1 00:00:00,012 --> 00:00:08,377 >> Well good morning, or afternoon, or evening, as the case may be again. 2 00:00:08,377 --> 00:00:17,311 What I, we want to do today is just say a little bit about what is, and is not, a 3 00:00:17,311 --> 00:00:21,133 combinatorial game. Alright. 4 00:00:21,133 --> 00:00:23,878 No dice. No dice. 5 00:00:23,878 --> 00:00:32,444 No cards. No random, randomization at all. 6 00:00:32,444 --> 00:00:44,881 So, Monopoly has dice, it has you pick cards from the two piles. 7 00:00:44,881 --> 00:00:49,764 Monopoly isn't a combinatorial game. All right. 8 00:00:49,764 --> 00:00:54,898 There's two players, at least for what we're doing. 9 00:00:54,898 --> 00:01:02,944 The two players we called Left and Right, call A and B, Bob and Susie, whatever you 10 00:01:02,944 --> 00:01:06,166 want. There's just two of them. 11 00:01:06,167 --> 00:01:13,632 They're different. Solitaire is not a[INAUDIBLE] game because 12 00:01:13,632 --> 00:01:21,217 there's just one player. So[inaudible] games, there's two players, 13 00:01:21,217 --> 00:01:24,026 no random moves. No ties. 14 00:01:24,026 --> 00:01:28,157 Ooh, that looks like one word. No ties. 15 00:01:28,157 --> 00:01:32,525 All right. So, chess can actually have a draw. 16 00:01:32,525 --> 00:01:38,536 That's when neither black wins nor white wins, but, but they tie. 17 00:01:38,536 --> 00:01:43,635 And so according to this, chess is not a[INAUDIBLE] game. 18 00:01:43,636 --> 00:01:49,334 There, there are no, there's no dice in chess, there's no cards in chess. 19 00:01:49,334 --> 00:01:55,018 There are two players but there is a tie. Now eventually we'll weaken that 20 00:01:55,018 --> 00:01:59,967 assumption and, and allow for[INAUDIBLE] games with ties. 21 00:01:59,967 --> 00:02:04,188 But for the time being, let's, let's only look at moves. 22 00:02:04,189 --> 00:02:08,971 And ca, and games that are, that have no ties. 23 00:02:08,971 --> 00:02:15,361 So either blue wins or red wins. But not both, and not neither. 24 00:02:15,361 --> 00:02:21,101 Also it's finite. The game ends in a finite number of moves. 25 00:02:21,101 --> 00:02:26,536 We might not be able to say ahead of time how many that is. 26 00:02:26,536 --> 00:02:31,596 But we know that's finite. It doesn't go on forever. 27 00:02:31,596 --> 00:02:37,398 Not that this is different than saying that there's only a finite number of 28 00:02:37,398 --> 00:02:43,336 options available to each player. There can be, you can have games where 29 00:02:43,336 --> 00:02:49,446 there's infinite number of things the players can do, but it still has to end in 30 00:02:49,446 --> 00:02:54,906 a finite amount of[INAUDIBLE]. [inaudible] And so, we what we think of 31 00:02:54,906 --> 00:03:00,003 this in the following way. The two players alternate playing the game 32 00:03:00,003 --> 00:03:05,919 and when the first player that it's his or her turn and there's no move that, that 33 00:03:05,919 --> 00:03:12,581 player can make, then that player loses. So in terms of chess, once you're in 34 00:03:12,581 --> 00:03:17,246 checkmate, there's no move that you can make. 35 00:03:17,246 --> 00:03:24,154 And therefore you lose. Two other things that are, that are close 36 00:03:24,154 --> 00:03:31,131 to countertorial/g games, but not quite, which we'll look at are Go. 37 00:03:31,131 --> 00:03:34,971 And you can look up. The details of Go. 38 00:03:34,971 --> 00:03:41,379 Go is an interesting game. And also you may have, you may have played 39 00:03:41,379 --> 00:03:47,827 this game in grade school. And it's called, or maybe high-school or 40 00:03:47,827 --> 00:03:53,862 college, you know, you never know, it's called Dots and Boxes. 41 00:03:53,862 --> 00:04:00,246 So, so players alternate drawing lines, and, and when, when you, when a player 42 00:04:00,246 --> 00:04:06,218 completes a box you put your initials in the middle and you have to move again. 43 00:04:06,218 --> 00:04:09,786 The score, then, is the total number of boxes. 44 00:04:09,786 --> 00:04:15,465 And if you work out, if the number of boxes is odd then one player or another 45 00:04:15,465 --> 00:04:19,670 must win. So, in that case, its a score but we can 46 00:04:19,670 --> 00:04:26,258 think of it in terms of the score determining who wins, but again, there is 47 00:04:26,258 --> 00:04:30,224 no ties. Each box is either red or blue and if 48 00:04:30,224 --> 00:04:37,179 there's an odd number of boxes then one player or the other must have more boxes 49 00:04:37,179 --> 00:04:42,076 at the end. This is actually a fairly complicated game 50 00:04:42,076 --> 00:04:48,124 in terms of its analysis we hope to say a little bit about it towards the end of 51 00:04:48,124 --> 00:04:53,262 the, of the course. Okay, so these are Kind of what is and 52 00:04:53,262 --> 00:04:59,122 isn't a[INAUDIBLE] game. There, there are actually many things 53 00:04:59,122 --> 00:05:06,574 that, that are very close to[INAUDIBLE] games in terms of, of ties and, and, and 54 00:05:06,574 --> 00:05:11,864 whatever. These We'll eventually look at, but for 55 00:05:11,864 --> 00:05:16,784 the time being we want to keep things fairly simple. 56 00:05:16,784 --> 00:05:24,038 2 players, no random moves, finite, no tops, and that's where we're headed. 57 00:05:24,038 --> 00:05:24,190 .