 dros ymlaennau, ond yr oedd yn gallu'r sloff mwy arna, ond rhai wobon wnaeth cyd-deithas y ddechrau yn cyd-doedd. Ar fy ydych chi'n gweld ymlaen, ond ddaw yn rhaid i'di'n ddŷch. Rhywbeth ymlaen i'w rhai ddau yn gallu stryd oedd. Mae'n ddod i'n ddod i'u cyfwyng y stryd ymlaen, rwy'n rhaid i'w ddod i'n gallu ddau. Rhaid i chi yw bod yn gweld i chi wedi wir i'ch canfwynt o'i ddweithbyddu yn mynd i cydweithio yn gynllunio i ddweud. Felly y peth yn ymmwneud o fylam a gyntafoligwyr. Felly ym mwyaf o ddweud y AME, a bod ydych yn dweud y sydd ei bod yn ymwneud eich cydweithio neu'r anestr? Mae'r anestr wedi rhoi'r hyn yn ymddangos. Mae efallaistaeth yMesafoligwyr pedigangol. Mae hala ffordd bob amlaen nhw yw yn amlir. Mae ffordd hynny i ddweud ymwneud ymwneud ym mwyaf ymwneud ymwneud. Mae brackets yn ymwyaf ymer. Mae'n meddwl gweld yn y gallu ches ar ychydig i'w gŷnol yng Nghymru. Mae'r bobl yn y gêmau no chenffol. Mae'r cyffredin iawn y gêm yn oed, mae'r gêm yn gwneud, mae'n gweithio'n gyda'r gêm. Mae'r gêm yn gweithio'n gyda'r gêm, mae'n gweithio'n gweithio, mae'n gweithio'n gweithio'n gweithio'n gweithio. Mae'r gêmau maen nhw i'r gêm. Mae'n gweithio'n gweithio'n gweithio'n gweithio'n gweithio. ond yn fwy a i datblygu'r lleidio llyfrigeidau o ran oedol iaidio—cant y gallu cyfroedd yn y cywir. Mae'r lleidio bwysig, mae'r lleidio bwysig, a'r lleidio bwysig, a'r lleidio bwysig yw gynhyrch chi'n gen reply. Llyrdoedd y lleidio lŵr o'r lleidio digad arerythio gyd. Mae'r lleidio bwysig o erioed, o'r lleidio'r lleidio arerythio, os y ddod y gallwch yn包 arfer. Mae'r lleidio arerythio arerythio, os y ddod y gallwch yn lleidio gael, os y ddod y gwaith. Dwi'n cael ei wneud pan i'w gweithio'r pethau ynglynig? Ieithu'r pethau i'r pethau i mi ac i'n cael eu pethau i mi yn gweithio ddydd. Felly hwnna'n ddysgraff y ddyddwch i'n meddwl. Rwy'n cael ei wneud i'r ddymnu ac hynny corei ddymnu. Felly mae gweithio'r ddymnu arall. Rwy'n cael ei wneud hi i hynny? Rwy'n cael ei ddymnu arall a ddymnu arall, ac rwy'n cael eu ddymnu. Mae y rhai amryl gwaith yn ychwyllfa. Chump is a game played on a grid. The red square is poisoned. You choose a square as a move, so it's perfect information obviously. It's an impartial game because obviously there's no player one moves or player two moves. The first player is usually called left. The second player is usually called right. By convention, I won't give the end away. I choose any square. The first player chooses that square. The idea is all of the things to the right and up of the square that you choose disappear. First person is moved. Second person moves. First person moves again. Second person moves. Now we're running out of moves. Remember the person who can't move loses. I've forgotten the parity. First person moves. Second person moves. First person moves. Second person moves. First person moves. Second person moves. Now the first person is lost, right? He or she cannot move, so let's get this right. First person moves. Second, first, second, first, second. Now it's the first person's move but there's no legal moves. You can't eat the poisoned one so the first person loses. Is that clear? That's how we play these games. There's a proof and this is really interesting. It's not a constructive proof but it's a nice mathematical proof of the fact that the first player should always win. My play then was suboptimal. There is a winning strategy for the first player. The idea is if there was, let's assume for contradiction, that there is a winning strategy for the second player, the first player makes this move in the top right corner, then whatever move the second player takes in their winning strategy, the first player could have taken it, couldn't it, because it's still a legal move for him. It's a bit subtle. It's not a constructive proof of a winning strategy for the first player but it's called a strategy stealing argument. If there was a strategy for the second player, the first player could steal it by making this move and then if there's a response that player two can give, he could have done that in the first place and he can therefore steal the strategy. That's one neat way of using a logical argument to show that it should be a win for the first player even if you can't quite construct it. One interesting thing for you to think about after the talk is what about the game chumpfinity, where it's on an infinite board so you can choose any n and any m as your first block and then you obviously disappear all greater than that. Does that change the nature of the game or does it or does it not? So chumpfinity an exercise for the interest of the munger. The first game I want to talk about, this is a game that was first solved I think in the Victorian times and the game is called NIM, the rules are as follows, you have piles of coins and on any particular pile you can remove any number of them and you can take up