1 00:00:00,012 --> 00:00:08,349 >> Alright, welcome back. None of this, none of these, no dice, no 2 00:00:08,349 --> 00:00:17,111 cards, games without random moves. Now, now that we know from last time how 3 00:00:17,111 --> 00:00:26,747 to add things up, add up games, maybe we'll play them simultaneously in the ways 4 00:00:26,747 --> 00:00:32,421 we described last time. We want to talk about what it means for 5 00:00:32,421 --> 00:00:38,437 two games to be equal, which is going to mean something like, whenever we play 6 00:00:38,437 --> 00:00:43,803 these games together with some other game, the same thing happens. 7 00:00:43,803 --> 00:00:47,849 Now, I want to be much more precise about that in a minute. 8 00:00:47,849 --> 00:00:53,813 But that's esesentially what's giong on. And so, to start this, I want to start 9 00:00:53,813 --> 00:00:59,710 with a simple, a very simple case. And our definition of win a game is 0. 10 00:00:59,710 --> 00:01:07,354 This is a definition and mathematics definitions are prescriptive, that means 11 00:01:07,354 --> 00:01:10,493 they're, what is the case. Okay. 12 00:01:10,493 --> 00:01:22,013 So what does it mean for game to be zero and this means in best play, the first 13 00:01:22,013 --> 00:01:32,270 player to move, loses. So, just like the zero game, whoever moves 14 00:01:32,270 --> 00:01:41,051 first in the zero game, loses, because there's no moves. 15 00:01:41,051 --> 00:01:46,131 Let me, let me show you. The zero game happened to bush is this. 16 00:01:46,131 --> 00:01:51,372 Nothing to cut. The zero game in, in a heap, is a heap of 17 00:01:51,372 --> 00:01:55,902 zero coins, which you can see over here to your left. 18 00:01:55,902 --> 00:02:01,886 And so that's the zero game. And then, we'll say the game is equal to 19 00:02:01,886 --> 00:02:06,085 0, if in best play the first player to move loses. 20 00:02:06,085 --> 00:02:11,927 Okay, let's look at an example. Nim-heap of size 2 over here, nim-heap of 21 00:02:11,927 --> 00:02:16,569 size 2 over here. It just matters that there's two dice over 22 00:02:16,569 --> 00:02:22,113 here, two dice over here, the numbers on the dice don't change anything. 23 00:02:22,113 --> 00:02:27,631 So, so if, if left, whatever left does over here, right does over here. 24 00:02:27,631 --> 00:02:32,205 Whatever left does over here, right does over here, and left loses. 25 00:02:32,205 --> 00:02:37,343 So, if left goes first, left loses. Similarly, if right goes first, right 26 00:02:37,343 --> 00:02:41,226 loses. So, this game 2 nim-heaps, a nim-heap of 27 00:02:41,226 --> 00:02:47,524 size 2, which we now take this way, plus a nim-heap of size 2, is first player lose, 28 00:02:47,524 --> 00:02:49,950 so it's equal to 0. Okay. 29 00:02:49,950 --> 00:03:03,236 So, we know when a game is equal to 0. So now, let's look, let's try to see what 30 00:03:03,236 --> 00:03:08,691 a negative of a game is. We have a game, G. 31 00:03:08,691 --> 00:03:15,661 We can compute, have an associated game called minus G. 32 00:03:15,661 --> 00:03:24,465 Now, let me show you this for hackenbush. Say, this hackenbush game is G. 33 00:03:25,839 --> 00:03:36,541 Then, minus G is the same thing with left and right interchanged, okay? 34 00:03:36,541 --> 00:03:51,105 Now, let's look at other possibilities. In, in cutcake, remembering that right, 35 00:03:51,105 --> 00:03:57,811 cuts left to right, and left cuts up and down. 36 00:03:57,811 --> 00:04:06,987 Then, if that's a game of cutcake, if we'd simply interchange rows and columns, then 37 00:04:06,987 --> 00:04:15,633 whatever became a, a, a possibility for left is now a possibility for right, and 38 00:04:15,633 --> 00:04:24,101 whatever became a possibility for right is now a possibility for left, okay? 39 00:04:24,101 --> 00:04:28,917 So, the negative of this cutcake game is this game. 40 00:04:28,917 --> 00:04:35,072 The, the negative of, say nim-heap of size 3, is now interchanged the versions of 41 00:04:35,072 --> 00:04:40,834 left and right, but since left and right are, are, have the same moves in them, the 42 00:04:40,834 --> 00:04:44,231 negative of this is the same, same as, as that. 43 00:04:44,231 --> 00:04:49,759 So, the negative, let's do it, of nim-heap of size 2, is a nim-heap of size 2. 44 00:04:49,759 --> 00:04:53,858 Interchanging left and right, a nim doesn't do anything. 45 00:04:53,858 --> 00:04:59,516 What is it in chess? Well, the negative of, of, the negative 46 00:04:59,516 --> 00:05:05,647 chess game is a game, a chess game where black goes first, I guess. 47 00:05:05,647 --> 00:05:11,693 Or maybe it's the negative of a chess game is where instead of you having white, you 48 00:05:11,693 --> 00:05:14,881 have black. So it's the interchange of the, of the two 49 00:05:14,881 --> 00:05:18,000 sides. Left becomes right, right becomes left, 50 00:05:18,000 --> 00:05:23,182 black becomes white, white becomes black, whatever the, the moves in the game are. 51 00:05:23,182 --> 00:05:28,726 In Go the negative of the game, which is corresponding to a different player going 52 00:05:28,726 --> 00:05:34,618 first. So instead of white stones, you have black 53 00:05:34,618 --> 00:05:40,127 stones etc., etc. So, that's, that's how a negative of game 54 00:05:40,127 --> 00:05:47,162 is. And now, we're ready to, to define G 55 00:05:47,162 --> 00:06:01,083 equals H, means G minus H is 0, or G minus H is just an abbreviation for this, okay? 56 00:06:01,083 --> 00:06:12,218 We'll look at some examples. For next time, this is a short one. 57 00:06:12,218 --> 00:06:22,318 Let's just take a look at, I'll give you an example to, to work on yourself. 58 00:06:22,319 --> 00:06:38,308 Let's look at a cutcake and a hackenbush, and my claim is, are these equal? 59 00:06:38,308 --> 00:06:47,660 Try it out and we'll see you next time. Take care.