1 00:00:11,520 --> 00:00:16,710 It's a very great pleasure to be here, thank you, Alan, for those very kind words of introduction. 2 00:00:16,710 --> 00:00:24,570 Yes, I was here in Oxford in the 1980s where I did my my DPhil, and then I was a fellow at heart for college. 3 00:00:24,570 --> 00:00:26,490 And it's always wonderful to come back. 4 00:00:26,490 --> 00:00:34,650 It feels always like coming home when I come to Oxford, and it is a particular pleasure that I can give the Christmas lecture. 5 00:00:34,650 --> 00:00:40,530 I think mathematics is a wonderful subject and to combine it with Christmas is the best thing of all. 6 00:00:40,530 --> 00:00:47,280 So, yes, I have the somewhat mysterious title. Why does Rudolph have a shiny nose? 7 00:00:47,280 --> 00:00:57,150 And this lecture has a subtitle which I hope all of you can identify with, which is How can a mathematician survive Christmas? 8 00:00:57,150 --> 00:01:01,680 So we're going to look in this lecture at various ways that a mathematician can 9 00:01:01,680 --> 00:01:06,510 survive Christmas by looking at the mathematics behind Christmas and so on. 10 00:01:06,510 --> 00:01:10,410 This is meant to be light hearted so the mathematics won't be too deep, 11 00:01:10,410 --> 00:01:16,320 and I will warn you that I will need volunteers at various points in the order in the talk, 12 00:01:16,320 --> 00:01:20,730 particularly towards the end where we might have time for some dancing. 13 00:01:20,730 --> 00:01:28,680 So I just thought I'd warn you of that before we start. So we're in in the month of December, it's cold, it's wet. 14 00:01:28,680 --> 00:01:31,770 The trains aren't working very well, but we are, of course, 15 00:01:31,770 --> 00:01:40,920 in the Christmas season and this is a slightly more interesting Christmas season than usual because we have a number of events happening. 16 00:01:40,920 --> 00:01:48,870 You might have noticed that some various events are happening, which are not quite Christmassy, but are still labelled as Christmas events. 17 00:01:48,870 --> 00:01:55,710 And one of these is, of course, the election. So yes, we will be talking a bit about that. 18 00:01:55,710 --> 00:02:03,210 This is followed shortly. So that's the first election, which is on the 12th of December, and that involves voting. 19 00:02:03,210 --> 00:02:07,410 And then on the 14th of December, we have a much more important election, 20 00:02:07,410 --> 00:02:13,350 which will also discuss in some detail which is the Strictly Come Dancing final. 21 00:02:13,350 --> 00:02:18,570 I hope that people are not to do what strictly come dancing, otherwise we're in trouble. 22 00:02:18,570 --> 00:02:25,290 OK, then we move on to the slightly more traditional events and we'll talk a bit about Christmas Night. 23 00:02:25,290 --> 00:02:31,020 And this is where you will meet Rudolph. And then we'll have a look at Christmas Day. 24 00:02:31,020 --> 00:02:37,890 And following Christmas Day, we'll have a look at the mathematics of the twelve days of Christmas. 25 00:02:37,890 --> 00:02:48,420 And finally, the Christmas season comes to an end with a great sigh of relief, I'm sure, with New Year on 1st of January 2020. 26 00:02:48,420 --> 00:02:52,470 My daughter lives in Edinburgh, so we always go up for a big party then. 27 00:02:52,470 --> 00:03:00,690 So that is the Christmas season and we're going to have a look at the way mathematics helps us enjoy the Christmas season. 28 00:03:00,690 --> 00:03:03,990 So I think I'll start with a perfectly reasonable question. 29 00:03:03,990 --> 00:03:11,610 It's a slightly mathematical question, and that's this Why is Christmas on the 25th of December? 30 00:03:11,610 --> 00:03:21,000 It's perfectly reasonable question to ask. We've been celebrating Christmas on the 25th of December for a very long time, so why is it on this date? 31 00:03:21,000 --> 00:03:24,660 Well, one thing we certainly know is that it wasn't. 32 00:03:24,660 --> 00:03:31,530 This sort of happens on a certain day three three six A.D. was when Emperor Constantine set the date. 33 00:03:31,530 --> 00:03:38,730 So Christmas on the 25th of December started in three hundred and thirty six A.D. 34 00:03:38,730 --> 00:03:44,010 And the easily accepted theory is that the Christmas is in some way linked with 35 00:03:44,010 --> 00:03:50,730 the winter solstice and the Roman Midway to Winter Festival of Saturnalia. 36 00:03:50,730 --> 00:03:56,100 But I've never been completely satisfied with that because that's on the 21st of December. 37 00:03:56,100 --> 00:04:00,180 And, you know, even with rounding error, that's still quite a big difference. 38 00:04:00,180 --> 00:04:04,770 OK. Here's another one. And again, slightly mathematical. 39 00:04:04,770 --> 00:04:14,670 There's a Christmas Christian tradition that Mary was told that she was expecting her baby on the 25th of March. 40 00:04:14,670 --> 00:04:18,270 And then if you add nine to that, you get to the 25th of December. 41 00:04:18,270 --> 00:04:20,370 So that's a possible reason. 42 00:04:20,370 --> 00:04:29,610 And there is another Roman festival, the Natalia's Solis Evictee, The Birth, The Unconquered Sun, which was held on December the 25th. 43 00:04:29,610 --> 00:04:37,950 So these are all perfectly acceptable reasons why Christmas is on the 25th, but they all have the same disadvantage. 44 00:04:37,950 --> 00:04:42,870 And that is, I would say, none of them are mathematical reasons. 45 00:04:42,870 --> 00:04:53,580 So is there a mathematical thing? Is there a mathematical equation which somehow yields the answer Christmas Day? 46 00:04:53,580 --> 00:05:00,030 And the answer is, of course, yes. That's why we have one of the reasons I'm here. 47 00:05:00,030 --> 00:05:07,140 So but before we start with that, I just get this little joke aside. 48 00:05:07,140 --> 00:05:14,040 I don't know how many of you know the answer. Why can't a mathematician tell the difference between Christmas Day and Halloween? 49 00:05:14,040 --> 00:05:28,190 Then you will know. Yes, because Oct. 31 is the same as that's correct, December the 25th or 25 in decimal is 31 in October, 50 00:05:28,190 --> 00:05:32,180 and therefore they are identical to a mathematician. 51 00:05:32,180 --> 00:05:36,870 Okay, so that's a perfectly good, acceptable reason. Here's another one. 52 00:05:36,870 --> 00:05:40,670 What's important function? Very important function. 53 00:05:40,670 --> 00:05:46,550 Since when mathematicians functions as a very famous function called the Raymond Zita function on the 54 00:05:46,550 --> 00:05:54,110 Raymond Z to function is conjectured to have all its non-trivial zeros on the line with real pass half. 55 00:05:54,110 --> 00:05:59,690 So the question is what important function has a zero on Christmas Day? 56 00:05:59,690 --> 00:06:04,250 OK, so there is an important function which has a zero on Christmas Day and are good. 57 00:06:04,250 --> 00:06:10,700 I would argue that's a very good mathematical reason why we have Christmas Day on Christmas Day. 58 00:06:10,700 --> 00:06:16,090 Anyone know about function is. Well, here's the answer. 59 00:06:16,090 --> 00:06:28,240 Is the function called the equation of time, the equation of time very important function has the property that it has a zero on Christmas Day. 60 00:06:28,240 --> 00:06:33,640 And I think that's a very good mathematical reason why we celebrate Christmas on Christmas Day. 61 00:06:33,640 --> 00:06:39,370 Well, let me tell you about the equation of time. There are two types of day you might have thought. 