1 00:00:00,012 --> 00:00:07,604 God morning, this is Games Without Chance. I'm Tom Morley. 2 00:00:07,605 --> 00:00:18,556 Let's look a second over here is an ace. Over here is an ace Over here is an ace 3 00:00:18,556 --> 00:00:22,446 and then we have one more. Okay. 4 00:00:22,446 --> 00:00:32,516 So today, this week, we have some modules and the first one is new ways of 5 00:00:32,516 --> 00:00:40,144 simplifying games. So let's look at two examples to start 6 00:00:40,144 --> 00:00:44,634 with. Here's the first game and the second game. 7 00:00:44,635 --> 00:00:51,224 The first game consists of a bunch of numbers for left options or left moves and 8 00:00:51,224 --> 00:00:57,872 a bunch of numbers of right options. The second is a up and if we recall from a 9 00:00:57,872 --> 00:01:05,330 while back Or you can look at one of the tabs on the left, for all the definitions, 10 00:01:05,330 --> 00:01:12,788 up is 0, is the only left option removed, star is the only right option remove and 11 00:01:12,788 --> 00:01:18,144 star is 0, 0. We've previously shown that up is positive 12 00:01:18,144 --> 00:01:22,906 but less than any 1 over 2 n, for any interger end. 13 00:01:22,907 --> 00:01:31,247 So, let's start with the first one and see how far we get. 14 00:01:31,247 --> 00:01:40,197 If you look at this gain, left has many options, if three. 15 00:01:40,197 --> 00:01:45,616 Three is many. But, the bigger the number is the better 16 00:01:45,616 --> 00:01:51,778 it is for left so, seven is always better for left than either one or two. 17 00:01:51,778 --> 00:01:58,154 So, there's no reason the left would ever play one or two, even if this game is in 18 00:01:58,154 --> 00:02:01,991 context of a much larger game and just a small. 19 00:02:01,991 --> 00:02:10,094 Peace So, left, as far as left is concerned, the left only will ever play 7. 20 00:02:10,094 --> 00:02:17,234 Now on right, in right's moves, right wants to make numbers as small as 21 00:02:17,234 --> 00:02:22,481 possible. Smaller is better for right and 8 is less 22 00:02:22,481 --> 00:02:26,974 than 10, so. Right will always choose 8, even if this 23 00:02:26,974 --> 00:02:32,430 is part of a larger context. So, these two are never used, these, this 24 00:02:32,430 --> 00:02:37,459 one is never used in best play. So, this is the same as 7/8 and the 25 00:02:37,459 --> 00:02:41,282 simpliest number between 7 and 8 is 7 and a half. 26 00:02:41,283 --> 00:02:45,518 Path. So, we've simplified getting here by using 27 00:02:45,518 --> 00:02:50,503 dominated moves. 7 dominates 1 and 2 because it's bigger 28 00:02:50,504 --> 00:02:57,237 and it's a le-, left option or left move. 8 dominates 10 because it's the right 29 00:02:57,237 --> 00:03:02,309 option that's less. So you, for domination we want. 30 00:03:02,310 --> 00:03:10,031 On the left side we want bigger and on the right side we want smaller. 31 00:03:10,031 --> 00:03:14,668 Ok, lets take a look at this game, up and 2. 32 00:03:14,669 --> 00:03:20,044 We remember that, that up is positive and, and less than 1 over 2 to the n. 33 00:03:20,044 --> 00:03:24,694 So, and, since 1 over 2 to the n is less than 2, up is less than 2. 34 00:03:24,694 --> 00:03:29,440 All of the left options are, are less than the right options. 35 00:03:29,440 --> 00:03:36,714 By claim, this is, this is a number and It's the simplest number that's in between 36 00:03:36,714 --> 00:03:43,658 up and 2 and that's 1 but, if you want to prove it directly, all you have to do is 37 00:03:43,658 --> 00:03:49,507 show that up 2 minus 1 is 0 and let's see if we can't argue that. 38 00:03:49,508 --> 00:03:56,895 If left moves first, left moves to there are no left moves in minus 1, so left 39 00:03:56,895 --> 00:04:03,593 moves to up and then we have up minus 1 which is negative to right wins. 40 00:04:03,593 --> 00:04:10,061 So left moves first, right wins. If right moves first, right can either 41 00:04:10,061 --> 00:04:15,950 move to Two, in which case you have 2 minus 1, which is one and left wins. 42 00:04:15,950 --> 00:04:19,245 And then you have to check the other cases. 