1 00:00:15,610 --> 00:00:21,580 I'm very excited to be here at Northumbria University today, and it's great to see lots of you. 2 00:00:21,580 --> 00:00:32,290 So what do we think about when we think about maths? I think maths is about truth and certainty and unchanging truth, that sort of certainty. 3 00:00:32,290 --> 00:00:42,130 Over time, we get through proof. I think the mathematics is about precision and logic and rigour, and I think the maths is about all of those things. 4 00:00:42,130 --> 00:00:48,970 But I also think the mathematics is about play and creativity and curiosity. 5 00:00:48,970 --> 00:00:54,760 It's about inspiration and frustration. It's about enjoyment. 6 00:00:54,760 --> 00:01:01,870 And I think sometimes when we look at a maths textbook that's not necessarily apparent, we read about Pythagoras theorem, 7 00:01:01,870 --> 00:01:06,850 about the sides of a right angle triangle and we learn Pythagoras Theorem, 8 00:01:06,850 --> 00:01:12,610 which is a fantastic theorem which has been known to be true for more than two thousand years. 9 00:01:12,610 --> 00:01:14,680 That unchanging certainty. 10 00:01:14,680 --> 00:01:22,930 But we don't see the journey that Pythagoras and his colleagues took to finding Pythagoras is there the roots of exploration, 11 00:01:22,930 --> 00:01:28,060 the adventure that they went on exploring right tangled triangles to reach that theorem? 12 00:01:28,060 --> 00:01:35,350 For me, it's really important that mathematics is done by humans. I'm interested in that process by which we do mathematics. 13 00:01:35,350 --> 00:01:42,490 How do we find new truths? How do we explore and come to understand pieces of mathematics? 14 00:01:42,490 --> 00:01:48,760 And I think that as people, we can all enjoy mathematics so we can all have those mathematical adventures. 15 00:01:48,760 --> 00:01:53,350 I think that as humans, we have an instinct for pattern infrastructure. 16 00:01:53,350 --> 00:01:58,330 We are curious about patterns. I think that's something that we have as people, and that's very mathematical. 17 00:01:58,330 --> 00:02:02,830 So for example, I've got here some scarves that I knitted. 18 00:02:02,830 --> 00:02:07,690 I like knitting in my spare time, I made some scarves. I'll tell you a bit more about them later on. 19 00:02:07,690 --> 00:02:12,520 But even without my telling you about the underlying mathematics, 20 00:02:12,520 --> 00:02:18,790 I think we can maybe already just see some pattern straight away and maybe wondering what's different between the two. 21 00:02:18,790 --> 00:02:23,530 So it's sort of blue, the green one at the top of the purple one. And there are some aspects that are kind of similar. 22 00:02:23,530 --> 00:02:31,030 And yet you sort of look slightly different. If you slightly unfocused your eyes, you can sort of see some patterns there maybe be what's going on. 23 00:02:31,030 --> 00:02:37,660 I'm kind of I'm kind of intrigued by be curious about that or also I like crochet recently, 24 00:02:37,660 --> 00:02:40,180 so I'm very excited in the mathematical potential of crochet. 25 00:02:40,180 --> 00:02:48,610 So I made some mathematical crochet blankets and again, without my telling you what's going on. 26 00:02:48,610 --> 00:02:53,950 I think our instincts as humans is to start looking and see, well, what is that structure? 27 00:02:53,950 --> 00:02:56,800 What are the patterns, what's going on there? 28 00:02:56,800 --> 00:03:03,220 So somebody who captured this really nicely, in my opinion, is Frances Sue, who's a mathematician in the US. 29 00:03:03,220 --> 00:03:10,990 Frances says mathematics makes the mind its playground. We play with patterns and within the structure of certain axioms. 30 00:03:10,990 --> 00:03:14,860 We exercise freedom in exploring their consequences. 31 00:03:14,860 --> 00:03:21,340 Joyful as any truths we find on this mathematics for Human Flourishing essay, it's available online. 32 00:03:21,340 --> 00:03:26,230 I strongly recommend. I think it's a beautiful description of what it is to do. 33 00:03:26,230 --> 00:03:32,490 Mathematics and mathematics for human flourishing itself is a phrase I think is really interesting. 34 00:03:32,490 --> 00:03:39,180 So I thought before I tell you any more about mathematics, maybe I'll tell you about my summer holiday because in September, 35 00:03:39,180 --> 00:03:44,100 I spent a week on the Isle of Skye of the northwest coast of Scotland. 36 00:03:44,100 --> 00:03:51,060 And I like going for walks. And for me, the experience of walking up a hill is a lot like going for a mathematical adventure. 37 00:03:51,060 --> 00:03:58,590 For me, there are a lot of parallels between setting out on a journey, and you're not quite sure what you're going to take. 38 00:03:58,590 --> 00:04:03,600 You're not quite sure what you'll find when you reach the top or indeed whether you'll reach the top. 39 00:04:03,600 --> 00:04:08,040 So this is called Ben Chiana vague in the sky. 40 00:04:08,040 --> 00:04:13,480 It's not a very big hill. For those of you who know the Isle of Skye, there were some very serious mountains, the killings. 41 00:04:13,480 --> 00:04:17,130 They're very beautiful. I'm not that serious a mountain walker. 42 00:04:17,130 --> 00:04:23,670 Also, the weather was not fantastic, so I didn't really want to set off into the clouds. 43 00:04:23,670 --> 00:04:28,140 So I picked Ben. She had a bag. So I've got a couple of pictures of this hill from different angles, 44 00:04:28,140 --> 00:04:33,840 and I decided to set myself the challenge of climbing this hill and see what I find along the way. 45 00:04:33,840 --> 00:04:37,560 And for me, this was just the right level of challenge for the day. 46 00:04:37,560 --> 00:04:45,060 It was large enough that it was feeling like a challenge, but I felt like I had a chance of reaching the top. 47 00:04:45,060 --> 00:04:51,480 This is a hill that lots of people have climbs before. There are guides online, so before I set out on my walk, I looked online. 48 00:04:51,480 --> 00:04:57,450 I sort of checked out the path. I got my trusty Ordnance Survey map, so I'd look to the routes and all of those kinds of things. 49 00:04:57,450 --> 00:05:01,860 So I guess some mathematicians are like mountaineering pioneers. 50 00:05:01,860 --> 00:05:06,300 They are the first people to climb Everest or to find a new route up some particular mountain. 51 00:05:06,300 --> 00:05:13,290 That's where mathematical research is about. But I think that we can all have mathematical adventures like me climbing Ben Big. 52 00:05:13,290 --> 00:05:21,510 It doesn't matter that other people have been up there before. For me, this was my adventure, so I had a map and an online suggestion of a route. 53 00:05:21,510 --> 00:05:27,960 What I'd like to do today is to invite you to come on a mathematical adventure with me, and I will be your guide. 54 00:05:27,960 --> 00:05:34,170 So what is this mathematical adventure? Well, I have a particular problem in mind, and to do this, 55 00:05:34,170 --> 00:05:41,340 I'd really like a volunteer to give me five whole numbers between between one and 30, let's say. 56 00:05:41,340 --> 00:05:47,560 So is there somebody who'd like to volunteer to give me five whole numbers, please? Yes. Twenty six. 57 00:05:47,560 --> 00:05:53,080 Keep going, I will five of them. 15. Three. 58 00:05:53,080 --> 00:06:03,550 Seven. Eight perfect, thank you so much. 59 00:06:03,550 --> 00:06:08,670 Bring somebody else, please give me five whole numbers between one and 30. 