62 00:06:39,370 --> 00:06:44,110 There's only one talk to date, but there's actually two types of date. 63 00:06:44,110 --> 00:06:51,460 There's the day that we generally use as a day, and that's a day which is 24 hours long. 64 00:06:51,460 --> 00:06:57,770 So I bought a clock here. I anticipated that it would be hard for me to read the clock at this distance and with my bad eyesight. 65 00:06:57,770 --> 00:07:04,660 So that's my own clock, and that clock goes all the way around when it goes around twice because it's 12 hours, 66 00:07:04,660 --> 00:07:12,640 but it goes around in 24 hours, and those 24 hours is measured using a crystal inside the clock. 67 00:07:12,640 --> 00:07:18,430 So that's what we often think of a day, but that actually is what we call a mean day. 68 00:07:18,430 --> 00:07:23,290 It's an average day, so the average length of the day is 24 hours. 69 00:07:23,290 --> 00:07:33,170 And that's why we talk about Greenwich Mean Time. But there's another way to look at the length of a day, and that is to say what is the time? 70 00:07:33,170 --> 00:07:44,300 We could say midday is when the sun is highest in the sky that is noon and then a day lasts from noon to noon tomorrow. 71 00:07:44,300 --> 00:07:48,830 OK, so that is another very good way of defining a day. 72 00:07:48,830 --> 00:07:54,740 That's how ancient people measured days. That's how most people until quite recently measured days. 73 00:07:54,740 --> 00:08:02,900 And that is called a solar day. But a solar day isn't always 24 hours. 74 00:08:02,900 --> 00:08:14,870 So the Earth is tilted on its axis. It's tilted at about 23 degrees to the ecliptic and goes around the sun in an elliptical orbit. 75 00:08:14,870 --> 00:08:19,820 And the combination of tilting on its axis and going rounds in an orbit means 76 00:08:19,820 --> 00:08:27,200 that the length of the solar day varies from day to day throughout the year, 77 00:08:27,200 --> 00:08:32,060 and as a result, the length of the solar day varies. 78 00:08:32,060 --> 00:08:40,370 And here's Greenwich, and the equation of time is the difference between the time at noon at Greenwich, 79 00:08:40,370 --> 00:08:45,730 which you would typically measure on a sun dial and 12 clock. 80 00:08:45,730 --> 00:08:53,710 And that time varies throughout the year. And here's a nice sort of graphic, which I stole, 81 00:08:53,710 --> 00:09:01,990 which shows that the the change in time over here due to the elliptical eccentricity that's the tail to the Earth, 82 00:09:01,990 --> 00:09:13,030 means that the difference between the time measured on a sundial and time measured on a clock varies throughout the year. 83 00:09:13,030 --> 00:09:19,960 So there's a lovely Sundial if you have a chance to look around Oxford, if you're visiting or if you live in Oxford. 84 00:09:19,960 --> 00:09:25,600 This is the Sundial at All Souls College, and it's very lovely Sundial. 85 00:09:25,600 --> 00:09:32,410 And on the 3rd of November, Sundial's all 16 minutes and 33 seconds fast. 86 00:09:32,410 --> 00:09:37,120 So that's quite a lot. Actually, you've missed plenty of buses with that. 87 00:09:37,120 --> 00:09:45,490 OK. On the 12th of February, a sundial is 14 minutes and six seconds slow again. 88 00:09:45,490 --> 00:09:50,650 Buses would be missed, but here's the great thing. 89 00:09:50,650 --> 00:09:57,640 It's exactly right on Christmas Day, so if you want to know when Christmas Day is, 90 00:09:57,640 --> 00:10:02,350 you go along, says All Souls College, and you look at the sundial and you look at a watch. 91 00:10:02,350 --> 00:10:06,970 And if they're saying the same time, it's Christmas Day. 92 00:10:06,970 --> 00:10:14,680 Apart from the fact it could be those of days as well, but Christmas Day is one of them and that's good enough for me. 93 00:10:14,680 --> 00:10:20,950 The main problem, of course, is that Sundial tend not to work in Christmas Day because it's raining or something like that. 94 00:10:20,950 --> 00:10:25,450 But anyway, that is for me, a very good reason why Christmas is on Christmas Day. 95 00:10:25,450 --> 00:10:31,240 It's the only day in the year apart from those other days when your Sundial actually works. 96 00:10:31,240 --> 00:10:43,540 OK. Right? Well, having established what Christmas is on the 25th of December, let's move on into the Christmas season as I advertised in my opening. 97 00:10:43,540 --> 00:10:51,760 And as I said, it will not have failed to pass your attention that on Thursday, something is happening. 98 00:10:51,760 --> 00:11:00,460 The Great Christmas vote. Now I'm on the purge of my university has forbidden me to say anything political, 99 00:11:00,460 --> 00:11:05,890 so I think it's probably best to avoid talking about voting about parties. 100 00:11:05,890 --> 00:11:09,580 So we will have a vote just to see how we get on. 101 00:11:09,580 --> 00:11:17,110 But we'll vote on a very contentious and important issue, which I can see lots of you are helping me with in the audience. 102 00:11:17,110 --> 00:11:23,650 And I'm sure anyone with children will know that Christmas is a great time for arguments and general disagreements and so on. 103 00:11:23,650 --> 00:11:31,390 So what I'm going to do is tell you how to resolve this. So here's the Great Christmas vote, which is the best Christmas jumper. 104 00:11:31,390 --> 00:11:34,420 So here you can see one. His his jumper. 105 00:11:34,420 --> 00:11:43,510 No, I you can see my Christmas tree in the background that we all have a few presents on Christmas jumper a his Christmas jumper. 106 00:11:43,510 --> 00:11:48,520 No, be there we are. And number three Christmas jumper. 107 00:11:48,520 --> 00:11:52,360 No. See, I'm also wearing this is Christmas jumper No. 108 00:11:52,360 --> 00:11:57,730 D, but we won't bring that into the election. OK, so so we're going to have a vote now and see how we get on. 109 00:11:57,730 --> 00:12:01,270 So there we are. Christmas jumper. No. A Christmas jumper. 110 00:12:01,270 --> 00:12:06,100 No be Christmas jumper. No. See, I hope you've all absorbed those. 111 00:12:06,100 --> 00:12:10,270 So in the spirit of democracy and audience participation and all of that. 112 00:12:10,270 --> 00:12:15,280 And if you're like a can you shout a if you like B, can you shout B? 113 00:12:15,280 --> 00:12:22,480 And if you like C, can you say this is how I do my lectures, by the way, in case you want to? 114 00:12:22,480 --> 00:12:30,690 It's modern teaching methods. OK, so after three a b or C one two three. 115 00:12:30,690 --> 00:12:36,160 I couldn't really hear that one try again. One, two, three. No, it could be any of them. 116 00:12:36,160 --> 00:12:43,360 So what we're going to have to do at this stage is bring in our expert panel of judges, 117 00:12:43,360 --> 00:12:53,230 which is how we will resolve this and we'll do a bit of mathematics on this. So my expert panel of judges on the left of this is Monty. 118 00:12:53,230 --> 00:12:57,400 This is Josh, my third expert panel member. 119 00:12:57,400 --> 00:13:01,630 Is this one, which is Cleopatra the hamster? 120 00:13:01,630 --> 00:13:07,360 I like big names for small animals, so these are my three friends. 121 00:13:07,360 --> 00:13:15,400 And so I ask their opinion on the on the jumpers and the again, this is something which often happens at Christmas. 