1 00:00:00,012 --> 00:00:03,953 Welcome back. None of these, none of these. 2 00:00:03,953 --> 00:00:07,798 You know the rules by now, no random moves. 3 00:00:07,798 --> 00:00:13,263 I'm Tom Morley. Last time we were looking at these two 4 00:00:13,263 --> 00:00:16,946 games. The left is a cutcake. 5 00:00:16,947 --> 00:00:23,961 And I always have to remind myself, right cuts this way. 6 00:00:23,961 --> 00:00:31,831 Left cuts up and down. And the right game is a Hackenbush. 7 00:00:31,831 --> 00:00:38,082 It's a very simple game. Left has one move left and cut this left 8 00:00:38,082 --> 00:00:40,236 branch. Right has no moves. 9 00:00:40,236 --> 00:00:44,581 Left goes first, left wins, right goes first, left wins. 10 00:00:44,581 --> 00:00:50,280 Left always wins in this game here. These are two games G and H. 11 00:00:50,280 --> 00:00:58,750 We want to know, is G equal to H? And to do so, we have to look at G minus H 12 00:00:58,750 --> 00:01:04,486 and see who wins. If this is 0, then the games are equal. 13 00:01:04,486 --> 00:01:12,928 To say that this is 0 means that and minus H is, of course, an abbreviation for G 14 00:01:12,928 --> 00:01:19,446 plus the negative of H. We'll go ahead and just write it that way. 15 00:01:19,446 --> 00:01:25,773 And, and, say this is equal to 0, just means that, that whoever moves first in 16 00:01:25,773 --> 00:01:30,871 this game loses. So, we have to analyze the strategies in 17 00:01:30,871 --> 00:01:34,828 best play in G minus H. So, let's look at G minus H. 18 00:01:34,828 --> 00:01:43,887 G minus H is cut take over here. And the Hackenbush over here. 19 00:01:43,888 --> 00:01:49,741 But the negative of the Hackenbush is this. 20 00:01:49,741 --> 00:01:58,677 Now you, let's look at right, what happens when right goes first. 21 00:01:58,678 --> 00:02:01,970 Okay. So, let's look at right going first in 22 00:02:01,970 --> 00:02:06,104 this game. The best move for right is to chop down 23 00:02:06,104 --> 00:02:12,929 the cherry tree in order for Presidents' Day, which is coming up, except by the 24 00:02:12,929 --> 00:02:19,964 time you see this it's already passed left's best move at this point is to chop 25 00:02:19,964 --> 00:02:25,251 this in half. Right then has to take one of these 2 by 26 00:02:25,251 --> 00:02:30,481 2s, it doesn't matter which one and chop it in half. 27 00:02:30,481 --> 00:02:34,111 And now look, look and see what's going on. 28 00:02:34,111 --> 00:02:39,076 Left has one, two three moves and right doesn't have much. 29 00:02:39,076 --> 00:02:45,067 So actually, there's a little bit more work to be done here, but left has lots 30 00:02:45,067 --> 00:02:49,316 more moves here than right ultimately does have. 31 00:02:49,316 --> 00:02:53,946 And so, it turns out that, that, that in fact, right loses. 32 00:02:53,946 --> 00:03:01,740 And there's a, there's a bit more to, in, in, in the whole process, but you can 33 00:03:01,740 --> 00:03:06,525 puzzle that out and, and, and try it yourself. 34 00:03:06,526 --> 00:03:13,535 You also might want to try if right, if left goes first. 35 00:03:13,536 --> 00:03:24,016 Left goes first the, the, the easiest move for, the best move for left going first is 36 00:03:24,016 --> 00:03:31,105 to chop up and down in the middle but still, left loses. 37 00:03:31,105 --> 00:03:42,744 So in this combination game of this cut-cake and this hackenbush, the sum is 38 00:03:42,744 --> 00:03:50,617 first player lose. The sum is zero and therefore this game 39 00:03:50,617 --> 00:03:56,676 here happens. Okay so we've done a couple of simple 40 00:03:56,676 --> 00:04:01,835 examples. One thing that we haven't done which you 41 00:04:01,835 --> 00:04:07,411 might think about is, is so that any game is equal to itself. 42 00:04:07,411 --> 00:04:13,041 That is to say any gain minus its negative equals zero. 43 00:04:13,041 --> 00:04:22,347 And the basic idea of showing this is what Conway and, and friends call Tweedledum, 44 00:04:22,347 --> 00:04:31,854 Tweedledee principle. If you have a complicated game over here, 45 00:04:31,854 --> 00:04:44,067 that it's negative over here then whatever one player does over here, the other 46 00:04:44,067 --> 00:04:52,095 player will do the opposite over here. Whatever one player does over here, the 47 00:04:52,095 --> 00:04:55,248 first, the other player will do the opposite over here. 48 00:04:55,248 --> 00:04:59,600 So, what that says the second player to move always has the advantage because 49 00:04:59,600 --> 00:05:04,197 there's always, always will be a counter. Whatever you do in one of these, the other 50 00:05:04,197 --> 00:05:06,644 player does the opposite in, in the other one. 51 00:05:06,644 --> 00:05:10,585 The branches over here that are right branches over here are left branches. 52 00:05:10,585 --> 00:05:14,304 The branches over here that are left branches are over right branches. 53 00:05:14,305 --> 00:05:20,331 So, if left moves over here, then right moves over here, for instance. 54 00:05:20,332 --> 00:05:26,523 So in general, this says any game is equal to itself, a not terribly surprising 55 00:05:26,523 --> 00:05:30,018 thing, but that's how we'll end the first week. 56 00:05:30,018 --> 00:05:36,914 And we'll come back check out some problems for you to work on in the, the 57 00:05:36,914 --> 00:05:43,698 homework/quiz, it will let you now how you're doing in the course and we'll go 58 00:05:43,698 --> 00:05:47,161 over them afterwards. So, take care.