43 00:04:19,245 --> 00:04:25,110 But in all cases if right moves first right loses if left moves first left 44 00:04:25,110 --> 00:04:29,077 loses. Therefore, this is zero and therefore this 45 00:04:29,077 --> 00:04:30,400 game is. One. 46 00:04:30,400 --> 00:04:37,574 So here we've reduced to a number, even though it wasn't a kind of standard form 47 00:04:37,574 --> 00:04:42,447 for numbers. Now if you go back a long time ago, we 48 00:04:42,447 --> 00:04:47,718 looked at men back, way back in the introduction video. 49 00:04:47,718 --> 00:04:53,055 We had three coins, we had two coins, and we had one coin. 50 00:04:53,055 --> 00:05:01,554 And the rules of Nim are play, each player in turn can remove one or, must remove one 51 00:05:01,554 --> 00:05:09,570 or more coins from one of these piles. Now this is a example of what's called an 52 00:05:09,570 --> 00:05:17,199 impartial game which means that, that The rules are the same for both players. 53 00:05:17,199 --> 00:05:21,880 If you, the negative of this game is the same as this game. 54 00:05:21,880 --> 00:05:27,966 And one of the things I asked you to think about, in terms of this game in the 55 00:05:27,966 --> 00:05:31,518 introduction video, was who wins this game? 56 00:05:31,518 --> 00:05:37,129 And, in terms of what we do now who wins this game is the second part. 57 00:05:37,129 --> 00:05:43,988 Player, whoever moves first loses, okay? So let's ask if extra coins help here. 58 00:05:43,988 --> 00:05:48,483 So, so I have this nym game with these 3 nym piles. 59 00:05:48,483 --> 00:05:55,326 Let's see I have them, that's a pile 1, that's a pile 2 and this is a pile 3 over 60 00:05:55,326 --> 00:05:59,039 here. I'll put them under their numbers. 61 00:05:59,039 --> 00:06:04,516 Star integer is just a notation for nim heap of that size. 62 00:06:04,516 --> 00:06:08,485 Nim heap of size one, size two, and size three. 63 00:06:08,486 --> 00:06:15,235 Now, if someone gives you some extra quarters, does that help you in any way? 64 00:06:15,235 --> 00:06:23,081 So we already know that if you remove any of these then the player, who's next move 65 00:06:23,081 --> 00:06:29,346 it is, has a winning move. So, the only possible winning move is to 66 00:06:29,346 --> 00:06:36,489 perhaps to put more coins on here. So, I've put a bunch more coins on there 67 00:06:36,489 --> 00:06:40,922 and it's a lot, you can count them if you want. 68 00:06:40,923 --> 00:06:48,129 Now, what is the response to this? The response to this, is the other player 69 00:06:48,129 --> 00:06:55,144 just removes the coins that you, that you just put on and puts a $1.25 in his 70 00:06:55,144 --> 00:06:59,853 pocket. Now you're left back where you started. 71 00:06:59,853 --> 00:07:04,206 It's still your move and now you loose again. 72 00:07:04,206 --> 00:07:10,426 So, if, if it's your move and instead of removing a coin from here, you put on 73 00:07:10,426 --> 00:07:16,440 extra coins, then the other player just removes them and he still wins. 74 00:07:16,440 --> 00:07:23,101 So the extra coins don't do you any good. Now in terms of what we our vocabulary for 75 00:07:23,101 --> 00:07:29,833 talking about games, this is called reversible move because once you make this 76 00:07:29,833 --> 00:07:36,690 move, the other player will just reverse it and we're left back where we started. 77 00:07:36,690 --> 00:07:40,369 Okay, this is the end of Module 1. Thank you.