to all of them if you want so for instance I can take three on the left hand side row or I can take just one off the most right hand side and the idea is again we're in normal form, the player that can't move loses. So a NIM game with one pile is really boring because what is the winning strategy for the NIM with one pile? Take them all, the objective is to stop the other person being able to move, remember we're in normal form so yeah you're right, take them all, a little bit more subtle is the winning NIM strategy for two piles I'll let you think for a couple of seconds, go on, take all but one off one is not the correct answer but it's a good try and thanks for volunteering so quick, it's a non-trivial problem but I know if possible you have a good instinct maybe, is that a second guess? Again, equalise them, did I imagine that or did I hear it? Yes okay awesome is the right answer, I was going to say if I just hallucinate in the right answers then this is going to be really good only it might not actually be true so you all understand great yeah so yeah you equalise them and then whatever they do you can do on the other one so if she takes one or she takes, you take two off the right so she's left with three and three she takes two off the left one, you take two off the right one and then they're always balanced then and if they're always balanced at some point that goes to zero, you take the other one to zero, they've lost so we can solve NIM for two games now, two piles now excuse me so now a more complicated NIM position have these come in the wrong order? Yes, a more complicated NIM position so there's a little bit of subtlety here NIM can be considered as the sum of several different small games if you treat the sum of two games to be not like chess boxing where you have to do chess and then boxing or boxing then chess or whatever is the most difficult way round it's where at any point in the game there's two games so there's draft here and there's chess there and you choose for this turn which game am I going to play in I'll play in drafts and then I make my move and then you make your move and you get the choice which game do I play in NIM is somehow, it's four different games of NIM potentially you can choose am I playing in this one or am I playing in this one and I'll go into this in a bit more detail in a minute but the idea is if I was playing in the game with just these two I've got a strategy haven't I and if I was playing in the game with just these two if I was one I'd have a strategy so a good idea might be well I'll keep those two and I'll equalize these two and then presumably if my opponent makes a move in these two I'll do the winning game if my opponent makes a move in these two without this as the single ones I'll make a move in that winning game so somehow I've decomposed this four pile into two games that I know I can win and if my opponent plays in one of them I give the winning response in it does that make sense brilliant ok so much harder now whenever you read about this they always talk about binary arithmetic but I find it easier to actually think about it in terms of powers of two so the idea is you want the sum the binary sum without carrying to equal zero so it's binary addition but without it's modulo two it's you don't you don't carry so there's a one here there's a two here there's a four here and there's a two here and a one so the idea is the one and the one cancels the two and the two cancels and the best move for me is in fact to remove the four so I do a worked example in a minute but does it sound plausible that removing this four is a winning move yeah it does because if you think about what's left if I remove these there's three, two, one think about what the opportunities are for the opponent if the opponent takes this one you equalize those two if the opponent takes this one those two are equal and you can remove all of them if the opponent takes both of these you're in this one and this one and you can equalize if the opponent takes one of these these are equal and you can remove this one if the opponent takes two of these these are equal and you can remove that one if the opponent takes three of these these are not equal and you can equalize and then win so that does sound plausible that removing the four there is a win so yeah I meant to do that without you know what I mean, didn't you? You imagined that the fourth was gone then when I was speaking didn't you? Even though I was speaking fast. Ffadda mi? I'm a little deaf, I was unwell on Friday whenever I've got a cold, this ear completely closes so it's going to be the worst bit of audience participation ever. So let's say you had 17, 25 and 13 in a pile and they usually do it with binary 0 and 1 notation but I think it's much clearer. Yn what we're looking for, we're going to say the 16s cancel. We're going to say the ones