122 00:13:15,400 --> 00:13:19,930 You have something you're trying to decide on and disagreements. 123 00:13:19,930 --> 00:13:24,400 And there we are. Monty said that he preferred A to B to C. 124 00:13:24,400 --> 00:13:32,290 Josh said he preferred B to C to way, and Cleopatra said that she preferred C to be to eight. 125 00:13:32,290 --> 00:13:37,840 So here's the question they clearly don't agree. They clearly don't agree. 126 00:13:37,840 --> 00:13:44,690 And which is the winner? OK, so. 127 00:13:44,690 --> 00:13:50,030 Again, this is a good thing. I mean, this is voting, this is what we, you know, we should be having voting like this, we expect. 128 00:13:50,030 --> 00:13:56,060 Anyway, that's another story. OK. So one way to resolve disagreements is to use the mathematics, 129 00:13:56,060 --> 00:14:03,260 and this is actually what we do in my house where my children don't agree on what television programme to watch or something like that. 130 00:14:03,260 --> 00:14:13,820 And normally my wife wins anyway. But if you want to note, this is a message which I quite like, we can use a thing called the border method. 131 00:14:13,820 --> 00:14:22,550 And the border method is a way of resolving disputes. It's used in one country in the world to do its elections, and that is Slovenia. 132 00:14:22,550 --> 00:14:29,270 So the border method works that if you put a first, you give it to be second. 133 00:14:29,270 --> 00:14:34,190 You give it one c last, you give it zero. 134 00:14:34,190 --> 00:14:38,720 And if you do that, that's Monti's choices. That's Josh's choices. 135 00:14:38,720 --> 00:14:51,110 That's Cleopatra's choices. And if you add them up, you find that the winner clearly is b just to show you that one was the panel of judges. 136 00:14:51,110 --> 00:14:57,440 So those of you who voted for B, you agree with the panel of judges, even though it wasn't clear from that. 137 00:14:57,440 --> 00:15:01,490 So mathematics can be used in elections. 138 00:15:01,490 --> 00:15:11,120 It won't be useful. First, I'm sure who knows what will happen. However, this method of voting will be used in this next item of the Christmas season, 139 00:15:11,120 --> 00:15:21,110 which I claim is even more important than the election, which is the Strictly Come Dancing final on the 14th of December. 140 00:15:21,110 --> 00:15:28,220 Just a quick sanity check who here is going to watch the Strictly Come Dancing final on the 14th of December. 141 00:15:28,220 --> 00:15:36,230 Excellent. I noticed more of the younger members of the audience and the older, but I should certainly be watching it. 142 00:15:36,230 --> 00:15:44,340 I shall be in Holland at the time, but I'll watch anyway. So here we have a panel, Anton and his partner. 143 00:15:44,340 --> 00:15:51,050 And for those of you who don't know how strictly come dancing works, you have a celebrity guest with a professional partner. 144 00:15:51,050 --> 00:16:00,500 They do dances. And then there's a panel of judges and the judges award points, just like we did for all our Christmas jumpers. 145 00:16:00,500 --> 00:16:06,350 And there are four judges, and each judge gives each dance a mark from 10 to one. 146 00:16:06,350 --> 00:16:13,670 OK, so that's how it works. So just like we voted and gave numbers to the jumpers. 147 00:16:13,670 --> 00:16:20,630 So the panel judges do so that you then add them up and that gives you a ranking. 148 00:16:20,630 --> 00:16:27,170 So if each judge gives a mark of 10, which is the top mark you can get, you get 40. 149 00:16:27,170 --> 00:16:38,060 Having got that, you, you take the judges ranking and the audience emails in or whatever their scores and you get a ranking 150 00:16:38,060 --> 00:16:45,470 from the audience and you add the judges ranking to the audience ranking to give you the final rank. 151 00:16:45,470 --> 00:16:49,970 And that's how the Strictly Come Dancing final works. 152 00:16:49,970 --> 00:16:56,390 And the glorious thing about this system is that close to being random, it's about as bad as you could get for any voting guesses. 153 00:16:56,390 --> 00:17:04,730 So I thought I might demonstrate that for those of you who are going to watch Strictly Come Dancing on Saturday night. 154 00:17:04,730 --> 00:17:12,050 OK, so let's see why it's so crazy. So here, by the way, if you don't know them, are all four judges. 155 00:17:12,050 --> 00:17:23,150 These are the official four judges, and they voted eight, eight, eight and eight for one of the dances. 156 00:17:23,150 --> 00:17:27,140 And here are two interesting combinations. 157 00:17:27,140 --> 00:17:33,900 This is the eight eight eight eight, which the judges who just voted up there and four times eight is thirty two. 158 00:17:33,900 --> 00:17:38,870 So the winning the couple have scored thirty two. 159 00:17:38,870 --> 00:17:47,840 And another pair gets the votes nine nine nine one, which adds up to twenty eight. 160 00:17:47,840 --> 00:17:53,780 And this is kind of interesting from a mathematical point of view, and this could easily happen on the floor. 161 00:17:53,780 --> 00:17:58,400 By the way, when I watch Strictly Come Dancing. I don't watch the dances, only watch the voting. 162 00:17:58,400 --> 00:18:00,860 It's much more fun. 163 00:18:00,860 --> 00:18:08,990 I apply exactly the same principle to the Eurovision Song Contest, OK, which is actually slightly fair in the way it does its voting. 164 00:18:08,990 --> 00:18:13,460 So but do watch ignore the dances. The voting is much the best bet. 165 00:18:13,460 --> 00:18:18,680 I'm not sure if you noticed what's going on here. This is the winner. This is the winner. 166 00:18:18,680 --> 00:18:24,790 This one. Three out of the four judges prefer. 167 00:18:24,790 --> 00:18:28,990 That candidate, not that pair. And yet, it's the second one, that's one. 168 00:18:28,990 --> 00:18:37,120 So according to Strictly Come Dancing voting rules, it's possible to win when most of you don't, like most of the judges, don't like you. 169 00:18:37,120 --> 00:18:45,280 And so the second pair has won. Despite I put this in because it's a sort of vague Oxford connexion. 170 00:18:45,280 --> 00:18:50,410 The fight aside being preferred by the three judges are not the first judge. 171 00:18:50,410 --> 00:18:55,840 The one that gave here the one has actually dictated the whole result. 172 00:18:55,840 --> 00:19:02,080 And that is called being a dictator. So it's a kind of a fair system. 173 00:19:02,080 --> 00:19:09,730 And if you think that's bad, it gets worse when you bring in the audience. So this is how the audience effect is. 174 00:19:09,730 --> 00:19:18,490 So here's a case where you have how many seven dancers and the judges have ranked the dancers in this order, 175 00:19:18,490 --> 00:19:23,950 giving seven to the top couple and one to the final. 176 00:19:23,950 --> 00:19:35,350 The worst couple, and they've given seven to couple D and a couple of D has got the top rank for the judges. 177 00:19:35,350 --> 00:19:42,280 The audience have rung in and they've given sevens a couple of C, so that's the top ranking for the audience. 178 00:19:42,280 --> 00:19:47,350 But if you add the ranking four, the judges, I'm ranking from the audience. 179 00:19:47,350 --> 00:19:52,120 This is the couple that wins Capital F with 10 votes. 