60 00:06:08,670 --> 00:06:13,930 Yes. Thirty. One. 61 00:06:13,930 --> 00:06:26,690 Three. Eighteen. Twenty seven, thank you very much. 62 00:06:26,690 --> 00:06:33,790 Somebody else, please give me five whole numbers between one and 30. Yes. 63 00:06:33,790 --> 00:06:39,870 13, 13, three. Well, 12. 64 00:06:39,870 --> 00:06:55,140 Twenty three. And 30, thank you very much. 65 00:06:55,140 --> 00:07:02,840 Somebody else, please give me five phone numbers, yes. Two, three, five. 66 00:07:02,840 --> 00:07:08,810 Seven, 11. I feel like I might have come across this number somewhere before. 67 00:07:08,810 --> 00:07:19,380 Thank you very much. Let's have a couple more examples. 68 00:07:19,380 --> 00:07:25,200 Yes. Nine. Fourteen. 69 00:07:25,200 --> 00:07:40,920 Twenty one. Twenty seven. And 11, thank you very much. 70 00:07:40,920 --> 00:07:47,780 And finally, let's have one more set of phone numbers. Yes, right to the back. 71 00:07:47,780 --> 00:07:56,380 Two, four, six, eight, 10. You see mathematicians love patterns. 72 00:07:56,380 --> 00:08:01,560 Thank you very much. So. What do I do? 73 00:08:01,560 --> 00:08:04,290 You give me five numbers. 74 00:08:04,290 --> 00:08:13,860 And then a colour in three of them, and I add them up, hopefully correctly arithmetic in public is no my favourite activity in the world. 75 00:08:13,860 --> 00:08:19,440 And I put the title on the right hand side on those titles have something in common. 76 00:08:19,440 --> 00:08:22,200 Those titles are all multiples of three. 77 00:08:22,200 --> 00:08:26,540 So maybe some of you know this sort of quick way to check whether the numbers are multiple of three, you can add the digits. 78 00:08:26,540 --> 00:08:30,030 So fifty seven, five plus seven, that's 12. 79 00:08:30,030 --> 00:08:36,180 Although I happen to know that 12 is a multiple of three already or I do the same thing, one plus two gets me three, that's a multiple of three. 80 00:08:36,180 --> 00:08:44,220 So you can check that these are multiple three. So each time you've given me five whole numbers, I've picked three that add up to a multiple of three. 81 00:08:44,220 --> 00:08:49,700 I call it the main. So our question, our pension of egg for today, if you like, 82 00:08:49,700 --> 00:08:57,610 is can we pick five whole numbers so that there aren't three that add up to a multiple of three? 83 00:08:57,610 --> 00:09:04,000 And I put it a constraint that that the whole numbers had to be between one and 30, that's because I didn't want you to give me seven digit numbers. 84 00:09:04,000 --> 00:09:07,540 I was going to have to add up in real time in front of you. The problem? 85 00:09:07,540 --> 00:09:14,260 I'm interested in just any whole numbers. So they might be positive numbers. They might be negative numbers, they might be zero. 86 00:09:14,260 --> 00:09:22,560 Maybe I should just address one thing right away because sometimes when I ask this question, somebody says, Well, what about zero zero zero zero zero? 87 00:09:22,560 --> 00:09:25,590 And that's a kind of interesting case to explore. 88 00:09:25,590 --> 00:09:34,630 But if you pick your five phone number three zero zero zero zero zero, I'm going to choose three of them to be zero zero zero, which adds up to zero. 89 00:09:34,630 --> 00:09:39,540 And for the avoidance of doubt, I think the zero is a multiple of three because it's three times a whole number. 90 00:09:39,540 --> 00:09:42,150 So I just want to get that one cleared up right away. 91 00:09:42,150 --> 00:09:49,130 So the question can we pick five whole numbers so that there aren't three that add up to a multiple of three? 92 00:09:49,130 --> 00:09:58,260 And I like this question because it is not obvious what the answer is on the best questions in my experience, that did not have an obvious answer. 93 00:09:58,260 --> 00:10:04,320 So maybe I'll tell you a bit more about my walk while you mull over this problem in maybe your subconscious mind, 94 00:10:04,320 --> 00:10:08,880 my subconscious mind is sometimes a better mathematician than my conscious mind. 95 00:10:08,880 --> 00:10:15,150 I can go to sleep and wake up in the morning with an idea or a problem, or doing the washing up and looking at the garden. 96 00:10:15,150 --> 00:10:19,260 And I have an idea on a maths problem, so sometimes subconscious thoughts is good. 97 00:10:19,260 --> 00:10:23,940 So here's my roots up until a big well, so you can't really see it. This is where I parked my car. 98 00:10:23,940 --> 00:10:28,170 I always think Beach on sleep on the beach is a good sign at the start of a walk. 99 00:10:28,170 --> 00:10:31,020 So I park my car and the route goes behind the houses. 100 00:10:31,020 --> 00:10:37,830 You can just see there and then out along the headland and then you sort of turn up the hill and you can't see the hill very well from this view. 101 00:10:37,830 --> 00:10:44,670 So I'm setting out across the headland, and at this point I see I started to wonder whether this was such a good idea at all. 102 00:10:44,670 --> 00:10:51,300 In fact, I started to think this might not be a very good idea, although you can see some blue sky that was very thick clouds. 103 00:10:51,300 --> 00:10:55,230 What you can't tell from the photos that there was a very strong wind blowing. 104 00:10:55,230 --> 00:10:58,740 And where I parked my car was nice and themselves, and I thought, Well, this is lovely. 105 00:10:58,740 --> 00:11:02,790 And then I walked out along the headland and this very strong wind arrived. 106 00:11:02,790 --> 00:11:08,040 And what you can't see, but I read in the online guide is that when you start going up the hill here, 107 00:11:08,040 --> 00:11:12,810 you end up sort of walking along the edge of a cliff. The ground just drops away to the side. 108 00:11:12,810 --> 00:11:16,470 The wind's blowing this way and the ground dropping way this way. 109 00:11:16,470 --> 00:11:21,750 And I'm starting to wonder whether this is such a good idea and thinking, Can I really do this? 110 00:11:21,750 --> 00:11:25,440 And I tell by you, but I definitely have that experience, but I look at a massive problem. 111 00:11:25,440 --> 00:11:33,060 I definitely have that experience of looking at this thinking. I'm not sure I can do this or even I'm pretty sure I can't do this. 112 00:11:33,060 --> 00:11:39,520 So I'm interested in that. How do you overcome that reaction? My. 113 00:11:39,520 --> 00:11:45,520 I think you need some some sort of belief that you might be able to solve a massive problem in order to engage with it. 114 00:11:45,520 --> 00:11:52,030 And sometimes that comes from experience. I've seen a problem like this. I've got some ideas of what I could do or my friends. 115 00:11:52,030 --> 00:11:55,390 My teacher gave me this problem. They know what maths skills I have. 116 00:11:55,390 --> 00:12:05,010 I can tackle this. But actually, it's an interesting question to think when a mathematician solves a world famous unsolved problem. 117 00:12:05,010 --> 00:12:07,890 How did they have the bravery to embark on that? 118 00:12:07,890 --> 00:12:14,610 Did they set out on that problem thinking, Well, maybe I can do this, maybe I can be the one to reach the top of Everest first? 119 00:12:14,610 --> 00:12:19,650 Or maybe I've talked to some. Some mathematicians told me, well, what I was doing was exploring the foothills. 120 00:12:19,650 --> 00:12:23,610 I had some ideas. I just wanted to sort of play around and see what I could do with those hills. 121 00:12:23,610 --> 00:12:30,570 And then I happened to find this path up to the top and it got me there. But I'm kind of interested in where is that role of belief? 122 00:12:30,570 --> 00:12:33,780 Just believing I can solve a problem won't solve the problem, certainly. 