there cancel. They have to cancel with a different row. So the 16s cancel, the 8s cancel, the ones cancel. It'd be really good if the four plus one wasn't there. And that's exactly what the winning strategy is. So all of those things cancel. It'd be really good if the four and the one weren't there because then everything would cancel. And the winning strategy at that point is to take the 13 down to an eight. So if you keep that, if your strategy is always make the But that makes you to make the sum, the special, so-called nim sum, which is binary addition without carrying, make that zero all the time. If it is already zero, you are screwed if the other person knows how to play. But if they are not, you make it zero. It is this argument of, I don't need to get the best move. I just need to make sure I've always got to move.的是 You, so if I know that I just turn it to zero again and that's a lose and then you, if what you do, if I can turn it to zero and then we just go down and because we're always decreasing the piles, at some point we're going to get actually to there be nothing there and hopefully that's when you've got it and there's a kind of parity type argument. keeping it, keeping this nim sum zero is kind of what is preserved ac mae'r rywbeth yn gwneud. Felly mae hyn yn gweithio bod yn dod o gweithio. Efallai 60 oes yn dweud, a 45 o gweithio. Rwy'n gofio i'n meddwl. Yr un gweithio ar hyn o'r ffordd, ychydig i ni, fel y dyma gynnogau rhai, yw'r ffordd o'r ffordd gyflym yn ei gweithio, ond rydyn ni'n ffordd o ddiweddu o'r teulu. Gweithio'r ffordd, rydyn ni'n bywch chi'n gweithio. Haid yna 100 o gweithio ar y rhawn. Ac efallai i'w gyd yn dod yn ddweud am arwain i'r ddweud i'r ysgol? Oherwydd? Dyma, ble yna'n fwy iddo i'r ddweud i fewn fod yn ddweud, mae ti'n gyfer 50 coen o model ac mae'n sefydl o'r rhiwp bob pryd yn ofen, mae'r rhiwp bod roedd yn cyffredig 50 coen o'r rheid i ddweud, oedd yr yn y ddweud gan y pwysgol. Mae'n meddwl i'n holl arwain i'r ddweud, ac mae'n z sixteenol yn ddweud i gryfbeth ar hyn o'r cyflwyn i'n gwneud. нихum. I add an extra rule you just waste in time this concept of reviewing будd some move some move actually are an option which just either take you back to where you were or actually give you something were so there is another game called kells so the game here is you have got a bunch of piles again similar to Nim. But this time you can remove any one or two adjacent bowling pins so I can take these two and leave one I can take these two and leave two piles, each of one, etc. So you can actually take any two adjacent and you're allowed to split the piles in this game. So a very similar game and the theory says, and this has been known for a while, maybe I wasn't so explicit, but these games are all impartial, right? There's no such thing as moves that only the left player can make. So in these impartial games, there's a theory called the Spray Grundy Theorem that says they're all equivalent to Nim somehow. So there's a really nice, really nice theorem that comes later. And some things that aren't impartial, there might be a sneaky way of making it impartial. So North Cuts game is played with on a checker's board and you can move your pieces up and down arbitrarily. And if the other, except you can't jump over the other people, you can move up and down in your row arbitrarily. You've got two of them here, say. You can move up and down arbitrarily. I keep saying that as if you haven't heard me the first four times. And then so the idea is sometimes you retreat, sometimes you come closer. But if he comes closer, this gap goes down, doesn't it? If this person comes up, the gap goes down. If this goes backwards, nothing really happens. It's these gaps are basically your Nim, aren't they? And going backwards is just like adding. It's like that silly, silly extra rule of Nim that does nothing. So this is a partisan game. I can only move my pieces and she can only move her pieces. But actually it's just Nim in disguise. So Nim in disguise is going to be the big idea for when we get to play the childhood game of Dots and Boxes. Is everybody... How do you win? How do you win again when the other person can't move? So if I keep moving up here, eventually white can't move because they can't jump past me. If I keep moving up here, black can't go this way. So I'm just closing the gap on black. And if they retreat, I can follow them, right? But they can't retreat forever. Yeah? Can you decide? No, sorry if I wasn't clear. It's up and down is the legal moves in this game. Sorry, I almost certainly wasn't clear if everyone's confused. I didn't actually look in detail at this put, to be honest. I just googled for one. All the blacks are on one side and all the whites are on the other side. This is a legit game. I'm not sure it is. Which one's bothering you? The point is, though, that actually it's the gap between