 Hello, EMF. Hello. Yes, so I'm going to give you four very short talks, which I hope you will collectively form some useful maths. So yes, my first thing, I'm going to talk about winning a game that I made up to write in an article in a magazine called Chalk Dust, which is for people who love maths. And one game which you're probably familiar with is called Go, and when you play Go, you need to capture parts of the board and capture your opponent's pieces. And back in 2016, Google's AlphaGo used state-of-the-art machine learning and AI to beat Lee C. Dole, who was then the World Go champion, which was a huge accomplishment. But one feature about it was that AlphaGo played to maximise the probability that it would win the game, not to try and maximise its expected score. So yes, and in general, like if you're playing a game and the result is just a binary win or lose, then the margin of winning or losing is not important, right? So a common mistake that people play is that they really fear having a terrible score and so just play to maximise, you know, play pretty steadily the whole time through. But actually if you're in the lead in a game like this, then it's okay to kind of avoid risk. But if you're losing, that's the time in life and in a lot of games where you need to start taking big risks, even if the expected value of taking those risks isn't that high. So here's a game I made up. So it's played over 10 turns. Each turn, you choose a number of fair coins to flip. So you can choose to flip non or up to 10. And then once you've chosen that number, you flip them all. For every head, you gain a point. For every tails, you lose a point. And your opponent is just given half a point each turn. So at the end of 10 turns, your opponent will have five points. Your expected score is zero because every time you toss a coin, you're 50-50 to gain one or lose one. But a recursive formula where you start on, what would I do on turn 10? And given those actions, what would I do on turn nine? And given those, what would I do on turn eight? It actually gives the optimal strategy. So here's a little diagram that shows that. So on your first go, you should toss all 10. If the white line represents your opponent who's just being given half a point each turn. Once you start beating them, you can then play it safe. So as soon as you reach six, you can just stop playing because your opponent's going to finish on five. But apart from that, once you start, especially if you're going to negative scores, you should start gambling as heavily as possible. So your expected score has to be zero because you're just tossing fair coins. But ideally what you want is just to... If you win, you want to just win with six points. And if you lose, you want to lose as badly as possible in order to maximize how high there's probability of winning is. So actually, even though your expected score is zero, the game is better than fair. So you've got a probability of 0.51 of winning if you play optimally. So yes, so in summary, if you're behind gamble and it has applications in real life, so let's say that you're on a date and it's not going particularly well, but the binary thing is just whether you get to the second date or not. Now is the time to talk about your niche, crazy interests because there's a time... Whether you just fail to get the second date or whether you completely bomb, doesn't make a difference, right? There's no second date. If you do well, then you're through. So my second talk is about factor graphs and unsupervised joke generation. So these factor graphs can represent the factorization of probabilities. So let's think about the joint probability of rain on Monday and on Tuesday. So it's probability of rain on Monday and then rain on Tuesday given that there was rain on Monday. So we can factorize that. So the probability of rain on Monday only features Monday and the second one features Monday and Tuesday. So this factor graph represents that. And so unsupervised joke generation from Big Data was a paper written by two maths and computer science professors, Petrovich and Matthews, and he uses these factor graphs to generate jokes of the form, X, like I like my Y, Z. So the stereotype for this kind of joke is like I like my women, like I like my tea hot. And my wife, who was a teacher when I wrote this, came up with this one. I like my man, like I like my homework marked. So yes, so the probability that a joke of the form I like my X, like I like my Y, Z is funny. So we're trying to maximize this probability that one of them is funny, and it can be represented in this factor graph form. So let me go through what those different factors are. So the first factoring codes that dissimilar nouns lead to funnier jokes. So if you didn't assume this, then you would get an anti-joke. So for example, I like my tea, like I like my coffee hot. Doesn't make sense because tea and coffee, everything that describes tea can also