123 00:12:33,780 --> 00:12:42,690 But if I if I don't feel like I believe it, if I don't believe I might be able to solve the problem, how do I engage with the problem? 124 00:12:42,690 --> 00:12:50,760 One source of thinking I've done on this is around the role of mindset, so I'm sure some of you have come across the work of Carol Dweck, 125 00:12:50,760 --> 00:12:54,030 who is a psychologist in Stanford, 126 00:12:54,030 --> 00:13:02,970 and she and her research team did the initial research and distinguish between what they called a fixed mindset and a growth mindset. 127 00:13:02,970 --> 00:13:08,820 So she says here a growth mindset is about believing people can develop their abilities. 128 00:13:08,820 --> 00:13:16,650 So do I believe that I have a fixed capacity for mathematics and whatever I do, that's my level in mathematics. 129 00:13:16,650 --> 00:13:19,770 Or do I believe that that can change somebody else's work? 130 00:13:19,770 --> 00:13:26,520 So this is Joe Bhola, who's a professor of mathematics education at Stanford and has looked at this a lot in the context of mathematics. 131 00:13:26,520 --> 00:13:32,970 And Joe Paula says people with a growth mindset are those who believe that smartness increases with hard work, 132 00:13:32,970 --> 00:13:41,370 whereas those with a fixed mindset believe that you can learn things, but you can't change your basic level of intelligence. 133 00:13:41,370 --> 00:13:47,310 So my understanding of this is not that if we all work hard enough, we can all win Olympic gold medals, 134 00:13:47,310 --> 00:13:51,990 we can all when fields, medals kind of top prises in mathematics that we can all climb Everest. 135 00:13:51,990 --> 00:13:57,870 I don't think that's what this is saying, but I think this is saying that if we work hard, we put in the right kind of work. 136 00:13:57,870 --> 00:14:06,090 We take on board feedback or we can develop, we can all get better. And the research that people like Harold Récord Joe Paula have done suggests 137 00:14:06,090 --> 00:14:11,970 that people with a growth mindset do better than people with a fixed mindset. 138 00:14:11,970 --> 00:14:14,010 And I think that makes sense in a mathematical context. 139 00:14:14,010 --> 00:14:22,260 If you look at the problem and you think, I'm not sure about this trying it, having a go feels like taking a bit of a risk. 140 00:14:22,260 --> 00:14:26,970 And if you try a maths problem and you don't solve it and you have a fixed mindset, 141 00:14:26,970 --> 00:14:32,220 you might take that as evidence that your level of mathematical ability wasn't enough for that problem. 142 00:14:32,220 --> 00:14:36,990 Whereas if you have the growth mindset and you try a problem and you don't manage to solve it, 143 00:14:36,990 --> 00:14:41,310 you can use that as a learning opportunity to say, Well, what can I take away from this? 144 00:14:41,310 --> 00:14:48,090 And I think that's a really interesting kind of distinction. So I don't think that people are fixed mindset people or growth mindset people. 145 00:14:48,090 --> 00:14:50,310 It's not like everybody has one of these labels. 146 00:14:50,310 --> 00:14:57,400 I don't think it's even as simple as in the context of mathematics as a place to play the piano or playing football or being an artist or whatever. 147 00:14:57,400 --> 00:15:02,050 I don't think it's as simple as I have a growth mindset about maths or fixed mindsets about maths. 148 00:15:02,050 --> 00:15:08,880 I think lots of us have a bit of both. But sometimes when I'm working on something and I'm finding it difficult, 149 00:15:08,880 --> 00:15:14,430 I just kind of just take a moment to think, OK, I want to have this sort of growth attitude towards this. 150 00:15:14,430 --> 00:15:20,500 So let's go back to this problem with our five hole numbers, so I've got a few more examples here, 151 00:15:20,500 --> 00:15:25,110 but one of the things I really like about this problem is there's something concrete to try. 152 00:15:25,110 --> 00:15:30,120 Sometimes when I look at a maths problem, I think I don't really have any ideas. Where do I start? 153 00:15:30,120 --> 00:15:34,630 One of the things I love about this problem is that we can start by just trying some examples. 154 00:15:34,630 --> 00:15:40,290 And the question was, can we find five whole numbers so that there aren't three that add up to a multiple of three? 155 00:15:40,290 --> 00:15:43,710 What we could just test out some more examples. You gave me some numbers. 156 00:15:43,710 --> 00:15:46,770 It seemed like we could always find three numbers add up to a multiple of three. 157 00:15:46,770 --> 00:15:51,840 We could just keep testing some numbers and try to get a bit of a feel for what's going on. 158 00:15:51,840 --> 00:15:57,690 And one of the things is the way I phrased the problem doesn't tell us what the answer is. 159 00:15:57,690 --> 00:16:00,480 So there are sort of two possible ways out of this problem. 160 00:16:00,480 --> 00:16:05,880 One way out of this problem is, yes, there are five numbers so that you can't have three months for three. 161 00:16:05,880 --> 00:16:12,810 Here is an example of such a set of five numbers. Another way out of the problem would be No, it's impossible. 162 00:16:12,810 --> 00:16:18,480 Here's a proof that whatever five numbers I pick, I can always choose three that add up to a multiple of three. 163 00:16:18,480 --> 00:16:22,170 And the question is, though I phrased it doesn't tell us which it is, 164 00:16:22,170 --> 00:16:27,510 which is much more representative of the world of research mathematics than if I just sort of said, 165 00:16:27,510 --> 00:16:34,380 Well, let's show that the answer is this or show that the answer is this when we're doing research, maths and mathematics, we don't know the answer. 166 00:16:34,380 --> 00:16:38,580 But those questions where you don't know, am I trying to come up with an example? Say, You can't do this? 167 00:16:38,580 --> 00:16:46,000 Am I trying to find a proof? It can be hard to know what to do. And I was reading a book by the mathematician a bit little. 168 00:16:46,000 --> 00:16:52,020 Littlewood was one of the great British mathematicians of the first part of the 20th century, 169 00:16:52,020 --> 00:16:56,340 and Littlewoods expressed exactly this question in the context of research mathematics, 170 00:16:56,340 --> 00:17:06,900 he says complete indecision between yes and no in an exciting new problem is agonising when you go all out one way, either yes or no. 171 00:17:06,900 --> 00:17:11,710 The thought keeps nagging that it's an even chance that it ought to be the other way. 172 00:17:11,710 --> 00:17:17,500 The difference when you do know when, for example, we are looking for a new proof is enormous. 173 00:17:17,500 --> 00:17:24,230 And I think mathematical history bears this out. Sometimes you have this really difficult question a lot of people have thought about. 174 00:17:24,230 --> 00:17:29,290 Finally, somebody sold said, and we know this thing is true because we've got a proof. 175 00:17:29,290 --> 00:17:32,980 And then there's a flurry of more proofs over the coming years or decades. 176 00:17:32,980 --> 00:17:39,610 Somehow, you know, for sure, it's true because there's one proof and that kind of liberates you to find other ways of thinking about the problem. 177 00:17:39,610 --> 00:17:44,050 Whereas while I'm not quite sure it can be really hard to fully commit to, I'm going for it. 178 00:17:44,050 --> 00:17:54,650 I'm going to try to find the proof. Whatever it may be. So I think that in maths, sometimes we are very fixed on getting the answer to the problem. 