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It's a very great pleasure to be here, thank you, Alan, for those very kind words of introduction.
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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.
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And it's always wonderful to come back.
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It feels always like coming home when I come to Oxford, and it is a particular pleasure that I can give the Christmas lecture.
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I think mathematics is a wonderful subject and to combine it with Christmas is the best thing of all.
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So, yes, I have the somewhat mysterious title. Why does Rudolph have a shiny nose?
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And this lecture has a subtitle which I hope all of you can identify with, which is How can a mathematician survive Christmas?
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So we're going to look in this lecture at various ways that a mathematician can
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survive Christmas by looking at the mathematics behind Christmas and so on.
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This is meant to be light hearted so the mathematics won't be too deep,
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and I will warn you that I will need volunteers at various points in the order in the talk,
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particularly towards the end where we might have time for some dancing.
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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.
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The trains aren't working very well, but we are, of course,
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in the Christmas season and this is a slightly more interesting Christmas season than usual because we have a number of events happening.
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You might have noticed that some various events are happening, which are not quite Christmassy, but are still labelled as Christmas events.
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And one of these is, of course, the election. So yes, we will be talking a bit about that.
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This is followed shortly. So that's the first election, which is on the 12th of December, and that involves voting.
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And then on the 14th of December, we have a much more important election,
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which will also discuss in some detail which is the Strictly Come Dancing final.
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I hope that people are not to do what strictly come dancing, otherwise we're in trouble.
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OK, then we move on to the slightly more traditional events and we'll talk a bit about Christmas Night.
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And this is where you will meet Rudolph. And then we'll have a look at Christmas Day.
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And following Christmas Day, we'll have a look at the mathematics of the twelve days of Christmas.
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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.
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My daughter lives in Edinburgh, so we always go up for a big party then.
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So that is the Christmas season and we're going to have a look at the way mathematics helps us enjoy the Christmas season.
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So I think I'll start with a perfectly reasonable question.
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It's a slightly mathematical question, and that's this Why is Christmas on the 25th of December?
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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?
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Well, one thing we certainly know is that it wasn't.
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This sort of happens on a certain day three three six A.D. was when Emperor Constantine set the date.
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So Christmas on the 25th of December started in three hundred and thirty six A.D.
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And the easily accepted theory is that the Christmas is in some way linked with
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the winter solstice and the Roman Midway to Winter Festival of Saturnalia.
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But I've never been completely satisfied with that because that's on the 21st of December.
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And, you know, even with rounding error, that's still quite a big difference.
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OK. Here's another one. And again, slightly mathematical.
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There's a Christmas Christian tradition that Mary was told that she was expecting her baby on the 25th of March.
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And then if you add nine to that, you get to the 25th of December.
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So that's a possible reason.
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And there is another Roman festival, the Natalia's Solis Evictee, The Birth, The Unconquered Sun, which was held on December the 25th.
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So these are all perfectly acceptable reasons why Christmas is on the 25th, but they all have the same disadvantage.
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And that is, I would say, none of them are mathematical reasons.
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So is there a mathematical thing? Is there a mathematical equation which somehow yields the answer Christmas Day?
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And the answer is, of course, yes. That's why we have one of the reasons I'm here.
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So but before we start with that, I just get this little joke aside.
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I don't know how many of you know the answer. Why can't a mathematician tell the difference between Christmas Day and Halloween?
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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,
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and therefore they are identical to a mathematician.
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Okay, so that's a perfectly good, acceptable reason. Here's another one.
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What's important function? Very important function.
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Since when mathematicians functions as a very famous function called the Raymond Zita function on the
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Raymond Z to function is conjectured to have all its non-trivial zeros on the line with real pass half.
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So the question is what important function has a zero on Christmas Day?
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OK, so there is an important function which has a zero on Christmas Day and are good.
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I would argue that's a very good mathematical reason why we have Christmas Day on Christmas Day.
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Anyone know about function is. Well, here's the answer.
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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.
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And I think that's a very good mathematical reason why we celebrate Christmas on Christmas Day.
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Well, let me tell you about the equation of time. There are two types of day you might have thought.
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There's only one talk to date, but there's actually two types of date.
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There's the day that we generally use as a day, and that's a day which is 24 hours long.
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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.
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So that's my own clock, and that clock goes all the way around when it goes around twice because it's 12 hours,
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but it goes around in 24 hours, and those 24 hours is measured using a crystal inside the clock.