180 00:19:52,120 --> 00:19:57,340 And it's gloriously won despite the fact that no one likes them. 181 00:19:57,340 --> 00:20:02,950 And if you've been watching Strictly Come Dancing Again, even better than the voting is, 182 00:20:02,950 --> 00:20:10,660 is the tears afterwards and all that sort of stuff when it's finally realised that the one that's one is the one no one actually likes. 183 00:20:10,660 --> 00:20:17,350 So do watch on Saturday and see if we hope this doesn't happen, of course. 184 00:20:17,350 --> 00:20:20,680 So, so that's the next event in the Christmas season. 185 00:20:20,680 --> 00:20:30,930 So we'll move on now to kind of the core of the lecture, which brings us to Rudolph, which is Christmas Night. 186 00:20:30,930 --> 00:20:39,210 So this is a slightly famous calculation, I apologise if you've seen it before, but one has to include it in any Christmas lecture about mathematics, 187 00:20:39,210 --> 00:20:47,910 which is a discussion of how Santa, given the 24 hours of darkness that you get around the world and the whole of Christmas night. 188 00:20:47,910 --> 00:20:54,480 So I know it's not 24 hours in the UK, although it seems like it, but it is 24 hours if you include New Zealand all the way round. 189 00:20:54,480 --> 00:20:55,590 Well. 190 00:20:55,590 --> 00:21:06,270 And the question is, can centre get round the world in time to deliver all of the presents that he needs to on Christmas Day and Christmas Night? 191 00:21:06,270 --> 00:21:14,220 Now we know various things about the delivery algorithm announces that Santa will only 192 00:21:14,220 --> 00:21:20,930 or will I think it's both necessary and sufficient deliver presents to good children. 193 00:21:20,930 --> 00:21:29,760 Okay, so you've got to be a good child. I'm never quite sure how good and child defined in in my house. 194 00:21:29,760 --> 00:21:35,100 My daughter, who's twenty seven, absolutely expects a stocking. Even the dog gets a stocking. 195 00:21:35,100 --> 00:21:38,070 OK, so there we go. So here's the thing. 196 00:21:38,070 --> 00:21:46,860 The world's population at last count was nearly eight billion, and I would say as a conservative estimate, it's difficult to tell. 197 00:21:46,860 --> 00:21:53,730 But there's about a billion how homes that have good children in them. 198 00:21:53,730 --> 00:21:59,910 I'm sure in this audience we have far more good children than that will be, but just just a rough, very rough estimate. 199 00:21:59,910 --> 00:22:08,940 It's only to get what we call an order of magnitude. OK, so we're going to work out how far Santa has to go to deliver this. 200 00:22:08,940 --> 00:22:18,150 And to do that, we need to know what the average distance of part of the houses are with good children in them. 201 00:22:18,150 --> 00:22:30,180 So to do this will assume that the houses are fairly regularly arranged and each is an average distance away from its nearest neighbour. 202 00:22:30,180 --> 00:22:34,110 This is mathematical modelling. We're making approximations, but you know it's Christmas. 203 00:22:34,110 --> 00:22:40,170 OK, so we're going to assume that the houses are a distance apart. 204 00:22:40,170 --> 00:22:47,940 OK, so the area occupied by a house before you reach another house is h times h or h squared. 205 00:22:47,940 --> 00:22:55,260 So the total area occupied by houses is a is and times h squared. 206 00:22:55,260 --> 00:23:03,810 Now we know a few other things go. Children, for example, don't live in the sea and they tend not to live at the North Pole. 207 00:23:03,810 --> 00:23:09,480 That's why Santa lives, so we only need to consider the continents. 208 00:23:09,480 --> 00:23:17,560 I don't believe children live in Antarctica, either, so the surface area of the main continents is this. 209 00:23:17,560 --> 00:23:22,110 And that's what it is. According to Wikipedia, OK. 210 00:23:22,110 --> 00:23:26,520 And we know that this is equal to end times eight squared. 211 00:23:26,520 --> 00:23:35,830 So if you're quite this, it turns out that the average distance between two houses with good children in them is four hundred and seventy five metres. 212 00:23:35,830 --> 00:23:37,830 OK, so this is a modelling thing even. 213 00:23:37,830 --> 00:23:45,810 I mean, it's the average distance because, you know, not not many good children live in the desert, for example, or jungle or whatever. 214 00:23:45,810 --> 00:23:52,200 So there we are. That's the average distance between homes with good children, and Santa has to visit every one. 215 00:23:52,200 --> 00:23:56,730 So the distance he has to go is n times that. 216 00:23:56,730 --> 00:24:01,500 So he has to travel in 12 hours, at least because there may be some. 217 00:24:01,500 --> 00:24:06,180 If the houses are less regular, you actually get a long distance, 218 00:24:06,180 --> 00:24:12,130 but the distance he has to travel is at least four hundred and seventy five billion metres. 219 00:24:12,130 --> 00:24:16,860 OK, so that's how far he has to go in 24 hours. 220 00:24:16,860 --> 00:24:23,040 OK, so from that, we can get an estimate of his speed. So here he is, travelling around the houses. 221 00:24:23,040 --> 00:24:27,460 There we go. I'm indebted to my son for the graphics here. 222 00:24:27,460 --> 00:24:35,280 Uh, I'm afraid to say my son studied at Cambridge, but he did do mathematics, though. 223 00:24:35,280 --> 00:24:42,510 And if we take that, the speed is one centimetre divided by 2014, 224 00:24:42,510 --> 00:24:52,080 three is 5.5 million metres per second, or if you prefer, that's about 10 million miles an hour. 225 00:24:52,080 --> 00:24:55,920 OK, now let's have a look at some comparisons. 226 00:24:55,920 --> 00:25:03,000 The speed of sound is three hundred and seventy five metres per second, so he's going comfortably faster than the speed of sound. 227 00:25:03,000 --> 00:25:10,130 We get onto that in the next slide. Light speed is 300 million metres per second. 228 00:25:10,130 --> 00:25:16,530 Now, this is very reassuring because we can do it. 229 00:25:16,530 --> 00:25:25,950 He can do it. He doesn't have to go faster than the speed of light in order to deliver the planet the presence to the good children. 230 00:25:25,950 --> 00:25:30,950 And of course, if you have a lot more good children, but you'd have to have 60 times as many. 231 00:25:30,950 --> 00:25:36,530 To prevent injury so he can actually do it, which is very, very reassuring. 232 00:25:36,530 --> 00:25:46,580 However, he does have to go very, very fast. So we now get on to the central question of the lecture, which is why does Rudolph have a shiny nose? 233 00:25:46,580 --> 00:25:51,560 Well, we've built up the basic mathematical model we can now predict from that. 234 00:25:51,560 --> 00:25:59,210 So Santa is going significantly faster than the speed of sound he's going at what we call hypersonic speeds. 235 00:25:59,210 --> 00:26:06,740 So there's a bullet bullet has a hyperbolic shockwave, so that's bullet going. 236 00:26:06,740 --> 00:26:13,460 Hypersonic fast has to be the leading reindeer in the sleigh. 237 00:26:13,460 --> 00:26:19,640 It's pulling from flight, which is, of course, Rudolph the bit of Rudolph, which is at the front, 238 00:26:19,640 --> 00:26:26,030 which is causing the main kind of entry into the air is, of course, the nose. 239 00:26:26,030 --> 00:26:29,420 As you go through the air, there are greater experts than me in the audience about that. 240 00:26:29,420 --> 00:26:40,550 The air will run past the nose. And due to friction and various things will heat the nose up and therefore the nose glows due to the high speeds. 