179 00:17:54,650 --> 00:18:02,120 Maybe especially in kind of educational context, there's a lot of pressure. I've got to get the answer to the problem and that makes a lot of sense. 180 00:18:02,120 --> 00:18:06,380 But I think that we could lose sight of the process by which we get to the answer. 181 00:18:06,380 --> 00:18:09,650 And I think that the process there is a really important part of doing mathematics. 182 00:18:09,650 --> 00:18:13,940 Maybe, maybe I don't get the right answer, but maybe I don't quite reach an answer at all. 183 00:18:13,940 --> 00:18:18,110 But maybe I've still led to a whole lot. Maybe I've enjoyed being curious about mathematics. 184 00:18:18,110 --> 00:18:25,730 Maybe I've being creative in my approach, taking making the most of the process regardless of the outcome. 185 00:18:25,730 --> 00:18:29,040 So I guess that feels to me a bit like going for a walk up a hill. 186 00:18:29,040 --> 00:18:35,390 The goal is not to be at the top of the hill. If the goal was to be at the top of the hill and I had a whole lot of money, 187 00:18:35,390 --> 00:18:41,030 maybe I'd just get somebody with a helicopter to just fly me in and drop me at the top of the hill. 188 00:18:41,030 --> 00:18:42,350 But that wouldn't do the job right. 189 00:18:42,350 --> 00:18:48,110 It's not the same as the struggle of trying to climb up the hill and the winds trying to blow me off the cliff and try to fall off, 190 00:18:48,110 --> 00:18:52,820 not fall off the cliff. And I try not to get lost in the fall and all of those struggles. 191 00:18:52,820 --> 00:18:58,640 That's part of the process, and I enjoy the walking up. I think those of us who like [INAUDIBLE] walking kind of enjoy that. 192 00:18:58,640 --> 00:19:03,650 So I thought I'd take a couple of pictures of things I enjoyed as I was walking up the hill. 193 00:19:03,650 --> 00:19:08,090 There's an extinct volcano and this one on the islands next door. I thought that was an exciting moment. 194 00:19:08,090 --> 00:19:11,900 I found a big furry caterpillar who doesn't love a big furry caterpillar. 195 00:19:11,900 --> 00:19:16,190 I'm enjoying the moment. I don't have a mathematical analogy for Big Furry Caterpillar. 196 00:19:16,190 --> 00:19:23,030 I'm not quite sure how to translate that. So just embracing the process for a moment. 197 00:19:23,030 --> 00:19:27,980 You gave me your five numbers. I didn't have prior knowledge of what those were. 198 00:19:27,980 --> 00:19:35,750 I typed them into my computer and fairly quickly I could find three that added up to a multiple of three. 199 00:19:35,750 --> 00:19:42,680 That is not because I am amazing at mental arithmetic, I am not amazing with magic, I am not terrible at it, but I'm not amazing at it. 200 00:19:42,680 --> 00:19:46,520 The slow bit of the process was me acting up the three numbers, 201 00:19:46,520 --> 00:19:51,020 but I have since added up all of the other combinations of three numbers to find once the work. 202 00:19:51,020 --> 00:19:55,880 So how was I checking those examples quickly? 203 00:19:55,880 --> 00:19:58,850 So I want to illustrate this with pictures of bars of chocolate. 204 00:19:58,850 --> 00:20:04,610 I like this way of thinking about it, so I've got three pictures here representing three types of numbers. 205 00:20:04,610 --> 00:20:12,700 So the picture on the left. Without counting the squares, I can tell the remainder when I divide the number of squares by three. 206 00:20:12,700 --> 00:20:18,010 Right, because their columns are three, I've got three rows I can see the thing on the left is a multiple of three at least remained two zero. 207 00:20:18,010 --> 00:20:21,940 When I divide by three, I don't have to count the squares and the one in the middle. 208 00:20:21,940 --> 00:20:24,700 These are made to one when I divide by three and the one on the right. 209 00:20:24,700 --> 00:20:31,980 These are the two when I divide by three, and every type of number in the world is one of these three types. 210 00:20:31,980 --> 00:20:39,120 For the purposes of this problem, I don't care how many columns in this bar of chocolate are only care about the remainder. 211 00:20:39,120 --> 00:20:46,080 So what I'm doing is I'm taking them notes you're giving me and I'm replacing them by zero or one or two. 212 00:20:46,080 --> 00:20:51,340 And I'm looking for patents within those zeros and ones and twos. And that makes the numbers much more manageable. 213 00:20:51,340 --> 00:20:53,970 Small, I can cope with zeros and ones and twos. 214 00:20:53,970 --> 00:21:02,520 So for example, if I had these five numbers, I've just replaced by zeros, ones and twos that I've picked out the first three. 215 00:21:02,520 --> 00:21:07,260 And I've got one of each type here and I see looking at the pictures of bars of chocolate. 216 00:21:07,260 --> 00:21:09,690 We can see that if we've got one of each type, 217 00:21:09,690 --> 00:21:15,740 we're going to be able to use the remainder one square and sort of fill in this gap and make it multiple of three. 218 00:21:15,740 --> 00:21:21,770 That's a really useful strategy that gives us a way of tackling this problem that feels much more manageable. 219 00:21:21,770 --> 00:21:25,820 How many possible combinations of five whole numbers are that in the world? 220 00:21:25,820 --> 00:21:31,250 Well, like loads. I mean, officially infinitely many. There are a lot of whole numbers in the world. 221 00:21:31,250 --> 00:21:34,850 I can't check them all with this way of thinking. 222 00:21:34,850 --> 00:21:41,600 This problem is a manageable kind of small problem now because I'm only I only need to think about zeros and ones and twos. 223 00:21:41,600 --> 00:21:46,640 There's no point in my writing a million. I'll replace it by one. So let's just focus on zeros, ones and twos. 224 00:21:46,640 --> 00:21:51,290 This is like a really good moment. Whichever roots of the problem we're trying to go for. 225 00:21:51,290 --> 00:21:57,650 So if we're trying to find an example of five whole numbers, this gives us a way of kind of focussing in and looking at zeros, ones and twos. 226 00:21:57,650 --> 00:22:02,630 Just sort of think about that if we try to prove there's no combination of five numbers. 227 00:22:02,630 --> 00:22:07,160 This might give me a use where thinking about it. So I feel like this is a good moment. 228 00:22:07,160 --> 00:22:12,680 I feel like we should feel happy with ourselves at this point. Just a cautionary tale, though. 229 00:22:12,680 --> 00:22:21,920 So this is the mathematician Julia Robinson, very distinguished mathematician in the US and 20th century and somebody I think an administrator who had 230 00:22:21,920 --> 00:22:28,010 sort of file a report or something asked her one day to describe her typical week as a mathematician. 231 00:22:28,010 --> 00:22:34,800 So Julia Robinson described her typical week, like this Monday try to prove theorem. 232 00:22:34,800 --> 00:22:43,480 Tuesday tried to prove theorem. Wed try to prove their. 233 00:22:43,480 --> 00:22:49,980 Thursday. Try to prove theorem. Friday. 234 00:22:49,980 --> 00:22:55,730 Theorem false. Somebody told me once I shouldn't use this quote because it was too depressing. 235 00:22:55,730 --> 00:23:02,750 I don't find this quite depressing, and the reason I don't find it depressing is that Julia Robinson was a really very distinguished mathematician. 236 00:23:02,750 --> 00:23:08,210 And if somebody is distinguished as Delia Robinson could spend a week trying to prove something that turned out not to be correct, 237 00:23:08,210 --> 00:23:14,330 that makes me feel a whole lot better about all the times I've spent in my life trying to prove things that have turned out not to be correct. 