the pieces. It doesn't matter whether black's on top or bottom. It's the gap between the pieces is important. And both players have got a choice, basically. Am I closing the gap? Am I increasing the gap? And someone's going to get trapped at either edge. It isn't the blacks on the bottom and whites on the top. Does that make sense? You can just close the gap. And if I get closer to you, your only move is to escape. And I just follow you up to the edge. Does that make sense? Can we just call this dealt with? I vote for just calling this dealt with. I googled for knock-out game, took an image. Didn't look at the context of the bounding page. Trust me, knock-out game is equivalent to NIM because Spray Grundy Theory says that all impartial two-person games of perfect information are equivalent to NIM. So I'm glad you all understand that. So moving swiftly on. There's a whole class of games called octal games. And they're described by a potentially infinite octal number like this. And the idea is a bit like file permissions in Unix. If the nth position says you can take n things off your pile and if you're allowed to leave zero piles, then you add one to the total. If you're allowed to leave one, then you add two to the total. If you're allowed to split and leave two piles, you add four to the total. So Cale's, if you remember, you're allowed to take one or two pins and you're allowed to leave zero, one or two piles, won't you? You could take the last of a pile, you could leave two piles or you could leave one pile. So Cale's is actually the game 0.77 because I can take one and leave zero, one or two piles or I can take two and leave zero, one and two piles. Audience participation. I want you to shout loud if you know the answer because I can't actually hear. What's the decimal number? What's the octal game number, sorry, for Nim? One, what makes you say one? It's a bloody good guess, but it's not quite true. It's very, very, it's very, very nearly true. So can you leave two piles in Nim? No, we had no splitting, did we? Can you leave zero piles in Nim? I think you can, can't you? You can take some and leave none, yeah? Can you leave one? You can take some and leave some, can't you? So it's definitely three, but can you take, how many pieces can you take? In Cale's, you could only take one or two. In Nim, there's no limit, is there? So Nim is 0.3333 recurring. Yeah, so you can take any number and you can leave either zero or one pile, yeah? You can't split in Nim. And Spray Grundy says every impartial two-person game is equivalent to Nim. And the way you solve them is you find the Nim equivalent. You play in Nim. It's like you're playing two games at once. A move in this game is a move in Nim. You find the winning move in Nim and translate it back into the new game. And it's bloody brilliant. And there's the Nim equivalent values for Cale's. It's an unsolved question just yet. And this is, if anyone is wanting to get involved in some mathematics. After this, this is 72 plus 12. After that, it cycles and all of the octal games of actually anyone that's been explored has actually eventually become periodic in terms of its Nim equivalent numbers. So do you see what I mean? A pile of N Cale's is equivalent to these Nim piles. And you can add them the way I described, which is choose which one you're playing. And then you can take a move in any of the choices. So after 72 plus 12, after 84, I guess, the same cycle repeats the same. And everyone they've tried cycles, but they haven't proved that it always cycles. It's quite interesting. So another game called Cale, so-called because it's cold, which means what I'll explain in a minute. Red and blue are the players. Blue is left, but I might have forgotten that when I made the slides. So a move is to colour in a region and Cale, the rules are, if I colour a region, I cannot colour an adjacent region. So if red plays the outside, then he's kind of reserved this for blue because red can't go there, that makes sense. So when blue then goes, blue's kind of reserved this for herself, but she can't go there. So now this is kind of reserved for red, but what is a good move for red in fact in this case? Take the white one exactly because then blue's got nowhere to go. Oh, no, I'm lying, aren't I? Actually, blue can go in the one that's reserved for blue. Wow. So blue's response is this, takes it. Ah, no, you're right. I forgot to colour in. Yes, no, no, I didn't. I've just colour blind as well. I'm looking at this. So see how it goes gray? Gray is not usable by either. It's a difference between turquoise and gray, excuse me. It's traditional to do cross-hatches of the ones that you can't go on, but I just didn't have the gimp food to do it. So it's confusing me, excuse me. So yeah, it's reserved for blue. Now no one can go in that space and yeah, the man there is exactly right. That's a winning move