241 00:26:40,550 --> 00:26:44,870 And so the speed heats up the nose till it close. 242 00:26:44,870 --> 00:26:59,020 And there we are. So. And fluid mechanics shows us why Rudolph has a shiny nose and only he has the shiny nose because 243 00:26:59,020 --> 00:27:05,020 he's at the front and therefore the shockwave goes round him and all the other reindeer are safe. 244 00:27:05,020 --> 00:27:09,910 But we are faced with another question to do with the good children. 245 00:27:09,910 --> 00:27:18,910 We know that Santa is a gentleman of large proportions and the good children live in houses with chimneys. 246 00:27:18,910 --> 00:27:24,440 So the question is how does he get down the chimney being large or small? 247 00:27:24,440 --> 00:27:28,710 So the first possible theories about this? 248 00:27:28,710 --> 00:27:37,300 And one theory which is quite popular, is Einstein's theory of relativity applied to Santa. 249 00:27:37,300 --> 00:27:41,920 What's that got to do with this problem of how we get down the chimney? 250 00:27:41,920 --> 00:27:45,640 It's a thing called the Lorenz contraction, which basically means, 251 00:27:45,640 --> 00:27:50,800 according to Einstein special theory relativity, the faster you go, the smaller you get. 252 00:27:50,800 --> 00:27:56,200 So we're now actually have our first one. We've had a few mathematical formula, but here's a decent one. 253 00:27:56,200 --> 00:28:02,290 This is what's called the Lorenz contraction, which down, he says. 254 00:28:02,290 --> 00:28:11,050 Oh, sorry, if you got to speed V and C is the speed of light, which is three times twenty eight metres per second. 255 00:28:11,050 --> 00:28:15,580 Then your length going in that speed is this fact. 256 00:28:15,580 --> 00:28:23,600 The thing here, which is smaller than one and gets closer to zero as V gets closer to C times the original. 257 00:28:23,600 --> 00:28:28,450 So this contract you so also I say so in lay terms. 258 00:28:28,450 --> 00:28:34,720 The faster you go, the smaller you get. OK, so that's that's the Lorenz contraction. 259 00:28:34,720 --> 00:28:42,220 So it's perfectly reasonable model. But as I always tell my students, I don't care how good your formulae are. 260 00:28:42,220 --> 00:28:54,700 Test them on real numbers. One of my students did a study, very manly problem, which was the you know, how urine is extracted from the male body. 261 00:28:54,700 --> 00:28:59,740 And he came up with the formula and I said, Have you put your numbers in there? 262 00:28:59,740 --> 00:29:07,780 He said, No. I said, Let's work it out. And it worked out, and we found that it was coming out significantly faster than the speed of light. 263 00:29:07,780 --> 00:29:17,890 And I said, Well, you may be able to do it, but I can't. So it's always good to put the formula in the numbers in so we know how fast the space going. 264 00:29:17,890 --> 00:29:25,930 It's going at 5.5 million metres per second. And if you substitute that into that, we get nought point nine nine nine eight three. 265 00:29:25,930 --> 00:29:30,340 It's the contraction. So he's smaller, but not a lot smaller. 266 00:29:30,340 --> 00:29:33,940 OK. So it's not enough. 267 00:29:33,940 --> 00:29:40,810 And in fact, he'd have to go to speed of zero point nine nine times the speed of light to contract enough to get down the chimney. 268 00:29:40,810 --> 00:29:45,910 And even Santa probably can't go that speed. If you did that, the chimney would explode and stuff like that. 269 00:29:45,910 --> 00:29:54,370 So we have to come up with other theories as well. I'm not sure I have a good theory, but I have a kind of cute way to make it possible. 270 00:29:54,370 --> 00:30:09,340 So the idea is this we may not be able to get Santa down a chimney, but possibly we might be able to get him through a piece of paper. 271 00:30:09,340 --> 00:30:17,070 Does anyone think I can get him through a piece of paper? Well, we have some hands going up. 272 00:30:17,070 --> 00:30:23,070 I may need you one of you to volunteer to be Santa. Would anyone like to volunteer to be Santa? 273 00:30:23,070 --> 00:30:26,700 See, people are hiding. Well, don't worry, you don't have to. 274 00:30:26,700 --> 00:30:34,530 I might volunteer someone in due course. The way we can get Santa through a piece of paper is to use. 275 00:30:34,530 --> 00:30:38,550 A fracture. So this is quite a nice demonstration again. 276 00:30:38,550 --> 00:30:46,350 Chris Paul Christmas is all about doing party tricks, and one party trick you can challenge is can you cut a hole in a piece of paper, 277 00:30:46,350 --> 00:30:52,620 which is large enough for everyone to fit through what you two fit through? 278 00:30:52,620 --> 00:31:01,050 So let's see if we can do that. And the way we do it is kind of fun is we're going to cut in black. 279 00:31:01,050 --> 00:31:06,780 That sort of set of lines on this piece of paper. 280 00:31:06,780 --> 00:31:20,150 OK. I hope this goes right. So we start by cutting sort of a hole like this through in the middle. 281 00:31:20,150 --> 00:31:27,860 By the way, she's perfectly normal paper. OK. 282 00:31:27,860 --> 00:31:36,980 Littering the place up there, you can see saw a hole in there, but it's probably not quite big enough for me to sit around and we fold it up 283 00:31:36,980 --> 00:31:47,810 and we cut one cut like this at end and another cut like this at the other end. 284 00:31:47,810 --> 00:32:00,740 OK. You can see that. Can you see that? And you do the same cuts regularly along on. 285 00:32:00,740 --> 00:32:08,470 To. I guess I hope this doesn't all go horribly wrong. 286 00:32:08,470 --> 00:32:24,110 Knowing me it well for five, so you can do this on Christmas night or in the Sherry party and then you cut the other way like this. 287 00:32:24,110 --> 00:32:32,370 And oh, it's all kind of coming apart to another one like this. 288 00:32:32,370 --> 00:32:41,790 And another one like this. And another one like this. 289 00:32:41,790 --> 00:32:50,590 And another one like this. There's another one like this. 290 00:32:50,590 --> 00:32:59,320 And providing I haven't completely marked this up if we open this up. 291 00:32:59,320 --> 00:33:03,610 We will get some like this with a hole in it. 292 00:33:03,610 --> 00:33:10,480 Would anyone like to volunteer to try and get through this whole? Would anyone like to volunteer, yes, would you like to come over here? 293 00:33:10,480 --> 00:33:15,720 Let's see if we can get you through this whole show. So you look very centrist to me. 294 00:33:15,720 --> 00:33:19,140 That's very nice. Thank you. What's your name? Georgia Here we are at. 295 00:33:19,140 --> 00:33:24,760 See if we can fit George with you to come on the stage. We'll see you better than you from Oxford, Georgia. 296 00:33:24,760 --> 00:33:32,940 Have you got it so we can fit you through, shall we? And we go. And there we go. 297 00:33:32,940 --> 00:33:37,740 Well done. Well, thank calculations. So. 298 00:33:37,740 --> 00:33:42,240 And you can fit any size through. You just have to do a few more cuts. 299 00:33:42,240 --> 00:33:50,130 And then we go. So that's how you fit Santa through a piece of paper a.k.a chimney on Christmas night. 300 00:33:50,130 --> 00:33:59,190 OK, so now I'm on the theme of cutting things. It allows me to get on Christmas Day and teach you a very beautiful piece of maths. 