238 00:23:14,330 --> 00:23:22,160 That is not time wasted, Littlewood said. There's this problem of, well, if I try to prove it, the answer is yes or no and. 239 00:23:22,160 --> 00:23:27,290 Julie Robinson reminds us that we might spend a week trying to predict the answers, yes, and it turns out the answer is no. 240 00:23:27,290 --> 00:23:32,690 Or vice versa. Another mathematician I was reading about recently. 241 00:23:32,690 --> 00:23:36,110 Turns out, also sees parallels between mathematics and mountain climbing. 242 00:23:36,110 --> 00:23:39,750 And I only discovered this quite after I planned this talk. I was really happy to find this. 243 00:23:39,750 --> 00:23:46,310 So this mathematician says there's a strong parallel between mountain climbing and mathematics research. 244 00:23:46,310 --> 00:23:51,710 Well, first attempts on a summit are made. The struggle is to find any route. 245 00:23:51,710 --> 00:23:53,120 Once on the top. 246 00:23:53,120 --> 00:24:02,300 Other possible routes up may be discerned, and sometimes a safer or shorter route could be chosen for the descent or for subsequent ascents. 247 00:24:02,300 --> 00:24:08,350 This is all my walk up and down arc. I thought I'd look back. Bottom here somewhere. 248 00:24:08,350 --> 00:24:10,480 The coolants, these famous mountain range, 249 00:24:10,480 --> 00:24:15,310 that's the dark shadow in the corner you can't see because they're in the clouds, just to put this in perspective. 250 00:24:15,310 --> 00:24:21,550 So this quote continues in mathematics. The challenge is finding a proof in the first place. 251 00:24:21,550 --> 00:24:30,610 Once found, almost any competent mathematician could easily find an alternative, often much better or shorter proof, at least in mountaineering. 252 00:24:30,610 --> 00:24:39,310 We know that the mountain is there and that if we can find a way up or reach the summit, we should triumph in mathematics. 253 00:24:39,310 --> 00:24:48,130 We do not always know that there is a result or if the proposition is only a figment of the imagination, let alone whether a proof can be found. 254 00:24:48,130 --> 00:24:53,290 So again, I don't want to be depressed by this, Kathleen all and so really interesting mathematician. 255 00:24:53,290 --> 00:24:58,750 Amazing person. I recommend going and reading about how if you haven't come across how previously 256 00:24:58,750 --> 00:25:03,010 she was also a very serious mountaineer in the way that I am not safe. 257 00:25:03,010 --> 00:25:09,580 But this idea that you could spend time trying to find a route up a mountain only to discover the mountain is not there. 258 00:25:09,580 --> 00:25:14,200 How do you have the bravery to just go for it? I think that's a really interesting question. 259 00:25:14,200 --> 00:25:18,160 I don't really have any good answers. I just think it's an interesting question. 260 00:25:18,160 --> 00:25:26,020 Let me go back to these bars of chocolate for a moment, because these bars of chocolate are a way of visualising something called modular arithmetic, 261 00:25:26,020 --> 00:25:29,950 which is one of my favourite ideas in mathematics, is immensely powerful. 262 00:25:29,950 --> 00:25:35,470 So you might have come across it in mathematics or computer science, coding, different kind of contexts. 263 00:25:35,470 --> 00:25:39,160 So here we're working March three and we're saying every number is a multiple of 264 00:25:39,160 --> 00:25:44,700 three or one more than a multiple of three or two more than a multiple of three. And this turns out to be an extraordinarily powerful idea. 265 00:25:44,700 --> 00:25:50,890 It's really useful tool for thinking about all sorts of things in number theory and cryptography in other contexts. 266 00:25:50,890 --> 00:25:56,890 So one nice thing that you can do with modular arithmetic is you can multiply. So I did when I was at primary school, 267 00:25:56,890 --> 00:26:00,230 maybe you had to do this to see these times tables you go one two three four five six one 268 00:26:00,230 --> 00:26:04,300 two three four five six and you fill in all of the kind of times tables of the Great Times, 269 00:26:04,300 --> 00:26:09,220 table practise or something. So it's kind of interesting to do that and the world of modular arithmetic. 270 00:26:09,220 --> 00:26:13,660 So in the world of Model three, there are only three different types of numbers. 271 00:26:13,660 --> 00:26:22,720 So that's why I've got a three by three square. So I got two two zero one two zero one two just fill in the Times table, so zero times zero is zero. 272 00:26:22,720 --> 00:26:27,760 If I take multiple three multiplied by multiple three, I get a multiple of three zero times. 273 00:26:27,760 --> 00:26:36,460 One is one zero two, I get zero times, two is zero one time zero is zero. 274 00:26:36,460 --> 00:26:43,540 One times one is one one times to is. This still feels quite familiar to times zero zero g times. 275 00:26:43,540 --> 00:26:54,500 One is two. Two times two. Sort of is full, but we're thinking in a more free world, that means it's the same as what if I do it three times to, 276 00:26:54,500 --> 00:26:59,600 I guess, a number of the middle type with these three pictures here. So I'm going with a one here. 277 00:26:59,600 --> 00:27:10,490 So this is like a multiplication table and the Model three world and I got up one day decided what I need to do is make a crochet version of this. 278 00:27:10,490 --> 00:27:16,270 So here is my crochet multiplication table. What three? 279 00:27:16,270 --> 00:27:20,200 It seemed like a good idea at the time. What is less clear to me, in hindsight, 280 00:27:20,200 --> 00:27:28,180 is why it seemed like a good idea at the time to make multiplication tables in crochet mode two three four five six seven eight nine 10. 281 00:27:28,180 --> 00:27:32,410 So each square is about this big. You know, each of the nine here is about this big. 282 00:27:32,410 --> 00:27:37,600 It takes me between 10 or 15 minutes to make a square. 283 00:27:37,600 --> 00:27:43,810 But I did make quite lots of them, and those pictures I showed you earlier on are crochet multiplication tables. 284 00:27:43,810 --> 00:27:49,630 So I think the one on the left is more seven and the one on the right is more eight. 285 00:27:49,630 --> 00:27:57,790 And what I love about this visualisation is that we can start to see differences fundamental structural differences between multiplication, 286 00:27:57,790 --> 00:28:02,860 WhatsApp and multiplication mode age. So it's slightly tricky to say. 287 00:28:02,860 --> 00:28:06,700 I'm afraid the Greens have sort of slightly merged in colour has a slight deficiency, 288 00:28:06,700 --> 00:28:10,510 but there are differences in pattern like in the mode seven table. 289 00:28:10,510 --> 00:28:12,910 But there's this sort of Sudoku property. 290 00:28:12,910 --> 00:28:21,730 So apart from all the zeros around the top and the left hand column, each row in each column has all of the numbers in it. 291 00:28:21,730 --> 00:28:26,380 This is a bit like a sea stock that doesn't happen on the right, on the right. 292 00:28:26,380 --> 00:28:32,330 There's a row, for example, that just alternate 040 for that sort of green green, green green. 293 00:28:32,330 --> 00:28:38,410 It's something fundamentally different that turns out to boil down to the fact the seven is prime and eight is not. 294 00:28:38,410 --> 00:28:41,380 So there's lots to explore there. And I'm not going to go into all the details now. 295 00:28:41,380 --> 00:28:48,910 But I love the visualisation and I say modular arithmetic helps us make sense of the scarves I showed you earlier on. 296 00:28:48,910 --> 00:28:57,390 These are my prime distribution scarves. So the deal is if we look at the top one, the top one, there are six rows on this stuff. 