57,030 That's. No, I did that one time. Yeah. Oh, okay. He's the true mathematician in the room. 624 00:56:58,050 --> 00:57:00,420 So I thought it was 238. But that's the interesting thing. 625 00:57:00,420 --> 00:57:06,490 You see, I might well have missed a path around here that that would give you a more efficient way. 626 00:57:06,510 --> 00:57:10,979 So anyway, if you want to find out more about some of these problems, I'm selling copies of my book, 627 00:57:10,980 --> 00:57:14,250 The Number Mysteries, which is about some of these monolingual problems. 628 00:57:14,890 --> 00:57:18,000 There were five problems. One has been solved. There's still £4 million. 629 00:57:18,000 --> 00:57:23,190 You can win if you buy a copy of the book. Anyway, thank you very much for coming along and.