301 00:33:59,190 --> 00:34:08,970 It's one of my favourite pieces of maths, and that is to help answer the other perennial question of What do you put on the tree? 302 00:34:08,970 --> 00:34:14,640 So we've now got two Christmas Day. We realised that there aren't enough decorations on the tree. 303 00:34:14,640 --> 00:34:20,910 You are required to produce a Christmas decoration. The question is, how can you do it? 304 00:34:20,910 --> 00:34:34,440 And I'm going to show you a very beautiful mathematical way of producing a Christmas decoration, and it relies on a result due to this wonderful guy. 305 00:34:34,440 --> 00:34:36,690 This is guy called Eric Domain. 306 00:34:36,690 --> 00:34:48,120 I'm sorry this picture is a bit fuzzy, but this is drawn from his video on YouTube, and Eric Domain has the wonderful title of maths genius at MIT. 307 00:34:48,120 --> 00:34:57,870 It's an official, some sort of title, but he's one of the top mathematicians in America, and he's extremely famous for his work in origami. 308 00:34:57,870 --> 00:35:09,690 And he proves a theorem called the what's called the fold and Cut Theorem, and the folding cup theorem goes as follows if you take a piece of paper. 309 00:35:09,690 --> 00:35:20,580 Here we are, and you fold it in the right sort of way and you make a single cut, then you can make any shape you want. 310 00:35:20,580 --> 00:35:29,010 That's called the fold and cut theorem. And he's demonstrating this where he folded a piece of paper and within a single cup produced an angel fish. 311 00:35:29,010 --> 00:35:34,380 I couldn't find an angel, so an angel fish is close enough for today. 312 00:35:34,380 --> 00:35:39,570 So when emailing me, so I put that on. Now I need it natural. 313 00:35:39,570 --> 00:35:43,360 So this is the angel fish. It's not particularly Christmassy. 314 00:35:43,360 --> 00:35:52,410 So let's see if we can do something a bit more Christmassy. We're going to use this theorem to make a star, which I think is Christmassy. 315 00:35:52,410 --> 00:36:04,140 So the way you do it is, you fold the paper over here and you fold it in half and then you fold it in half again. 316 00:36:04,140 --> 00:36:14,660 So that's now folded into four. You unfold and you have a sort of crease there. 317 00:36:14,660 --> 00:36:23,690 You then fold it up like this. So that piece of paper is sort of up against that fold. 318 00:36:23,690 --> 00:36:30,890 I hope you can see that you then fold this piece in. 319 00:36:30,890 --> 00:36:35,320 I like that, so I've just folded that piece in. 320 00:36:35,320 --> 00:36:47,910 And you can see that and to give you that shape. And then you fold over again to produce what the Americans call a pizza slice. 321 00:36:47,910 --> 00:36:49,920 So all I've done is folded that up. 322 00:36:49,920 --> 00:37:03,440 Can you see I've done nothing more than fold that paper in these various ways and then taking my trusty pair of scissors, I cut along here. 323 00:37:03,440 --> 00:37:09,130 So there's a single cut, just one cup with a pair of scissors. 324 00:37:09,130 --> 00:37:15,590 And I get the shape, and hopefully if I am folded shape. 325 00:37:15,590 --> 00:37:24,310 Let's see what we get. We get. Let's start. 326 00:37:24,310 --> 00:37:30,520 You guys of so nice party trick at Christmas. 327 00:37:30,520 --> 00:37:43,510 And if you take the rest of the paper and I thrilled that you get other stuff, so that's how you make Christmas decorations on Christmas Day. 328 00:37:43,510 --> 00:37:51,100 If you're a mathematician, the story goes that this was how the American flag was done as well, right? 329 00:37:51,100 --> 00:37:52,870 Well, let's move on now. 330 00:37:52,870 --> 00:38:01,870 So we've got a past Christmas and we're now into the twelve days of Christmas and we all sing the song The Twelve Days of Christmas. 331 00:38:01,870 --> 00:38:11,770 But I wonder if any of you have ever paused to consider how many presents to my true love sent on Christmas Day? 332 00:38:11,770 --> 00:38:20,420 Many of you thought about this very important question. So here's the 12 days of Christmas. 333 00:38:20,420 --> 00:38:26,750 So let me tell you a bit about this. We have Bing Crosby to help us here. 334 00:38:26,750 --> 00:38:37,730 So on the 12 days of Christmas, on the first day of true love is going to be a partridge in a pear tree on the second day of Christmas. 335 00:38:37,730 --> 00:38:46,400 My true mortgage raising turn dogs and a partridge in a pear tree on the third day of Christmas. 336 00:38:46,400 --> 00:38:51,080 My true love get to be three French children. 337 00:38:51,080 --> 00:38:59,380 The party, which is apparently on the fourth day of Christmas special, looking to me for comment. 338 00:38:59,380 --> 00:39:03,870 I think I'll spare you the rest. You can sing it ahead. You want five? 339 00:39:03,870 --> 00:39:17,430 Oh oh, oh oh. Some people say, and do you want to sing along when we get to it on the fifth day of Christmas? 340 00:39:17,430 --> 00:39:27,210 For me? Oh. Seen. 341 00:39:27,210 --> 00:39:32,790 That's right, I'll stop there. 342 00:39:32,790 --> 00:39:37,110 Just out of interest, this song was originally written for children as a fourth hit song, 343 00:39:37,110 --> 00:39:42,930 which they had to sing along, and if one of them made a mistake, the other children had to tickle them. 344 00:39:42,930 --> 00:39:47,190 So I'll spare you that. OK, so let's have a think how this works. 345 00:39:47,190 --> 00:39:54,810 So the 12 days of Christmas on day one, we got a partridge in a pear tree on day two. 346 00:39:54,810 --> 00:39:58,380 We got to talk to doves and a partridge in a pear tree. 347 00:39:58,380 --> 00:40:03,090 So there's one present on the first day. How many on the second day? 348 00:40:03,090 --> 00:40:08,550 OK, on day three. Three French hens, two turtle doves and a partridge in a pear tree. 349 00:40:08,550 --> 00:40:16,080 Day four day five day six day seven eight nine day 10 to 11 day 12. 350 00:40:16,080 --> 00:40:25,410 So that's how many presents are given on each day. And these, if you don't know these things here, have a special name. 351 00:40:25,410 --> 00:40:31,650 These are called triangle numbers, so one is one one plus two is three one plus two plus three six. 352 00:40:31,650 --> 00:40:36,750 And if you add up any of them, then you get the number ntx plus one over two. 353 00:40:36,750 --> 00:40:41,610 So the professional mathematicians will know this, but these are called triangle numbers. 354 00:40:41,610 --> 00:40:48,060 And here they are, appropriately decorated with the animals in question. 355 00:40:48,060 --> 00:40:51,810 So our job is to work out the 12 days of Christmas. 356 00:40:51,810 --> 00:40:56,970 Some is to find a way of adding up the triangle numbers. 357 00:40:56,970 --> 00:41:04,950 So I'm going to show you two ways of doing this, one of which is not very Christmassy on the other one is very Christmassy. 358 00:41:04,950 --> 00:41:09,060 OK, so here's another Christmassy way on Christmas Eve. 359 00:41:09,060 --> 00:41:13,260 So then on Christmas, your way is the total number of presents. 360 00:41:13,260 --> 00:41:24,960 I know this is a Christmas lecture. We will get Christmas in a sec. Is this and then plus one you can express as a difference between two products. 