for red. So you will be able to beat me at all of these games because I'm actually shit. So Snort now is a hot game. Snort is different to Col in that any time you move into a segment, the adjacent segment is reserved for you rather than for the your opponent. So the thing with Snort is you want to play in that game because the more moves you make in that game, the better, whereas in Col, the more moves you make. You're just reserving space for your opponent. So hopefully this is the right way around and I'm not colour blind. So blue now is reserved for himself. Red's actually only got one choice. Now everything goes gray because blue's reserved, a bunch of them. Red's reserved a bunch of them. But now blue can't go, so blue loses. Is that clear? Yes, I forgot. You are paying attention. I forgot there was an outside. I originally wanted to make it the lozenge, but then I couldn't paste it into Google presentations. But you guys are bloody good. It's not the lozenge you're playing and of course the outside, yeah, you passed the test. So quite often with these games, there's like a parody type argument and you would always say, I don't need to find the best move. I just need to ensure I always have a move. Every time you move, if I've got a move, then I win because you run out eventually. So the parody always comes into it and it's the simplest possible game. It's like she loves me, she loves me now, she loves me, she loves me now. You're really asking what's the parody of the petals, aren't you? You're not actually doing anything other than that. So yeah, I just wanted to remind myself about the parody type argument. So now kids game in school, you may well remember dots and boxes. So you've got a grid and you draw lines. Every move someone does a line. Eventually something like this happens. This is what professionals call a blunder because you would never actually let someone complete a square at this stage. So as you all know, eventually without blunder, so level zero play like if you're a kid and you think about it for more than like a minute or so, you realise that you don't want to complete three sides on a square. You want to avoid that as long as possible. And something like this happens to the board. And then unfortunately someone's unlucky enough they have to open a chain, right? So we're calling these chains. Somebody at some point has to open a chain. And from then, either then... Shit! This is... A bit of a... Hello? Okay. Just rewind it. Yeah, yeah. We'll take it out in post-production, it's fine. So... So... Important mathematics. Yeah, you avoid blunders basically and at some point you open up, someone's forced to open up one of the chains. And now level one play, I said level zero play is blunder avoidance. Level one play is oh well, I've got to open one of the chains. I'll open the smallest one first. And then someone takes the chain, they open another chain, and then... So you can't kind of alternate between who takes the chains. But one nice little trick, which as soon as you start doing this to people, they get the trick straight away, is if someone opens that chain, I don't take them all. I take them all except the final two and say, you have those two? And they take those last two and then they're forced to open another chain for me. I gobble up all of that chain and leave them the last two. So that's like level two play kind of thing. So also, there's two kinds of things. There's chains and there's cycles. So in here is actually a cycle, whereas this starts at the end and finishes at the other end. This is a cycle. In order to be sneaky with this one, if someone opens it up, I can't leave two, but I can leave four. So I say, I don't want those four, thanks. And then they go... four, but then they're forced to open this. And this gives me, in fact, one, two, three, four. And in this, I would gobble up five, six, right? If there was other things going on in other parts of the board, I would say one, two, three, four. You have those two and open up somewhere else for me. Does that make sense? So this is called the double cross. So anytime you take two squares simultaneously in dots and boxes, especially if you play it against me, you're losing, okay? So you've been double crossed. So let's make sure I can remember these. If you see any of these, it means you're losing because you've got basically no choice you should take them. In fact, if somebody gives you these, you should take them, it's a no-brainer because you've got no choice, you may as well take them and get the extra squares, but it probably means you're losing. Now, this is a bit of a blunder. If this is part of a game and the other board is not yet finished, if you give me this, I've definitely won. I may not know how, but I've definitely won because if the rest of the game is a lose for the person to move, I go there and make you play in it. Does