297 00:28:57,390 --> 00:28:59,580 We're looking at the numbers, what six? 298 00:28:59,580 --> 00:29:08,220 And you can be a white number with a green frame or green number with a white frame and the white numbers with green frames and not prime numbers. 299 00:29:08,220 --> 00:29:11,330 And the green ones with white frames are prime numbers. 300 00:29:11,330 --> 00:29:16,770 So the prime numbers, the individual numbers, the numbers that can be divided by themselves and by one, but not by anything else. 301 00:29:16,770 --> 00:29:19,800 So if you start from this end and count down the rows, 302 00:29:19,800 --> 00:29:24,600 you go one two three four five six and then next column seven, eight, nine, 10, 11, 12 and so on. 303 00:29:24,600 --> 00:29:31,870 And you can see two three five seven 11, which somebody helpfully picks as an example earlier on our prime numbers. 304 00:29:31,870 --> 00:29:36,220 The bottom one same story, but I've got seven rows instead of six. 305 00:29:36,220 --> 00:29:42,430 So the purple scarf is working more seven. The green one, which is blue in real life, is working. 306 00:29:42,430 --> 00:29:47,510 What? Six? And what I love is that I can start to see differences again. 307 00:29:47,510 --> 00:29:53,840 What six? Apart from two and three, all the primes are lining up in the top row in the fifth row. 308 00:29:53,840 --> 00:29:58,700 What this scarf illustrates is the gorgeous fact that apart from two and three, 309 00:29:58,700 --> 00:30:03,830 every prime in the world is one more than a multiple of six or one less than a multiple six. 310 00:30:03,830 --> 00:30:07,700 Well, the bottom scarf illustrates, is this a whole lot more complicated note? 311 00:30:07,700 --> 00:30:14,780 Seven. So if the bottom row, apart from seven, has no primes because there are no prime multiples, the seven apart from seven, 312 00:30:14,780 --> 00:30:20,180 but the primes the kind of scattered throughout you might see sort of diagonals in the bottom one. 313 00:30:20,180 --> 00:30:23,270 I don't know whether you can sort of see these diagonal stripes, the problems. 314 00:30:23,270 --> 00:30:29,300 Those are, in fact, the same stripes from the Watch six one just kind of staggered because we're working about seven. 315 00:30:29,300 --> 00:30:32,270 So that was a little kind of diversion into the, well, too much of the arithmetic, 316 00:30:32,270 --> 00:30:38,210 which I love for its mathematical, mathematically intriguing properties is mathematically powerful. 317 00:30:38,210 --> 00:30:46,130 Properties has lots of applications, too, so let's try going back to our problem about our five hole numbers and see, 318 00:30:46,130 --> 00:30:50,270 can we use this sort of zero one two way of thinking this mode? Three way of thinking? 319 00:30:50,270 --> 00:30:56,300 So here's the numbers that once you gave me, these are the examples that I did afterwards. 320 00:30:56,300 --> 00:31:01,220 So what I can do is take each of these and replace each number by zero or one or two to reflect is 321 00:31:01,220 --> 00:31:07,090 it a multiple of three or one more than a multiple of three or two more than a multiple of three? And then? 322 00:31:07,090 --> 00:31:11,290 Well, I don't know about you, I find that very difficult to read because it's got a whole load of numbers crossed out, 323 00:31:11,290 --> 00:31:16,080 so there's a tidied up version with just zeros and ones and twos. 324 00:31:16,080 --> 00:31:27,280 And. Maybe I start to find some patterns, so things like in the not the bottom rope, but the one above, I've got three zeros. 325 00:31:27,280 --> 00:31:29,530 I've got three numbers that are multiple of three. 326 00:31:29,530 --> 00:31:33,970 I love it when people give me an example with three numbers in about four three because I know if I add three months, 327 00:31:33,970 --> 00:31:38,030 wheels see three, I get a multiple of three. So you give me three multiple three. 328 00:31:38,030 --> 00:31:42,460 I'm very happy. The bottom one, we've got one of each type. 329 00:31:42,460 --> 00:31:46,720 And we said before, if we put one of each type that going to add to give a multiple of three. 330 00:31:46,720 --> 00:31:49,420 What else have we got? So the fourth wrote down, We've got one one on one. 331 00:31:49,420 --> 00:31:54,130 I've got three numbers the rule one more than the multiple of three I can use. All of those one leftover squares make another column. 332 00:31:54,130 --> 00:31:57,760 I get a multiple of three. I think the top row I've got three numbers. 333 00:31:57,760 --> 00:32:04,750 There are two more than a multiple of three. So what I can see is that there are some particular combinations of zeros, ones and twos. 334 00:32:04,750 --> 00:32:08,980 And if I find those amongst my five numbers there, zero, one or two. 335 00:32:08,980 --> 00:32:13,330 I'm in good shape. So if I have three multiples of three, we're done. 336 00:32:13,330 --> 00:32:17,950 If we've got three numbers there, one more than a multiple of three. We're done. 337 00:32:17,950 --> 00:32:24,940 We got three numbers that are two more than a multiple of three. We're done. And if we got one of each type, we're done. 338 00:32:24,940 --> 00:32:30,070 This feels like a whole lot of progress I think I did about you, I'm feeling like we're pretty near the top of the hill at this point. 339 00:32:30,070 --> 00:32:35,590 So let me show you some pictures of being near the top of the hill. This blue sky, there's sunshine. 340 00:32:35,590 --> 00:32:38,980 The wind is blowing a gale off chasing the path, not right next to the cliff, 341 00:32:38,980 --> 00:32:43,030 but the path a bit farther in so that I don't fall over the edge, but I'm getting near the top. 342 00:32:43,030 --> 00:32:47,620 This is a good feeling. It's just there. So what's going on with us? 343 00:32:47,620 --> 00:32:51,820 Zeros, ones and twos? We got to pick five numbers that it will zero, one or two. 344 00:32:51,820 --> 00:32:59,560 And if we find one of these four bad combinations that we know that we're done, we've got three numbers to add up to multiple three. 345 00:32:59,560 --> 00:33:04,570 So I can just check, right? So here's a first bunch of checking. 346 00:33:04,570 --> 00:33:08,060 So I try to be really systematic about listing my five types of numbers. 347 00:33:08,060 --> 00:33:13,810 So starting with zero zero zero zero zero. So I've got five numbers through a multiple of three and I'm working through and 348 00:33:13,810 --> 00:33:20,860 I couldn't fit them all on the slide and still have slightly readable font size. But here's a first chunk and then I look at these, and for each one, 349 00:33:20,860 --> 00:33:26,770 I have like one of these bad combinations and I'm a very patient person and I did this for you. 350 00:33:26,770 --> 00:33:32,590 You can say thank you later. And each row I've been able to colour in. 351 00:33:32,590 --> 00:33:37,270 So each row here I've been able to find three numbers that add up to a multiple of three. 352 00:33:37,270 --> 00:33:39,700 And that's great. So how many combinations are that? 353 00:33:39,700 --> 00:33:43,570 Well, there are three possibilities for the first column and three for the second of three for the third and three. 354 00:33:43,570 --> 00:33:48,580 So that's three times three times, three times three times three. That's two hundred and forty three possibilities. 