361 00:41:24,960 --> 00:41:33,330 When you add up the sum the telescope you find that you get the total number of presents is the final product, 362 00:41:33,330 --> 00:41:40,080 which is 12 times 13 times 14, divided by six, which is three hundred sixty four. 363 00:41:40,080 --> 00:41:47,990 And so just to emphasise that three hundred sixty four is the total number of presents given. 364 00:41:47,990 --> 00:41:51,740 When you see this, there's always a question everybody asks, what's the question? 365 00:41:51,740 --> 00:41:58,990 Everyone asks? The question is what happened to the last present? 366 00:41:58,990 --> 00:42:06,490 OK, what happened to the last president? Because the three hundred and sixty five days in the airport for next year, which is Libya, 367 00:42:06,490 --> 00:42:11,730 what happened to the last presence and well, basically we're all allowed to make a mistake. 368 00:42:11,730 --> 00:42:17,470 OK, so there's my true love. This is Benji the dog. So that's a non Christmassy way of doing the song. 369 00:42:17,470 --> 00:42:23,260 But I thought I'd like to tell you about the Christmas stocking theorem, which is one of my favourite results, 370 00:42:23,260 --> 00:42:28,990 which is a Christmas way of adding up the number of presents in the twelve days of Christmas. 371 00:42:28,990 --> 00:42:33,220 And the inspiration of this comes from my chairman, Professor Gabriela. 372 00:42:33,220 --> 00:42:38,680 So some of you may not know, but Professor Gary Aly is in league with darkness. 373 00:42:38,680 --> 00:42:48,790 In this, he is the chief advisor to Moriarty, which is the arch villain in Sherlock Holmes and in particular produced memorialises Blackboard. 374 00:42:48,790 --> 00:42:56,170 So if ever you watch Sherlock Holmes the movie, you will see this blackboard, which I believe Professor Gore really designed. 375 00:42:56,170 --> 00:43:02,680 Well, he's got to do with Christmas. Well, if you look at this, if I was to do this one of my lectures, I think I get very low. 376 00:43:02,680 --> 00:43:06,760 Student appreciation is quite a lot of detail. Here's how to see what's going on. 377 00:43:06,760 --> 00:43:09,400 So I put a ring around the important thing. 378 00:43:09,400 --> 00:43:16,780 So this is the clue to the twelve days of Christmas is this bit of the blackboard here, which is called Pascal's Triangle. 379 00:43:16,780 --> 00:43:25,580 And there it is. So this is Pascal's triangle, and Pascal's triangle has everything to do with the 12 days of Christmas. 380 00:43:25,580 --> 00:43:30,450 Let me show you the way prosecco stronger works in case you don't know. 381 00:43:30,450 --> 00:43:39,140 There's lots of numbers in it. And if you take any two numbers in Pascal's triangle and you add them up, you get the number underneath. 382 00:43:39,140 --> 00:43:46,850 OK, so twenty eight plus fifty six is 84. And Canada two fifty five plus 11 is sixty six. 383 00:43:46,850 --> 00:43:57,080 And we can leave those decorations hanging on the tree. Now, if you look more closely down here, we have the days of Christmas one down to 12. 384 00:43:57,080 --> 00:44:04,220 There we are. And what's magical about this triangle is next to them of a number of presents that's given on the dice. 385 00:44:04,220 --> 00:44:09,860 On the first day we get one present two second day three presents six days 21 one presence. 386 00:44:09,860 --> 00:44:13,100 There are the triangle numbers. 387 00:44:13,100 --> 00:44:24,470 So in terms of Pascal's triangle, if we want to add up the presence to give in on the twelve days of Christmas, there are the triangle numbers. 388 00:44:24,470 --> 00:44:29,530 Those are the total number of presents given on each day and we have to add them up. 389 00:44:29,530 --> 00:44:34,160 Can anyone remember what the answer was to that? Three hundred sixty four? 390 00:44:34,160 --> 00:44:38,900 Can anyone see three hundred sixty four on that diagram? 391 00:44:38,900 --> 00:44:44,660 There it is. And what we do is we use what we love beautifully. 392 00:44:44,660 --> 00:44:50,240 Call a Christmas stocking to add them up. And then there is the answer. 393 00:44:50,240 --> 00:44:57,410 So if you go down the dark, no and then do a little stocking, that's the answer. 394 00:44:57,410 --> 00:45:01,490 And what's amazing about Pascal's triangle is that this always works. 395 00:45:01,490 --> 00:45:10,460 If I take any other diagonal and add them up, then I will always get the answer by just veering off to the left. 396 00:45:10,460 --> 00:45:17,750 So that's give a proof of this. I'm going to give two proofs one for the non Christmas mathematicians and then for the Christmas tree amongst you. 397 00:45:17,750 --> 00:45:27,500 So here's the Christmas tree proof. That's a statement of the Christmas stocking theorem not quite looking like a Christmas stocking, but that's it. 398 00:45:27,500 --> 00:45:34,040 And each of those you can express as another telescoping sum, you add them up and you get the answer. 399 00:45:34,040 --> 00:45:37,580 OK, I'm sure you've got all that. So that's the algebraic light. 400 00:45:37,580 --> 00:45:43,070 Let's look at the Christmas Seaway. We're going to prove it by induction. 401 00:45:43,070 --> 00:45:47,630 So the inductive proof is and you'll notice that we now have a proper Christmas tree 402 00:45:47,630 --> 00:45:54,020 here in our presence going down here and we claim that the sum of these is seventy. 403 00:45:54,020 --> 00:46:00,710 If you add them up, one plus four is five plus 10 is 15 plus twenty five plus 35 70. 404 00:46:00,710 --> 00:46:04,790 So yes, it works good, right? 405 00:46:04,790 --> 00:46:08,600 Well, let's add in the next one is the next one. 406 00:46:08,600 --> 00:46:20,230 And if you remember about Pascal's triangle, the property of Pascal's Triangle is that those two add up to give the one underneath, which is their. 407 00:46:20,230 --> 00:46:22,720 And now we've got that so we can get rid of that one. 408 00:46:22,720 --> 00:46:28,750 And we proved it for the next next number down the sequence, and if we do it again, we have them up. 409 00:46:28,750 --> 00:46:32,590 There it is. And then we go. Add them up. 410 00:46:32,590 --> 00:46:42,370 There it is. And that's how the Christmas stocking is proved by adding numbers successfully and building up the Christmas stocking. 411 00:46:42,370 --> 00:46:46,720 And that's the Christmas stocking theorem proved by induction. 412 00:46:46,720 --> 00:47:00,080 And I was I really like that. Proof is very beautiful proof. It has Christmas written all over it, and I think we can congratulate ourselves for that. 413 00:47:00,080 --> 00:47:10,730 And there we are. Now we've got a few more minutes left in the lecture, but I'm going to set you all a bit of homework. 414 00:47:10,730 --> 00:47:20,510 Here's the bonus question. What is the largest number of any one presence that was given during the 12 days so you can go and think about that. 415 00:47:20,510 --> 00:47:24,740 I won't tell you the answer to the end, but at the end of the lecture, I will give you the answer. 416 00:47:24,740 --> 00:47:29,210 OK, so what is the largest number of Christmas presents that were given any anyone days? 417 00:47:29,210 --> 00:47:30,980 I'll give you the answer at the end. 418 00:47:30,980 --> 00:47:42,510 So we finally get to the end of the Christmas season, which is New Year's Day and your chance to have a Kelly who goes on Christmas on New Year's Day. 419 00:47:42,510 --> 00:47:51,330 Well, few people, so I thought we finished this lecture with a mathematical Kelly, and I will need four volunteers for this. 420 00:47:51,330 --> 00:47:55,650 I really do need four volunteers. So can I have four volunteers for Christmas Kelly? 