1 00:00:00,012 --> 00:00:03,582 >> Games without chance. I did well this time. 2 00:00:03,582 --> 00:00:07,351 I just dealt that the self, I dealt that to myself. 3 00:00:07,351 --> 00:00:10,902 But that was a chance move, actually it wasn't. 4 00:00:10,902 --> 00:00:14,621 But this is games without chance. I'm Tom Morley. 5 00:00:14,621 --> 00:00:20,417 Here we are this week. We're now in the module, some games are 6 00:00:20,417 --> 00:00:25,861 numbers. So, let's go back and look at this 7 00:00:25,861 --> 00:00:31,923 alternative way we have of looking at games. 8 00:00:31,923 --> 00:00:37,070 Left options, right options. Okay. 9 00:00:37,071 --> 00:00:44,056 All of it, we can write down in terms of this notation. 10 00:00:44,056 --> 00:00:54,165 The games that we looked at last time. 0 is no options for left, no options for 11 00:00:54,165 --> 00:00:59,924 right. 1 is 0 for left and none for right. 12 00:00:59,924 --> 00:01:10,232 2 is 1 for left and none for right. And we can get some of the fractions this 13 00:01:10,232 --> 00:01:15,984 way. Okay, so here's the plan for this module. 14 00:01:15,984 --> 00:01:25,166 We want to look at games that are given in this form and try to figure out what 15 00:01:25,166 --> 00:01:31,826 numbers they are. Now the reason why we want to figure out 16 00:01:31,826 --> 00:01:40,554 what number a game is, is because we know that positive numbers are positive. 17 00:01:40,554 --> 00:01:47,355 And in positive games, left wins. Negative numbers are negative and in 18 00:01:47,355 --> 00:01:53,870 negative games, right wins. 0 games are 0, and in 0 games whoever 19 00:01:53,870 --> 00:01:59,234 moves next loses. So by, if once we know what number a game 20 00:01:59,234 --> 00:02:04,349 is, we know who wins. So, this is partly to do with, with 21 00:02:04,349 --> 00:02:11,515 connecting up games and numbers, but also partly to do with figuring out who wins. 22 00:02:11,515 --> 00:02:18,239 And if we have a sum of games that are all numbers, all we got to do is add them up. 23 00:02:18,239 --> 00:02:23,160 And, numbers add up like numbers. 1 plus 1 is 2. 24 00:02:23,160 --> 00:02:29,176 You can prove that, I bet. Okay, so let's take a look. 25 00:02:29,176 --> 00:02:36,629 So we know that 0 is, is this. But we also know that there are other 26 00:02:36,629 --> 00:02:45,541 games that are equal to 0. A game is 0 if it's first person lose. 27 00:02:45,541 --> 00:02:58,846 And any game that's 0, behaves like 0, like over here, even in a larger context. 28 00:02:58,846 --> 00:03:09,812 So let's look at the game minus 3, 1. My claim is that this is zero. 29 00:03:09,812 --> 00:03:17,104 Why? If left moves first, then left's only move 30 00:03:17,104 --> 00:03:25,962 is to minus 3. And now right wins, because minus 3 is 31 00:03:25,962 --> 00:03:31,178 negative. If right moves first, right moves to 1, 32 00:03:31,178 --> 00:03:36,598 that's the only option for right. And 1 is positive, so left wins. 33 00:03:36,598 --> 00:03:42,175 So if left moves first, right wins. If right moves first, left wins. 34 00:03:42,175 --> 00:03:45,932 That's a first player lose, so this game is 0. 35 00:03:45,933 --> 00:03:52,922 And I think you can actually get a much more general result, that if you have a 36 00:03:52,922 --> 00:04:00,512 negative number over here, and a positive number over here, then any game like that 37 00:04:00,512 --> 00:04:05,236 has got to be 0. Let's take a look at another example. 38 00:04:05,236 --> 00:04:09,661 Let's look at. Let's do 1, 4. 39 00:04:09,661 --> 00:04:22,856 My claim is that this is equal to 2. Now, what do I have to do. 40 00:04:22,856 --> 00:04:35,478 I have to show that 1,4 minus 2 is 0. That is, whoever moves first in this game 41 00:04:35,478 --> 00:04:41,162 loses. If left moves first, left has, remember 42 00:04:41,162 --> 00:04:46,719 minus 2 is nothing minus 1, that's minus 2. 43 00:04:46,719 --> 00:04:52,140 That's and then minus 2, left has no moves. 44 00:04:52,141 --> 00:04:55,863 So in this game my left has no moves in minus 2. 45 00:04:55,863 --> 00:04:59,951 So the, if left goes first left moves to 1, over here. 46 00:04:59,951 --> 00:05:06,256 What's left is minus 2 which is minus 1. So if left moves first, we, left moves to 47 00:05:06,256 --> 00:05:09,765 this 1, we still have this minus 2 over here. 48 00:05:09,765 --> 00:05:13,453 1 minus 2 is minus 1 and right wins minus 1. 49 00:05:13,453 --> 00:05:18,832 If right moves first, there's two moves to consider. 50 00:05:18,832 --> 00:05:24,427 Right either moves to 4 minus 2, which is this 4 minus 2. 51 00:05:24,427 --> 00:05:30,598 Or, 2 1, 4 minus 1. In either case, there's a little bit of 52 00:05:30,598 --> 00:05:38,316 work to do, but ri-, left wins. So there is a lot of games written this 53 00:05:38,316 --> 00:05:47,026 way that aren't the same, aren't the identical game to the Hackenbush games we 54 00:05:47,026 --> 00:05:53,591 looked at before, but nonetheless turn out to be numbers. 55 00:05:53,591 --> 00:06:02,251 And so here's one for you to try. And next time, we'll look at trying to 56 00:06:02,251 --> 00:06:08,461 figure out, if it is a number, what number is it? 57 00:06:08,461 --> 00:06:13,140 So, so try this one. Minus 4, minus 1. 58 00:06:13,140 --> 00:06:20,756 What ga-, what number is this? The answer is that it's minus 2. 59 00:06:20,756 --> 00:06:28,545 It fits in the middle and we'll see that's a key when the key stops. 60 00:06:28,545 --> 00:06:38,115 And so to actually prove this, you would have to show that minus 4, the game minus 61 00:06:38,115 --> 00:06:43,815 4, minus 1 plus 2 is 0. That is whoever moves first in this game 62 00:06:43,815 --> 00:06:45,057 loses. Okay. 63 00:06:45,057 --> 00:06:52,083 In 2, right has no move, so right has, if right goes first right has to move to this 64 00:06:52,083 --> 00:06:55,906 minus 1. We still have minus 1 plus 2 which is 1. 65 00:06:55,906 --> 00:07:01,802 Which is positive and so left wins. So there's half the argument and the other 66 00:07:01,802 --> 00:07:07,448 is a little bit more complicated. But I think y'all can handle that.