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So that's what we often think of a day, but that actually is what we call a mean day.
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It's an average day, so the average length of the day is 24 hours.
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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?
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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.
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OK, so that is another very good way of defining a day.
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That's how ancient people measured days. That's how most people until quite recently measured days.
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And that is called a solar day. But a solar day isn't always 24 hours.
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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.
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And the combination of tilting on its axis and going rounds in an orbit means
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that the length of the solar day varies from day to day throughout the year,
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and as a result, the length of the solar day varies.
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And here's Greenwich, and the equation of time is the difference between the time at noon at Greenwich,
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which you would typically measure on a sun dial and 12 clock.
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And that time varies throughout the year. And here's a nice sort of graphic, which I stole,
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which shows that the the change in time over here due to the elliptical eccentricity that's the tail to the Earth,
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means that the difference between the time measured on a sundial and time measured on a clock varies throughout the year.
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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.
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This is the Sundial at All Souls College, and it's very lovely Sundial.
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And on the 3rd of November, Sundial's all 16 minutes and 33 seconds fast.
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So that's quite a lot. Actually, you've missed plenty of buses with that.
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OK. On the 12th of February, a sundial is 14 minutes and six seconds slow again.
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Buses would be missed, but here's the great thing.
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It's exactly right on Christmas Day, so if you want to know when Christmas Day is,
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you go along, says All Souls College, and you look at the sundial and you look at a watch.
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And if they're saying the same time, it's Christmas Day.
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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.
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The main problem, of course, is that Sundial tend not to work in Christmas Day because it's raining or something like that.
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But anyway, that is for me, a very good reason why Christmas is on Christmas Day.
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It's the only day in the year apart from those other days when your Sundial actually works.
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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.
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And as I said, it will not have failed to pass your attention that on Thursday, something is happening.
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The Great Christmas vote. Now I'm on the purge of my university has forbidden me to say anything political,
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so I think it's probably best to avoid talking about voting about parties.
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So we will have a vote just to see how we get on.
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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.
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And I'm sure anyone with children will know that Christmas is a great time for arguments and general disagreements and so on.
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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.
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So here you can see one. His his jumper.
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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.
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No, be there we are. And number three Christmas jumper.
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No. See, I'm also wearing this is Christmas jumper No.
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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.
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So there we are. Christmas jumper. No. A Christmas jumper.
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No be Christmas jumper. No. See, I hope you've all absorbed those.
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So in the spirit of democracy and audience participation and all of that.
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And if you're like a can you shout a if you like B, can you shout B?
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And if you like C, can you say this is how I do my lectures, by the way, in case you want to?
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It's modern teaching methods. OK, so after three a b or C one two three.
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I couldn't really hear that one try again. One, two, three. No, it could be any of them.
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So what we're going to have to do at this stage is bring in our expert panel of judges,
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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.
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This is Josh, my third expert panel member.
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Is this one, which is Cleopatra the hamster?
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I like big names for small animals, so these are my three friends.
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And so I ask their opinion on the on the jumpers and the again, this is something which often happens at Christmas.
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You have something you're trying to decide on and disagreements.
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And there we are. Monty said that he preferred A to B to C.
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Josh said he preferred B to C to way, and Cleopatra said that she preferred C to be to eight.
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So here's the question they clearly don't agree. They clearly don't agree.
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And which is the winner? OK, so.
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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.
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Anyway, that's another story. OK. So one way to resolve disagreements is to use the mathematics,
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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.
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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.
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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.
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So the border method works that if you put a first, you give it to be second.
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You give it one c last, you give it zero.
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And if you do that, that's Monti's choices. That's Josh's choices.
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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.
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So those of you who voted for B, you agree with the panel of judges, even though it wasn't clear from that.
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So mathematics can be used in elections.
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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,
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which I claim is even more important than the election, which is the Strictly Come Dancing final on the 14th of December.
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Just a quick sanity check who here is going to watch the Strictly Come Dancing final on the 14th of December.
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Excellent. I noticed more of the younger members of the audience and the older, but I should certainly be watching it.
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I shall be in Holland at the time, but I'll watch anyway. So here we have a panel, Anton and his partner.
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And for those of you who don't know how strictly come dancing works, you have a celebrity guest with a professional partner.
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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.