describe coffee, right? The second factoring codes that the joke is funny if Z is often used to describe X. So if you have a joke like I like my tea pot, like I like my fireplace, crazy. You know, it doesn't make sense as well because crazy is never used to describe either of those. The third factoring codes that the joke is funny if Z, the adjective is not a particularly common one, otherwise you'd get a joke like I like my dogs, like I like my quality of life. Good, because it can just describe everything. And the final factoring codes that the joke is funny if Z has many different senses because the humor often comes from Z being used in one sense for X and another for Y. So without that you'd get a joke like I like my horror films like I like my murder, gruesome. There's only one meaning of gruesome, so it doesn't work. So yeah, so using Google Ngram data, which is just basically a large corpus of written text, they use Monte Carlo, so like random simulation to generate potential jokes that they reasoned had high probability of being found funny. And 16% of the jokes were found funny. So here are two of the jokes that it generated. I like my coffee like I like my war, cold. And I like my relationships like I like my source, open. All right, so yes, in conclusion, factor graphs can represent probabilities and they can be used to help generate jokes. So let me present, tell you about the longest game of chess. So, and so Sig Bovic is a paper, a journal that some, for recreational computer science, some of you may be familiar with. And in his 2020 paper, Tom Murphy asked, what is the longest possible game of chess? You may be thinking, can't a game just last forever with people moving the nights back and forth infinitely? But actually chess has two well-known rules in order to avoid infinite games. So in 1561, the 50 move rule was introduced, which said, if you go 50 moves with no pawns advancing and no pieces being taken off, then you can claim a draw. Technically, that only applies to games started after its introduction. However, any games that began before 1561, they could continue to be played. And also if the same position is repeated three times, then either player can claim the draw. So for example, if the 50 move rules, I'll move 63 of a game between Karpov and Kasparov in 1991. Black had two pieces, white had four pieces. So that move 63 were in this position, move 114. Again, the pieces have moved around, but nothing's been taken off, nothing's happening, and at that point Black claimed a draw. But this actually only gives you the option, the 50 move rule actually only gives you the option of having a draw. Both players can decide to carry on playing if they want. But in the current rules, so rule 9.6, if you want to check it out in the FIDE official rules, if you go 75 moves with no pieces being captured and no pawns advancing, then the game is over. You have to declare a draw. And if the same position is repeated five times, you have to declare a draw at that point. So I'm going to call a critical move, one that is either a pawn advance or a piece capture. So these are the moves that prevent us from hitting the 75 move rule. So there are 16 pawns. Each of them, assuming we start by moving them one square, they need six moves to get promoted. And apart from the pawns, there are 14 capturable pieces. So yeah, so for each of the critical moves, we've got 75 joke moves for one player, 74 for the other, and then the critical move. So for each critical move, we can have a maximum of 150 moves. And then the possible number of critical moves is 16 times 6 pawn advances, 14 non-pawn captures and 16 pawn captures. And in total that gives you an upper bound of 18,900 moves for a chess game. And so yeah, so 76% of these critical moves are pawn advances. And so we really don't want the pawns to block each other. So we want something like this where the pawns can just advance past each other. But in order to do that, we have to lose a few critical moves because some of the pawn advances coincide with taking pieces off. We have to do at least eight of these captures. So the upper bound is lowered by 1,200. So it's now 17,700. So yeah, so if we keep moving the same color, using the same color for critical moves, so let's say black is the first piece that moves a pawn or takes a piece off. If we keep using the same color each time, then we're maximally inefficient, which is good. So every time we change the color, we're guaranteed to lose a half move. So black can't do everything, so we have to change color at least once. Reducing the upper bound by 1. And Tom Murphy's solution, so we start by moving a couple of black pawns down, take a couple of pieces. Then the white pawns move around the black pawns like this. They get promoted, take off all the black pieces apart from the pawns. Then the black pawns advance, get promoted, take off all of the white pieces apart from a queen. Then the white