1 00:00:00,012 --> 00:00:05,620 >> Welcome back, still week three. What we want to do is take a look at an 2 00:00:05,620 --> 00:00:11,380 example, and see how we can use inequalities, to analyze games and how we 3 00:00:11,380 --> 00:00:14,982 can use inequalities to, to simplify games. 4 00:00:14,983 --> 00:00:21,061 Example I want to look at is this example here in Hackenbush. 5 00:00:21,061 --> 00:00:27,123 This is the game G. There's the, the ground here, the ground 6 00:00:27,123 --> 00:00:31,496 is green. The blue edge, a red edge, and a blue 7 00:00:31,496 --> 00:00:35,608 edge. This is Hackenbush, and if you go back to 8 00:00:35,608 --> 00:00:43,526 week one, or at some point there should be a, dictionary on the site, with, with how 9 00:00:43,526 --> 00:00:47,095 various games work. Players alternate. 10 00:00:47,095 --> 00:00:52,004 Red is, can only be cut by right. Blue can only be cut by left. 11 00:00:52,004 --> 00:00:58,666 And once an edge is cut, not only does that edge disappear, but any edge that's 12 00:00:58,666 --> 00:01:04,659 no longer connected to ground disappears, will also disappear. 13 00:01:04,659 --> 00:01:15,545 So if you look at the possible left moves, left can cut this blue edge on the top. 14 00:01:15,545 --> 00:01:24,385 In which case what is left is just, blue, red, like this. 15 00:01:24,385 --> 00:01:34,030 Or, blue can cut the edge down at the bottom, in which case what is left is just 16 00:01:34,030 --> 00:01:39,665 nothing. Now, what I want to do now is, is, give 17 00:01:39,665 --> 00:01:47,767 you another way of writing out games. Another way of writing out games, and 18 00:01:47,767 --> 00:01:55,472 you'll see this on any of the references we look at, is left curly bracket, and now 19 00:01:55,472 --> 00:02:04,212 list the possible left moves then a, pipe in unix vertical, vertical, verticals up 20 00:02:04,212 --> 00:02:11,883 and down and then list the right moves. So, the possible left moves, are chop off 21 00:02:11,883 --> 00:02:16,882 the bottom blue edge, and that's zero, this game is zero. 22 00:02:16,883 --> 00:02:22,619 Whoever moves first loses. Now the other possible opening move for 23 00:02:22,619 --> 00:02:26,361 blue is, or left, is to cut off the top here. 24 00:02:26,361 --> 00:02:31,172 And that leaves this. And awhile back we called this game 1 25 00:02:31,172 --> 00:02:35,048 half. And it turns out, this game, plus itself 26 00:02:35,048 --> 00:02:39,141 is equal to 1, so we might as well call this 1 half. 27 00:02:39,141 --> 00:02:46,116 Now the possible right moves are to cut off the red edge there, in which case, 28 00:02:46,116 --> 00:02:52,883 what's left is just one blue edge, and one blue edge is one move for left and that's 29 00:02:52,883 --> 00:03:00,007 the game 1, we'll call that the game 1. So this game up here could be represented 30 00:03:00,007 --> 00:03:07,303 abstractly as the game, where left moves are to 0 or 1 half, and the right moves 31 00:03:07,303 --> 00:03:12,899 are to 1. Now, you can prove, or, you can go through 32 00:03:12,899 --> 00:03:20,083 the details, or you probably would guess, that 0 is less than a half. 33 00:03:20,083 --> 00:03:23,404 That's true. Just look at 1 half. 34 00:03:23,404 --> 00:03:28,383 To say 0 is less than a game means, left wins the game. 35 00:03:28,383 --> 00:03:33,287 1 half, going first or second. And look at this game up here. 36 00:03:33,287 --> 00:03:39,435 1 half,, and we see left always wins. Left going first cuts the blue edge. 37 00:03:39,436 --> 00:03:44,914 Right has no move, right loses. Right going first cuts the red edge, then 38 00:03:44,914 --> 00:03:48,359 blue cuts the then left cuts the blue edge. 39 00:03:48,359 --> 00:03:53,180 And again left wins. So if 0 is left than a half which means 40 00:03:53,180 --> 00:03:57,230 that 1 half is always better, for left, than 0 is. 41 00:03:57,230 --> 00:04:03,702 So what that says, is there's no reason whatsoever, that left would ever choose 0. 42 00:04:03,702 --> 00:04:08,425 1 half is always better. So this game is the same as this, and now 43 00:04:08,425 --> 00:04:14,629 we've simplified the game, to an equivalent gain with fewer, fewer possible 44 00:04:14,629 --> 00:04:18,373 moves. We've deleted the moves that are no good 45 00:04:18,373 --> 00:04:23,464 because of inequality. Now in terms of right moves, there's only 46 00:04:23,464 --> 00:04:29,132 one right move here. But, if there were several right moves, we 47 00:04:29,132 --> 00:04:36,748 could delete all but the smallest, because right wants moves to be small, and left 48 00:04:36,748 --> 00:04:42,796 wants moves to be big. Okay, so let me end with a problem, and 49 00:04:42,796 --> 00:04:49,476 this is for you to work on. Here's two games, G and H. 50 00:04:49,476 --> 00:04:56,401 And see if you can prove that G is less than or equal to H. 51 00:04:56,401 --> 00:05:05,671 If you remember right, that says that H minus G wins the left going second. 52 00:05:05,671 --> 00:05:08,740 So you can work that out.