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And there are four judges, and each judge gives each dance a mark from 10 to one.
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OK, so that's how it works. So just like we voted and gave numbers to the jumpers.
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So the panel judges do so that you then add them up and that gives you a ranking.
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So if each judge gives a mark of 10, which is the top mark you can get, you get 40.
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Having got that, you, you take the judges ranking and the audience emails in or whatever their scores and you get a ranking
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from the audience and you add the judges ranking to the audience ranking to give you the final rank.
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And that's how the Strictly Come Dancing final works.
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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.
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So I thought I might demonstrate that for those of you who are going to watch Strictly Come Dancing on Saturday night.
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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.
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These are the official four judges, and they voted eight, eight, eight and eight for one of the dances.
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And here are two interesting combinations.
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This is the eight eight eight eight, which the judges who just voted up there and four times eight is thirty two.
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So the winning the couple have scored thirty two.
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And another pair gets the votes nine nine nine one, which adds up to twenty eight.
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And this is kind of interesting from a mathematical point of view, and this could easily happen on the floor.
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By the way, when I watch Strictly Come Dancing. I don't watch the dances, only watch the voting.
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It's much more fun.
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I apply exactly the same principle to the Eurovision Song Contest, OK, which is actually slightly fair in the way it does its voting.
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So but do watch ignore the dances. The voting is much the best bet.
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I'm not sure if you noticed what's going on here. This is the winner. This is the winner.
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This one. Three out of the four judges prefer.
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That candidate, not that pair. And yet, it's the second one, that's one.
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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.
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And so the second pair has won. Despite I put this in because it's a sort of vague Oxford connexion.
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The fight aside being preferred by the three judges are not the first judge.
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The one that gave here the one has actually dictated the whole result.
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And that is called being a dictator. So it's a kind of a fair system.
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And if you think that's bad, it gets worse when you bring in the audience. So this is how the audience effect is.
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So here's a case where you have how many seven dancers and the judges have ranked the dancers in this order,
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giving seven to the top couple and one to the final.
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The worst couple, and they've given seven to couple D and a couple of D has got the top rank for the judges.
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The audience have rung in and they've given sevens a couple of C, so that's the top ranking for the audience.
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But if you add the ranking four, the judges, I'm ranking from the audience.
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This is the couple that wins Capital F with 10 votes.
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And it's gloriously won despite the fact that no one likes them.
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And if you've been watching Strictly Come Dancing Again, even better than the voting is,
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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.
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So do watch on Saturday and see if we hope this doesn't happen, of course.
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So, so that's the next event in the Christmas season.
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So we'll move on now to kind of the core of the lecture, which brings us to Rudolph, which is Christmas Night.
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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,
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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.
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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.
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Well.
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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?
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Now we know various things about the delivery algorithm announces that Santa will only
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or will I think it's both necessary and sufficient deliver presents to good children.
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Okay, so you've got to be a good child. I'm never quite sure how good and child defined in in my house.
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My daughter, who's twenty seven, absolutely expects a stocking. Even the dog gets a stocking.
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OK, so there we go. So here's the thing.
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The world's population at last count was nearly eight billion, and I would say as a conservative estimate, it's difficult to tell.
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But there's about a billion how homes that have good children in them.
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I'm sure in this audience we have far more good children than that will be, but just just a rough, very rough estimate.
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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.
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And to do that, we need to know what the average distance of part of the houses are with good children in them.
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So to do this will assume that the houses are fairly regularly arranged and each is an average distance away from its nearest neighbour.
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This is mathematical modelling. We're making approximations, but you know it's Christmas.
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OK, so we're going to assume that the houses are a distance apart.
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OK, so the area occupied by a house before you reach another house is h times h or h squared.
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So the total area occupied by houses is a is and times h squared.
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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.
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That's why Santa lives, so we only need to consider the continents.
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I don't believe children live in Antarctica, either, so the surface area of the main continents is this.
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And that's what it is. According to Wikipedia, OK.
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And we know that this is equal to end times eight squared.
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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.
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OK, so this is a modelling thing even.
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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.
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So there we are. That's the average distance between homes with good children, and Santa has to visit every one.
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So the distance he has to go is n times that.
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So he has to travel in 12 hours, at least because there may be some.
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If the houses are less regular, you actually get a long distance,
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but the distance he has to travel is at least four hundred and seventy five billion metres.