421 00:47:55,650 --> 00:48:02,360 You won't have to do too much. This won't work if I don't have full volunteers. 422 00:48:02,360 --> 00:48:08,600 Yes, please, two the back, fantastic, would you like to come forward? Do we have one over there? 423 00:48:08,600 --> 00:48:16,010 Yes, please. Would you like to come for Christmas, Kylie, please? It will be elegant if we had a bloke now. 424 00:48:16,010 --> 00:48:23,660 Yes. Oh, sorry, yes. Apologies are fantastic. 425 00:48:23,660 --> 00:48:34,610 So what have you for? So if you'd like to step up where we can see you, I'm going to show you how mathematically we can do a. 426 00:48:34,610 --> 00:48:39,050 So is anyone his name? Begin with a second name. 427 00:48:39,050 --> 00:48:44,030 I don't know if you'd like to come over here. We are. 428 00:48:44,030 --> 00:48:48,620 You can have an A. Anyone name began with the second name. 429 00:48:48,620 --> 00:48:53,030 So name. Best friend's name. Go on. You'll be happy. 430 00:48:53,030 --> 00:48:57,500 Anyone with a C. I'll take you. Obviously there. 431 00:48:57,500 --> 00:49:03,050 And I'm afraid I left my day at home, so I had to make one new all day right? 432 00:49:03,050 --> 00:49:10,790 So can we show those to? Everyone can see them. So we're going to do a bit of square dancing with these letters. 433 00:49:10,790 --> 00:49:18,320 And when I say square dancing, I mean square dancing. So we're going to have a square. 434 00:49:18,320 --> 00:49:22,760 And don't worry, it's going to be OK. 435 00:49:22,760 --> 00:49:31,670 So there's a square, and those of you who have been properly brought up and know about our huge groups will know that squares have symmetries. 436 00:49:31,670 --> 00:49:38,480 One symmetry is to reflect the square across a diagonal, which gives you that square hope. 437 00:49:38,480 --> 00:49:43,940 You can see that, and that corresponds in dancing to ABCD. 438 00:49:43,940 --> 00:49:49,550 Going to a CBD, or what is known in technical language as an inner twiddle. 439 00:49:49,550 --> 00:49:55,310 Would you like to do an industrial? Would B and C like to swap over place? OK, that's called an inner tweddle. 440 00:49:55,310 --> 00:50:00,020 Can you see it? And would you like to go back again? There we go. That's an inner twiddle. 441 00:50:00,020 --> 00:50:11,120 You might try and guess what's going to happen next. If you reflect the square along a vertical, then the what you get is what we call outer twiddle. 442 00:50:11,120 --> 00:50:12,440 And then an outer twiddle. 443 00:50:12,440 --> 00:50:20,420 You change places and you change places that's called an outer twiddle, and she'll be to do it like, you're enjoying this good. 444 00:50:20,420 --> 00:50:25,550 That's what we want. So that's an inner trigger and an outer twiddle. 445 00:50:25,550 --> 00:50:31,580 And what I want you to do now is an inner twist, followed by an outer twiddle so you can have an inner turtle. 446 00:50:31,580 --> 00:50:40,580 Now do an outer twiddle. That's it. Oh yes. So we get see a deep I'm sorry, that's not quite visible, but you can work on it. 447 00:50:40,580 --> 00:50:50,210 So the clever bit is, if you do that, you get what happens if you rotate a square by 90 degrees. 448 00:50:50,210 --> 00:51:00,200 So in inner twiddle, followed by an outer twiddle, gives you that if you take a square and you rotate it by 90 degrees, you've got you have that. 449 00:51:00,200 --> 00:51:05,000 If you do it again, you get that. If you do it again, you get that. 450 00:51:05,000 --> 00:51:09,460 And if you do it again, you get back to the beginning and that's a dance. 451 00:51:09,460 --> 00:51:15,950 So and here's the dance done with letters, and we're going to do it with you guys. 452 00:51:15,950 --> 00:51:21,380 Could you go back to where you started? So let's see if this works. 453 00:51:21,380 --> 00:51:31,220 And this is mathematical square dancing for those of you really want to know, this is a really for and is a common figure and many, many dances. 454 00:51:31,220 --> 00:51:35,780 So I'm in a Twitter outer twiddle. 455 00:51:35,780 --> 00:51:50,540 I sit in a tweddle. Outer Twitter in a tweet, your answer, Twitter in a tweet all out of Twitter. 456 00:51:50,540 --> 00:51:57,030 And what do you notice? Now. 457 00:51:57,030 --> 00:52:01,050 I just need you to bear with me for a little bit more, so. 458 00:52:01,050 --> 00:52:07,860 Could we take the sea away? The bear was in the doorway, right? 459 00:52:07,860 --> 00:52:17,940 Can we put give you the eighth right? 460 00:52:17,940 --> 00:52:27,740 We're going to do the same, but hopefully if this all works with some music. 461 00:52:27,740 --> 00:52:31,590 Oh, I've never done it to this before, but this music before. 462 00:52:31,590 --> 00:52:38,190 Normally I do it to burn dance music, but as it's Christmas, we're going to do Jingle Bells. 463 00:52:38,190 --> 00:52:45,270 So here we go. Right? OK, I'll start again so soon as the music. 464 00:52:45,270 --> 00:52:49,890 Oh, shush, go. I know. Not it. OK, yeah. 465 00:52:49,890 --> 00:52:54,130 Go back to where you are. So we'll start. 466 00:52:54,130 --> 00:52:58,680 When the music starts, you start doing whatever it is. 467 00:52:58,680 --> 00:53:18,720 Dance, that's the word you got out of Twitter in a tweet out to Twitter in a Twitter outage, Rachel in a tweet a little after 12. 468 00:53:18,720 --> 00:53:23,950 Yay! Well done. Thank you. 469 00:53:23,950 --> 00:53:27,830 You've been great sports. I could write. Thank you without not. 470 00:53:27,830 --> 00:53:34,040 You have to do a dance at Christmas again. You can return to your. 471 00:53:34,040 --> 00:53:36,920 Right, so we're virtually there now. 472 00:53:36,920 --> 00:53:45,350 So I sent you all a bit of homework, which is what is the largest number of single presence that was given on any one day. 473 00:53:45,350 --> 00:53:50,780 And what is the answer to that is what is the answer? 474 00:53:50,780 --> 00:53:57,470 What is the answer? What is the numerical answer? 475 00:53:57,470 --> 00:54:05,650 No. 42. I sometimes wonder why bother? 476 00:54:05,650 --> 00:54:14,170 OK, so the answer, as we all know, is 42 in Douglas Adams The Hitchhiker's Guide to the Galaxy, we have the answer. 477 00:54:14,170 --> 00:54:18,760 We're never told what the question is. Now you know the question. It's the largest number of presents. 478 00:54:18,760 --> 00:54:26,980 This is given on any one date. So it's either six swans, a singing swimming for seven days or seven, whatever it is for six days is 42. 479 00:54:26,980 --> 00:54:37,390 Six times, 70 is 42. The answer so I think merely finish now by saying, Merry Christmas, talk to you all. 480 00:54:37,390 --> 00:54:46,590 And in honour of the title of the talk, we have a little bit of, I'm sure you know what's coming. 481 00:54:46,590 --> 00:54:51,160 Don't assume that we're just a second. 482 00:54:51,160 --> 00:55:04,900 We need to get to the punchline and monitoring hearing from the most famous reindeer of all Rudolph reindeer. 483 00:55:04,900 --> 00:55:10,590 All right. You might have heard somebody say right. 484 00:55:10,590 --> 00:55:31,924 I think we better stop.