that make sense? I close this two, leaving you these two squares. You take them, but then you have to play in the rest of the game, which is presumably a loss. If the rest of the game is a win, I just take them and then play in the winning game. Does that make sense? So, if you give me the bottom, then I've got control. I may not know what optimal play is. Which is actually quite an unsatisfying thing. It's knowing that you've won, but just not knowing how. Or knowing that you can win rather, but not knowing how is actually more annoying than losing sometimes. We've spoken about NIM because I said gaining control is important, keeping control is important, but I always give away two at the end of every chain and four at the end of every cycle. So, in the actual game of dots and boxes, it's a partisan game because actually the squares that are mine isn't there and the squares that are yours, and all our moves aren't necessarily the same. Also, I get an extra move whenever I complete a square. For those who haven't played dots and boxes recently, whenever you complete a square, you're forced to take an extra move, so I should have mentioned that probably. But what if it was impartial? If NIM string is impartial, the only difference between the version, if you're trying to play for whoever can't move losers, then do you remember here when I said, if this was the only game, I'd just gobble up all this and get the six points? In the normal form version, in NIM string, I take them all, leave you these four, you take that move, but then you have to go somewhere else and you can't move, so you lose. NIM string, the only difference in play between dots and boxes and NIM string is the final chain that I take, I don't gobble them all up and get the extra two points. I leave that bit for you and make you not have a move at the end. Does that make sense? It's a little bit subtle, so if you ignore the points in dots and boxes and play the equivalent game, I just want to control who is the first person who moves into a chain and then from then I win. You win in the normal game of dots and boxes or the squares, but you win in the impartial version by making them unable to move, which, by Spray Grundy theory, means is equivalent to NIM. So there is, when you're really, really good, now I'm not really, really good, you're not playing dots and boxes, you're playing, because the dots and boxes are kind of like a graph, and you know the graphs have a dual when you swap nodes for edges and edges for nodes, what really good dots and boxes play in it? It's a coins game, which is the dual of the dots and boxes, and they're no longer playing the game strings and coins even. They're playing a game called Dawson's Vines, which is a similar, you know we've played Dawson's Cales before, it's a similar game to that, so that's when you're stratosphericly good at dots and boxes, I'm not. I just like beating kids, and they're... I like winning against children. I'm sorry. I'm overrunning, I've taken too long. If you want to leave at the... Is it already gone? If you want to leave at the 8, was it? I won't be too offended, but I'll try and be quick. So another game is Domineering. It's played with dominos on a chess board, and the idea is one person is playing with dominos on a chess board, and the idea is one person left plays vertically, one person right plays horizontally, first and second player, and eventually you fill the board up, but there's certain strategies where if you look here, you see left has reserved herself this one, right can't move there, can she? So you've got... and also this part of the board is kind of separate to this part of the board, it's like two separate games, and would that be different to saying which of these games do you want to play in? If you play in this one, make a move in it, or you can play in this one, make a move in it. So we've got two disjoint parts of the game, which is kind of what our theory is about. So the next person makes a move who is right, the second player, left the first player, and now right moves, and now left moves and wins, in fact. So though there's two squares left, they're both zero, you can't move in them. So the idea is going to be the zero game where either the person to move loses or there is no moves, because then you've lost. Does that make sense? So these games here have number values. If left is the person who plays vertically, then that's one move for left clearly, isn't it? If that's part of the game anywhere, especially if it's isolated, I mean if it's isolated then that is a segment of the game. That's a move in the bank kind of thing, isn't it? And similarly this is a move in the bank for right, and this though is much more contested, isn't it? Either left or right can play there, they both want it. That's like a hot game, so we want it. This is super hot, isn't it? If left plays