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OK, so that's how far he has to go in 24 hours.
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OK, so from that, we can get an estimate of his speed. So here he is, travelling around the houses.
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There we go. I'm indebted to my son for the graphics here.
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Uh, I'm afraid to say my son studied at Cambridge, but he did do mathematics, though.
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And if we take that, the speed is one centimetre divided by 2014,
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three is 5.5 million metres per second, or if you prefer, that's about 10 million miles an hour.
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OK, now let's have a look at some comparisons.
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The speed of sound is three hundred and seventy five metres per second, so he's going comfortably faster than the speed of sound.
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We get onto that in the next slide. Light speed is 300 million metres per second.
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Now, this is very reassuring because we can do it.
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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.
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And of course, if you have a lot more good children, but you'd have to have 60 times as many.
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To prevent injury so he can actually do it, which is very, very reassuring.
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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?
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Well, we've built up the basic mathematical model we can now predict from that.
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So Santa is going significantly faster than the speed of sound he's going at what we call hypersonic speeds.
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So there's a bullet bullet has a hyperbolic shockwave, so that's bullet going.
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Hypersonic fast has to be the leading reindeer in the sleigh.
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It's pulling from flight, which is, of course, Rudolph the bit of Rudolph, which is at the front,
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which is causing the main kind of entry into the air is, of course, the nose.
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As you go through the air, there are greater experts than me in the audience about that.
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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.
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And so the speed heats up the nose till it close.
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And there we are. So. And fluid mechanics shows us why Rudolph has a shiny nose and only he has the shiny nose because
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he's at the front and therefore the shockwave goes round him and all the other reindeer are safe.
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But we are faced with another question to do with the good children.
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We know that Santa is a gentleman of large proportions and the good children live in houses with chimneys.
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So the question is how does he get down the chimney being large or small?
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So the first possible theories about this?
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And one theory which is quite popular, is Einstein's theory of relativity applied to Santa.
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What's that got to do with this problem of how we get down the chimney?
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It's a thing called the Lorenz contraction, which basically means,
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according to Einstein special theory relativity, the faster you go, the smaller you get.
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So we're now actually have our first one. We've had a few mathematical formula, but here's a decent one.
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This is what's called the Lorenz contraction, which down, he says.
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Oh, sorry, if you got to speed V and C is the speed of light, which is three times twenty eight metres per second.
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Then your length going in that speed is this fact.
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The thing here, which is smaller than one and gets closer to zero as V gets closer to C times the original.
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So this contract you so also I say so in lay terms.
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The faster you go, the smaller you get. OK, so that's that's the Lorenz contraction.
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So it's perfectly reasonable model. But as I always tell my students, I don't care how good your formulae are.
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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.
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And he came up with the formula and I said, Have you put your numbers in there?
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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.
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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.
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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.
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It's the contraction. So he's smaller, but not a lot smaller.
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OK. So it's not enough.
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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.
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And even Santa probably can't go that speed. If you did that, the chimney would explode and stuff like that.
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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.
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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.
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Does anyone think I can get him through a piece of paper? Well, we have some hands going up.
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I may need you one of you to volunteer to be Santa. Would anyone like to volunteer to be Santa?
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See, people are hiding. Well, don't worry, you don't have to.
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I might volunteer someone in due course. The way we can get Santa through a piece of paper is to use.
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A fracture. So this is quite a nice demonstration again.
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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,
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which is large enough for everyone to fit through what you two fit through?
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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.
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That sort of set of lines on this piece of paper.
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OK. I hope this goes right. So we start by cutting sort of a hole like this through in the middle.
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By the way, she's perfectly normal paper. OK.
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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
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and we cut one cut like this at end and another cut like this at the other end.
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OK. You can see that. Can you see that? And you do the same cuts regularly along on.
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To. I guess I hope this doesn't all go horribly wrong.
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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.
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And oh, it's all kind of coming apart to another one like this.
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And another one like this. And another one like this.
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And another one like this. There's another one like this.
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And providing I haven't completely marked this up if we open this up.
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We will get some like this with a hole in it.
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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?
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Let's see if we can get you through this whole show. So you look very centrist to me.
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That's very nice. Thank you. What's your name? Georgia Here we are at.
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See if we can fit George with you to come on the stage. We'll see you better than you from Oxford, Georgia.