355 00:33:48,580 --> 00:33:53,650 I've managed to fit seventy five on here, at least another hundred and sixty eight to go. 356 00:33:53,650 --> 00:33:58,090 I do it by you. I'm not very excited about the idea of checking another hundred and sixty eight cases. 357 00:33:58,090 --> 00:34:01,400 It wasn't that exciting checking those seventy five. 358 00:34:01,400 --> 00:34:08,600 So I feel like I can solve the problem because I feel like if I can contain my just be patient for long enough, 359 00:34:08,600 --> 00:34:11,280 I check my next hundred and sixty eight cases. 360 00:34:11,280 --> 00:34:16,820 And I bet based on those seventy five, I will be able to find three numbers add up to multiple three and then I'll go, 361 00:34:16,820 --> 00:34:22,250 Oh no, it's not possible to find five fold numbers so that there aren't three that add it to multiple three. 362 00:34:22,250 --> 00:34:27,260 But I feel a bit flat about the whole thing because I mean, yuck. 363 00:34:27,260 --> 00:34:33,110 It's horrible, isn't it? Just checking all those cases? I don't feel like I've understood what's going on behind the scenes. 364 00:34:33,110 --> 00:34:40,790 I feel like there is some underlying mathematics here, and I just have kind of tapped into that, so I'm not so keen on that. 365 00:34:40,790 --> 00:34:47,690 And actually, I go up to the top of the hill and it turns out it wasn't the top of the hill at all, and that's really kind of frustrating as well. 366 00:34:47,690 --> 00:34:57,200 So, yeah, we've still got a little way to go. And I don't really have a better way of summing up this up apart from whatever this emerges. 367 00:34:57,200 --> 00:35:03,330 So now I'm fed up because I'm not at the top of pantheon of egg and I've got a maybe way of doing the problem, but it's not very nice. 368 00:35:03,330 --> 00:35:08,840 And let's just take a pause from this and let's have a digression. 369 00:35:08,840 --> 00:35:14,490 This is really important to maths. Sometimes you have to know I just need to stop thinking about this problem for bit. 370 00:35:14,490 --> 00:35:16,820 I just need to think about maybe another massive problem. 371 00:35:16,820 --> 00:35:22,490 Maybe I just need to go for a walk, or maybe I need to go and have a cup of tea with my friend or whatever it may be. 372 00:35:22,490 --> 00:35:27,200 And there's a sort of fine balance between I have to spend long enough persevering on the problem. 373 00:35:27,200 --> 00:35:32,540 I have to kind of persuade enough for my subconscious to kind of absorb everything it can about the problem. 374 00:35:32,540 --> 00:35:36,290 But also, actually, I've just hit the point of frustration. I'm not making any progress. 375 00:35:36,290 --> 00:35:40,580 It's going to walk away. So let's have a digression. OK. 376 00:35:40,580 --> 00:35:45,110 This is maybe not the most exciting facts in the world, but one is equal to one squared. 377 00:35:45,110 --> 00:35:53,800 But if I do one plus three plus two squared and if I do one plus three plus five, that's three squared. 378 00:35:53,800 --> 00:35:58,780 And if I do one plus three plus five plus seven, that's four squared, we're seeing a pattern. 379 00:35:58,780 --> 00:36:05,020 So I just keep going and it seems like if I add up the first half of many odd numbers, 380 00:36:05,020 --> 00:36:10,510 I got a square number about a kind of nice patterns that I mean, you know, why not? 381 00:36:10,510 --> 00:36:16,580 There's a nice pattern. So I think I could go all the way up to five where I ran out of space on the side. 382 00:36:16,580 --> 00:36:23,910 So one plus three plus five plus seven plus nine plus 11+, 13 plus 15 plus 17 is nine squared. 383 00:36:23,910 --> 00:36:31,050 And at this point, I kind of I'm willing to bet that if I do one plus three plus five plus all the way up to 19, I get 10 squared, 384 00:36:31,050 --> 00:36:38,790 but I run out of room on the side and I didn't really feel like doing the calculation, so I thought I'd arrange my crochet blankets instead. 385 00:36:38,790 --> 00:36:41,070 Because having gone to the trouble of making them, why not? 386 00:36:41,070 --> 00:36:46,240 So one is this green slash blue square in the bottom right hand corner because I didn't make it one by one square. 387 00:36:46,240 --> 00:36:52,590 And look, here's three the remaining squares in the two by two square and then his five, which are the squares. 388 00:36:52,590 --> 00:37:01,950 We can still see them three by three square and his seven, a nine, 11 and 13, 15 and 17 and 19 and hollow adds up to 100. 389 00:37:01,950 --> 00:37:09,240 Isn't that lovely? That's not remotely my idea. I'm not sure whether anybody has been silly enough to make it out of crochet blankets before, 390 00:37:09,240 --> 00:37:13,470 but I just love this idea for the beauty, for the elegance. 391 00:37:13,470 --> 00:37:20,520 I can see why the result is true and that for me, it's a sign of a really lovely mathematical argument. 392 00:37:20,520 --> 00:37:25,410 There are lots of ways to saying that this is true. And some of them are nicer than other ways. 393 00:37:25,410 --> 00:37:34,560 For me, beauty is an important part of my work as a mathematician. So one person who famously wrote about this was a mathematician, James Hardie, 394 00:37:34,560 --> 00:37:38,580 who like Littlewoods, was one of the leading mathematicians of the first part of the 20th century, 395 00:37:38,580 --> 00:37:45,240 in fact, and which worked together a lot as a very famous collaboration and hardy in his book, Right? 396 00:37:45,240 --> 00:37:50,070 A mathematician like a painter or a poet is a maker of patterns. 397 00:37:50,070 --> 00:37:54,660 The mathematicians patterns, like the painters or the poets must be beautiful. 398 00:37:54,660 --> 00:37:59,850 The ideas like the colours or the words must fit together in a harmonious way. 399 00:37:59,850 --> 00:38:07,010 Beauty is the first test. There is no permanent place in the world for ugly mathematics. 400 00:38:07,010 --> 00:38:14,480 That's a pretty controversial statement. I think there's a whole bunch of ugly mathematics and probably does have a prominent place in the world. 401 00:38:14,480 --> 00:38:23,090 I think possibly this is a slightly extreme view. But for me, parts of the role of beauty is in helping me guide me towards a particular approach. 402 00:38:23,090 --> 00:38:26,060 If I'm looking at the [INAUDIBLE] I'm, I can see the way. 403 00:38:26,060 --> 00:38:33,140 If it involves climbing across a lovely green, smooth, grassy bank and the way that involves fighting through the bushes, 404 00:38:33,140 --> 00:38:38,630 it's the green smoothies bank all the way I want to have. 405 00:38:38,630 --> 00:38:44,000 I want to choose the beautiful route if I can, if I'm working on a problem and I can see different lines of attack. 406 00:38:44,000 --> 00:38:47,810 My instinct is to go for the one that I find aesthetically pleasing first. 407 00:38:47,810 --> 00:38:51,680 And maybe it doesn't work. Maybe I have to come back and try another tag. 408 00:38:51,680 --> 00:38:56,160 Or maybe I find an ugly solution, and then I try to find a nicer one later on. 409 00:38:56,160 --> 00:39:01,970 Like Kathleen, other and show said somebody finds a proof. Maybe you can then find another way. 410 00:39:01,970 --> 00:39:13,100 But but beauty can be powerful. I think beauty is a subjective in mathematics as it is in art or music or landscapes or cityscapes or whatever else. 411 00:39:13,100 --> 00:39:14,120 We might find beauty. 