here, then she's got a free move as well. If right plays here, then she's got a free move as well. So actually that's even hotter. So of these four games you would ignore these. You wouldn't even go for that one. You would get this one but you wouldn't even get the extra move. That's somehow more desirable. That's the hottest. I'm going to go rather quickly through something slightly more technical. John Conway, the guy who also discovered the game of life and I think was also involved in categorising one of the finite simple groups. So an incredibly prolific guy. Serial numbers are defined as X's is a set of numbers on the left and a set of numbers on the right and the condition is that everything in this set is less than everything in this set. That's a slight abuse of notation if you're going to be a little picky about the notation. But the idea is we haven't said anything yet. So this is like a list of things on the left and this is a list of things on the right and the guarantee is that each of them here is less than each of them here so that's fair-wise. How do we bootstrap it and as always we bootstrap it with the empty set. So actually inserted here is no element because it's the empty set. That's defined to be zero and then you say well okay so what if I have on the right zero and then on the left nothing and then on the left zero and on the right nothing and then zero zero here. This is for now not a number but I'll let you into a secret it is a game. So day 2 you have several different ways of organising all the numbers you've seen before and there's an addition defined on these kinds of objects and there's a multiplication and there's an exponentiation so these objects are behaving like numbers. I won't go into it but you have on day 2 1 minus 2 and a half fill in and in fact I think probably 3 over 2 is filled in as well I'll check in a sec. So you get the dyadic rational so you start getting on day n of these numbers that are being generated you get all the numbers of the form something over 2 to the n so you're slowly building up and that's I'm bounded by above it's much clearer if I show you the diagram so here you've got day 1, day 2 and the idea is you start filling out all of these and you only have the so-called dyadic rational so they're only the lower part is always a power of 2 so the rule for deciding what the numbers are is it's the simplest number that was born on the smallest day so between 0 and 1 the simplest thing is a half the number that is the value of this strange notation is the number that's the simplest so it's bigger than this number and it's smaller than this number and it's the simplest one in between so that's the rule for determining serial numbers this is a non-unique description I could write 2,0 or minus 2,0 here as well but remember it's always greater than all the ones in the left and less than all the ones in the right and it's the simplest so games are similar the games are the same as numbers there's some number of left moves so when you choose a move of all the possible positions there's some number of moves for right games are similar the only thing is on all the numbers in the left being less than all the numbers in the right and games have this special property where unlike numbers games can be fuzzy or confused so a game of 0 is the person to move loses so that's the 0 game that's in the game you don't want to play in if anything's confused with 0 then it's not positive or negative it's like a hot game somehow so you do want to play in it if a game's positive it's actually a win for the first player and if a game's negative it's actually a win for the second player so I don't have tons of time to go into how it works but I hope I've wet your appetite I've got some book recommendations we said here this is for left so this is just one extra move for left this is right so this would be minus 1 these are actually equivalent to numbers because I said some games were numbers this is the same thing here this is 0 0 because left can take this and leave the 0 game for right right can take this and leave the 0 game for left this is our first it's an infinitesimal number it's smaller it's confused with 0 and it's smaller than all of the positive numbers it's confused with 0 from below as well it's greater than all the negative numbers somehow right close to 0 neither positive nor negative so these are the strange numbers that come in this one do you remember I said this was super super hot in the game you want to get into this if I say well if I'm left I can actually go here and leave 1 for myself whereas right can go here and leave 1 for herself so it's 1 and minus 1 this game is confused with all the numbers between 1 and minus 1 that with how confused it is is somehow the how hot it is and how much you want to play in it and even I was struggling to finish these slides that should say