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Have you got it so we can fit you through, shall we? And we go. And there we go.
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Well done. Well, thank calculations. So.
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And you can fit any size through. You just have to do a few more cuts.
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And then we go. So that's how you fit Santa through a piece of paper a.k.a chimney on Christmas night.
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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.
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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?
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So we've now got two Christmas Day. We realised that there aren't enough decorations on the tree.
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You are required to produce a Christmas decoration. The question is, how can you do it?
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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.
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This is guy called Eric Domain.
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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.
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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.
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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.
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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.
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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.
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I couldn't find an angel, so an angel fish is close enough for today.
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So when emailing me, so I put that on. Now I need it natural.
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So this is the angel fish. It's not particularly Christmassy.
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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.
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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.
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So that's now folded into four. You unfold and you have a sort of crease there.
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You then fold it up like this. So that piece of paper is sort of up against that fold.
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I hope you can see that you then fold this piece in.
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I like that, so I've just folded that piece in.
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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.
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So all I've done is folded that up.
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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.
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So there's a single cut, just one cup with a pair of scissors.
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And I get the shape, and hopefully if I am folded shape.
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Let's see what we get. We get. Let's start.
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You guys of so nice party trick at Christmas.
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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.
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If you're a mathematician, the story goes that this was how the American flag was done as well, right?
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Well, let's move on now.
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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.
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But I wonder if any of you have ever paused to consider how many presents to my true love sent on Christmas Day?
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Many of you thought about this very important question. So here's the 12 days of Christmas.
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So let me tell you a bit about this. We have Bing Crosby to help us here.
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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.
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My true mortgage raising turn dogs and a partridge in a pear tree on the third day of Christmas.
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My true love get to be three French children.
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The party, which is apparently on the fourth day of Christmas special, looking to me for comment.
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I think I'll spare you the rest. You can sing it ahead. You want five?
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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?
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For me? Oh. Seen.
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That's right, I'll stop there.
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Just out of interest, this song was originally written for children as a fourth hit song,
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which they had to sing along, and if one of them made a mistake, the other children had to tickle them.
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So I'll spare you that. OK, so let's have a think how this works.
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So the 12 days of Christmas on day one, we got a partridge in a pear tree on day two.
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We got to talk to doves and a partridge in a pear tree.
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So there's one present on the first day. How many on the second day?
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OK, on day three. Three French hens, two turtle doves and a partridge in a pear tree.
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Day four day five day six day seven eight nine day 10 to 11 day 12.
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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.
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These are called triangle numbers, so one is one one plus two is three one plus two plus three six.
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And if you add up any of them, then you get the number ntx plus one over two.
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So the professional mathematicians will know this, but these are called triangle numbers.
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And here they are, appropriately decorated with the animals in question.
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So our job is to work out the 12 days of Christmas.
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Some is to find a way of adding up the triangle numbers.
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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.
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OK, so here's another Christmassy way on Christmas Eve.
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So then on Christmas, your way is the total number of presents.
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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.
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When you add up the sum the telescope you find that you get the total number of presents is the final product,
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which is 12 times 13 times 14, divided by six, which is three hundred sixty four.
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And so just to emphasise that three hundred sixty four is the total number of presents given.
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When you see this, there's always a question everybody asks, what's the question?
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Everyone asks? The question is what happened to the last present?
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OK, what happened to the last president? Because the three hundred and sixty five days in the airport for next year, which is Libya,
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what happened to the last presence and well, basically we're all allowed to make a mistake.
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OK, so there's my true love. This is Benji the dog. So that's a non Christmassy way of doing the song.
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But I thought I'd like to tell you about the Christmas stocking theorem, which is one of my favourite results,
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which is a Christmas way of adding up the number of presents in the twelve days of Christmas.
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And the inspiration of this comes from my chairman, Professor Gabriela.
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So some of you may not know, but Professor Gary Aly is in league with darkness.
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In this, he is the chief advisor to Moriarty, which is the arch villain in Sherlock Holmes and in particular produced memorialises Blackboard.
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So if ever you watch Sherlock Holmes the movie, you will see this blackboard, which I believe Professor Gore really designed.
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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.
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Student appreciation is quite a lot of detail. Here's how to see what's going on.
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So I put a ring around the important thing.
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So this is the clue to the twelve days of Christmas is this bit of the blackboard here, which is called Pascal's Triangle.
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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.