412 00:39:14,120 --> 00:39:21,620 So I don't think it's the case that because mathematicians don't all agree about what's beautiful, the mathematics doesn't count the speed to. 413 00:39:21,620 --> 00:39:26,090 For me, that sort of feeling of what is beautiful mathematics feels a little bit like, 414 00:39:26,090 --> 00:39:30,830 what do I think of when I think of a beautiful piece of music or a beautiful landscape? 415 00:39:30,830 --> 00:39:36,740 I sometimes worry that we might use beauty to exclude people from mathematics, and it's really important to me that we don't do that. 416 00:39:36,740 --> 00:39:41,120 If you look at a piece of mathematics and somebody is telling you it's beautiful and you don't think it's beautiful, 417 00:39:41,120 --> 00:39:45,650 that doesn't mean you're not a mathematician. Different people have different tastes. 418 00:39:45,650 --> 00:39:49,610 Also, I can't find a piece of mathematics beautiful until I've understood it. 419 00:39:49,610 --> 00:39:53,750 And I mean, spent quite a long time understanding it before I can appreciate the beauty. 420 00:39:53,750 --> 00:39:58,430 But I think the beauty is important in mathematics. One of the things about maths, 421 00:39:58,430 --> 00:40:03,550 so maybe you have this experience is that you work on a problem for a long time and you don't really feel like 422 00:40:03,550 --> 00:40:08,770 you're getting anywhere and you've tried writing out seventy five combinations of zeros and ones and twos. 423 00:40:08,770 --> 00:40:14,140 So this is not really kind of doing it for you. Oh, I don't know what to do sometimes. 424 00:40:14,140 --> 00:40:19,960 Sometimes the answer comes at just one of those moments of, Oh OK, I see what to do now. 425 00:40:19,960 --> 00:40:25,420 It's not always like that. Sometimes it's really slow and incremental. Sometimes you have that moment. 426 00:40:25,420 --> 00:40:27,490 So the question you're working on, let me just remind you, 427 00:40:27,490 --> 00:40:34,000 was can we pick five phone numbers so that there aren't three that up to a multiple of three? 428 00:40:34,000 --> 00:40:41,560 We haven't yet found an example, and we tried a bunch of examples and we've systematically checked 75 cases and it seems like we can't. 429 00:40:41,560 --> 00:40:47,800 And the way we were doing this was we think about zeros and ones and twos, so I can think about having five numbers that will zero, one or two. 430 00:40:47,800 --> 00:40:54,700 I'm picking out these bad combinations. If I've got three that are multiple of three, I have three that add up to three and so on. 431 00:40:54,700 --> 00:41:00,640 So if I take five numbers that it will zero or one or two. 432 00:41:00,640 --> 00:41:05,800 What might happen? Well, I might have a zero and one and a two, I might have all three types, 433 00:41:05,800 --> 00:41:10,390 and if I have all three types, it's game over because that was one of the bad combinations. 434 00:41:10,390 --> 00:41:14,620 So if I go two zero and a one and a two, those are going to adds up to give me a multiple of three. 435 00:41:14,620 --> 00:41:24,560 We can stop right away. What happens if we don't have a zero to one and a two, so I don't have all three types represented amongst my five numbers. 436 00:41:24,560 --> 00:41:28,100 So that means we've got almost two types in them, but maybe they're all the same type. 437 00:41:28,100 --> 00:41:37,220 Maybe that's split between two types, but we've got a most two types of number. So we got five numbers split between these two types. 438 00:41:37,220 --> 00:41:40,430 And that means we must have three that are the same. 439 00:41:40,430 --> 00:41:46,250 If I've got five numbers that fall into two categories, I must have at least three in the same category. 440 00:41:46,250 --> 00:41:54,530 And that's the other type of bad combination. So whatever five whole numbers we pick that are always three adults, a multiple of three, 441 00:41:54,530 --> 00:42:04,010 and I don't know about you, but I think that's a really lovely argument. And I think it's a much more lovely argument than let's check 243 cases. 442 00:42:04,010 --> 00:42:10,490 So that's one of those moments you think, Oh yes, that satisfaction when you reach the top of the hill? 443 00:42:10,490 --> 00:42:17,180 That's a good feeling. There's a trigger point. It was properly the top of the hill and how many coins to the top of the hill. 444 00:42:17,180 --> 00:42:24,530 I wanted to say for the moment, I wanted to take some photos. I was taking these photos on my phone one handed with my other hand, 445 00:42:24,530 --> 00:42:30,590 hanging on to the trigger point because the wind's trying to blow me off the hill. I'm not really into selfies, 446 00:42:30,590 --> 00:42:35,300 but I thought I should take a selfie to prove to you that it was me who climbed to top of the hill and also 447 00:42:35,300 --> 00:42:41,400 to demonstrate that this walk the winds meant this walk really was more hair raising than I had expected. 448 00:42:41,400 --> 00:42:48,600 So at this point, I'm thinking, OK, where next? What I'm thinking is I got to climb down this hill very carefully, not being blown off the cliff, 449 00:42:48,600 --> 00:42:54,510 and I'm going to get my car and I'm going to find a nice, warm cafe where next with our maths problem? 450 00:42:54,510 --> 00:42:57,860 Well, we could think about varying and ways. 451 00:42:57,860 --> 00:42:59,280 Some mathematicians do this all the time. 452 00:42:59,280 --> 00:43:04,800 We solve a problem and then we're like, Oh, OK, now I can ask all these other questions so we could think of. 453 00:43:04,800 --> 00:43:11,390 The problem we just talked about is take three from five. I pick five numbers. Are there three that add up to a multiple of three? 454 00:43:11,390 --> 00:43:19,670 So we could think, well, what if we do take two from three, if I pick three numbers, are there ways to the add up to a multiple of two? 455 00:43:19,670 --> 00:43:28,790 That's a really nice question. Or this is me planning my more adventurous clients in the crew islands when the weather is not covering them in cloud. 456 00:43:28,790 --> 00:43:31,370 What could I do? Take four from something? 457 00:43:31,370 --> 00:43:38,660 Are there some number of numbers so that if I pick four, I can pick four of them that add up to a multiple of four, or I should keep going. 458 00:43:38,660 --> 00:43:44,690 I could just kind of keep thinking, Well, there's lots to explore here. I have many mountains to climb. 459 00:43:44,690 --> 00:43:47,540 We have many mathematical problems to climb. 460 00:43:47,540 --> 00:43:55,100 So I don't think that having a mathematical adventure needs to be restricted to people whose job title is research and mathematics. 461 00:43:55,100 --> 00:43:59,330 I don't think that mathematical adventuring has to mean being the first person 462 00:43:59,330 --> 00:44:02,810 to climb Everest or being the first person to reach a particular summit. 463 00:44:02,810 --> 00:44:10,560 I think that everybody can enjoy the experience a mathematical adventure can enjoy the play and the creativity and the curiosity along the way. 464 00:44:10,560 --> 00:44:16,040 So I hope that perhaps you might be inspired to tackle your own mathematical adventures. 465 00:44:16,040 --> 00:44:20,480 I just want to say thank you to my friends and reds for introducing me to this problem. 466 00:44:20,480 --> 00:44:23,910 Take three, for five and for letting me share it with you today. I love this problem. 467 00:44:23,910 --> 00:44:32,480 It's such a beautiful problem. And I also wanted to show you my complete sets of crochet blankets just to give you some more patterns to think about. 468 00:44:32,480 --> 00:44:53,770 Thank you very much.