queen takes off all of the black pieces and it finishes in checkmate, actually, for white. So yes, the upper bound was 17,699 half moves. Tom used three color changes, so the solution was 17,697. Can it be done with fewer parity changes? I don't think so. I think I've discussed this with some people before and we're pretty convinced that three is the minimum, but we'll see if you can do better. So yes, and I made a Twitter bot, chess bot one, which plays a move from this game every four hours. So if you enjoy watching pieces move aimlessly around the board, you'll really enjoy it. It's been running for a year, so it only has seven years left to run. So yeah, longest chess game bot. And yes, and now a tiny bit of number theory. So everyone loves prime numbers. I think we can agree that some of us prefer the exact opposite. Numbers with loads of factors. I'm talking about satisfying numbers, like 24 and 360. Numbers which are used when you might need to divide something, right? Ten is great, but you can't divide by three. So I'm just throwing that out there. So let S of n be the sum of the proper factors of n. So for example, S of 10. So the factors of 10 are one, two and five. So the sum of the factors is eight. S of 11 is just one. And the factors of 12 are one, two, three, four, and six, giving a total of 16. Yep, so as you can see for any prime, we have this terrible, the thing that I don't really like when the sum of the factors is just one. But an abundant number is one of the best ones where the sum of the factors is actually bigger than the number itself. So 12 is the first abundant number because the sum of its factors is 16 and that's bigger than 12, and that's the first number that happens for. So 12, 18, 20 and 24 are your first abundant numbers. Yep, and if a is abundant, then any multiple of a is also abundant. So 24 is abundant, but we could have shown that by knowing that 12 is abundant and 12 times two is 24. So basically, here's a formal proof if you want to have a look, but yeah, basically if, let's think about 12, each of its factors, some of its factors is bigger than 12, but for 24, if you think about the factors of 12 and just double them, all of those are factors of 24, and so the sum of the factors of 24 must also be bigger than 24. So yeah, so because we've got at least one abundant number and any multiple of an abundant number is also abundant, there are therefore infinitely many abundant numbers, but you may be thinking the first four that you've shown me are all even abundant numbers, are there any odd ones? And the answer is yes, and the first one is 945, and so any odd times an odd is odd, so 945, 945 times three, 945 times five, and so on are all odd abundant numbers, and so there's infinitely many of them. So yeah, so A's abundant means that N times A is also abundant, so this gives us this definition of primitive abundant numbers, so a primitive abundant number is one that's not a multiple of a smaller one. So yeah, so the first primitive ones are 12, 18, 20, not 24, and then 30, and Erdos in 1934 proved that there are infinitely many primitive abundant numbers. So the question then is, if we look at bigger numbers, do abundant numbers become more common? So let's let A of N be the number of abundant numbers less than or equal to N. What happens to the proportion of, so A of N over N is what proportion of numbers less than or equal to N are abundant? What happens as we look at that proportion with huge numbers? So let's let capital A be this limit as we get to huge N, so I'm going to let you vote on this. Which of these three statements is true? So is A less than or equal to a half? Is it less than one but bigger than a half? Or the limit does not exist? So you know that there are infinitely many abundant numbers, you know that there are infinitely many primitive abundant numbers, you know there's infinitely many odd abundant numbers. Okay, votes for number one. One person. Votes for number two. Okay, quite a few, votes for number three. Oh, the majority. Okay, well, I have produced my first ever image that I contributed to Wikipedia. And I will show you it in GIF form. So here is this proportion over time. So you can see that it starts to race up. But is it going to clear the half mark, as a lot of you thought? Unfortunately for those who thought it was bigger than a half, it just flat lines. So it's actually flat lines at a quarter. Yeah, it's just under a quarter. It was shown to exist in 1933 and it took 65 years to prove that it was less than a quarter. But we still don't know it exactly, but it's within that range, 0.2474 to 0.2480. Yeah, so here are the main results. Any multiple of an abundant number is itself abundant. There are infinitely many odd and even abundant numbers. And abundant numbers make up about a quarter of the natural numbers, the positive whole numbers. Yeah, I hope you'll agree that is very useful. Thank you very much. Thank you very much.