equals up arrow so we said star was and star coincidentally is nim again so a 1 a 1 star for nim because when you do nim addition and you say a nim heap of size 8 is like 8 star and size 6 is 6 star so you say 5 star plus 6 star etc so I probably didn't mention that I forgot excuse me but this is the same as 1 as a nim heap of size 1 which if you think about it is right isn't it because a nim heap of size 1 I can take it or they can take it and that's all they can do do you see how this piece of the game would be if you were playing it alongside nim and that's the whole point with the game addition playing nim plus this plus chess and you just choose for this move which game am I playing in make a move in it if this is a hot game then the person probably wants to reply in that game and what you do is you play the hottest game first is the best strategy and then you play the less hot game and all the cold games that no one wants to play in or if that games if you imagine the two games one game is the person to move wins and one person is the person to move person to move losers which is a zero game adding that game doesn't make a difference because whoever wins in this game makes the other person play in the losing games that makes sense that's why it's called the zero game you add it to another game and there's no point if you've got a game of nim with two a pile of two and a pile of two and another game is chess no one wants to play enough because it's a losing game right so we play in chess till we're done whoever wins chess forces chess isn't a normal form excuse me playing something else that's normal form excuse me sorry just imagine I didn't say chess imagine I said something else anyway I've waffled enough I hope you're well I'm worried if you're as excited as I am because I mean because I can't I don't think I've sold how awesome this idea is as well as hours worth of reading amazing books would do or doing a longer course or something so here's some references the dots and boxes gave if you want to get to the point where you're transcendently playing Dawson's Vines instead of the actual game there's been a recently recent book published the chapy coincidentally now works at Twitter a combinatorial game theme this is the sort of graduate studies one on numbers in games John Horton Conway's initial sort of 1970s description of the serial numbers there's amazing four volume work called winning ways for your mathematical plays which is incredibly fun there's one called I've forgotten lessons in play which is also excellent like a simpler introduction and there's one by Donald Knuth of the art of computer programming fame who actually he was the chap who named them the serials Conway always wanted to just call them numbers and they are the maximal ordered fields so it's defensible to just call them numbers and yeah I don't have a conclusion I hope you enjoyed it I really enjoyed giving it thanks for the answers sorry I couldn't hear 80% of them have you got time for questions or do we have to pack up and go find out cool thank you I'm not joking you will have to shout if you ask a question we've got a question absolutely I should have had a slide on it it's been used in chess end games and in go end games so there's a game by a book by David Wolfe Chappy wrote one of the books I mentioned which is lessons in play he's a co-author of that he's got a book called you know what I'm saying about things being hot there's a concept called chilling and there's a book called mathematical go chilling gets the last point or something and the idea is in a very close game it becomes amenable to this style of analysis so chess end games and go end games in particular and of course the surreal numbers are interesting mathematical field in themselves they do have an analysis within them and stuff like that so and it's similar to a concept called dedicated cuts so it's not without precedent to do that kind of thing for numbers I moved very quickly through that stuff but it was more to sort of wet your appetite I hope that answers your question no Wikipedia is actually very good for it so and the Knuth book is more like a Socratic dialogue between two people to find some oh sorry my style in this was to go from games to numbers and just dip into numbers because the treatment going from numbers to games has been done in on numbers in games by Conway and the surreal numbers book by Knuth just deals with the numbery aspects and it's like a Plato style dialogue, Socratic dialogue type thing I don't recall it being a love story but it is I remember one of them being female but I don't remember ha ha ha ha ha I'm probably 12 years off I've read the book I don't think I remember it's the tablets arrive and you go on this nice journey I don't remember any actual sex any last question anyone no ok another big hand for Tom Hall thanks for your indulgence