1 00:00:00,012 --> 00:00:04,134 Okay, welcome to Games without Chance. None of that. 2 00:00:04,134 --> 00:00:10,445 Okay, I'm Tom Morley and let's see where we are this week, we're doing the 3 00:00:10,445 --> 00:00:14,694 following. Some numbers are games and that's where we 4 00:00:14,694 --> 00:00:17,696 are right now. Some numbers are games. 5 00:00:17,697 --> 00:00:27,308 Okay, so let's look at Hackenbush. This was a game we looked at from the 6 00:00:27,308 --> 00:00:31,887 beginning. Here's our ground. 7 00:00:31,888 --> 00:00:36,709 Here's a blue edge. Blue is left. 8 00:00:36,709 --> 00:00:41,306 Left can cut blue. Here's a red edge. 9 00:00:41,306 --> 00:00:50,050 Red edges can be cut by right. Now, this was a game that we looked at and 10 00:00:50,050 --> 00:01:00,368 we computed a while back, that two of these, copies of these the other with one 11 00:01:00,368 --> 00:01:05,462 red is 0. Whoever moves first in this game loses. 12 00:01:05,462 --> 00:01:13,988 If blue moves first, cutting blue edges, one at a time, and alternates with right, 13 00:01:13,988 --> 00:01:22,028 who cuts red edges then if left moves first, left loses, if right moves first, 14 00:01:22,028 --> 00:01:25,047 right loses. This is a zero game. 15 00:01:25,047 --> 00:01:37,727 This is minus 1 here, so might as well call this 1 half. 16 00:01:37,727 --> 00:01:50,901 1 half plus 1 half minus 1 is 0. So this game here, the number. 17 00:01:50,901 --> 00:02:03,729 Now, suppose we have two of these. Two, we have blue, but two red on top. 18 00:02:03,729 --> 00:02:11,677 Now there's no reason for right ever to cut the bottom red, so right might as well 19 00:02:11,677 --> 00:02:16,519 cut the top red. And it turns out that, it turns out that 20 00:02:16,519 --> 00:02:21,343 it's a four-letter acronym four-letter acronym. 21 00:02:21,343 --> 00:02:29,063 It turns out that this behaves like 1 4th, you take two of these and they add up to 22 00:02:29,063 --> 00:02:34,163 minus a half. There's a half minus a half is reverse 23 00:02:34,163 --> 00:02:39,031 colors. Now in general, we have the following 24 00:02:39,031 --> 00:02:42,976 games, so, so here's 0. There's 0. 25 00:02:42,976 --> 00:02:47,291 There's the zero game, right there. Okay. 26 00:02:47,291 --> 00:02:53,115 A picture of the zero game. I should hang this up on my wall. 27 00:02:53,115 --> 00:02:57,985 There, there's the zero game. Let's see. 28 00:02:57,985 --> 00:03:02,975 Here's so that's zero, this is 1, this is 2. 29 00:03:02,975 --> 00:03:08,884 Run out of ground. Let's put some more ground over here. 30 00:03:08,884 --> 00:03:15,220 Here's 3, one, two, three. We have the negatives of these. 31 00:03:15,220 --> 00:03:24,206 This, for instance, is minus 2. We have some fractions here's 1 half. 32 00:03:24,207 --> 00:03:30,287 I'm going to do a bunch of these wait. Here's a half. 33 00:03:31,554 --> 00:03:35,821 This is a quarter. This is an eighth. 34 00:03:35,821 --> 00:03:42,756 This is a sixteenth. And, you can guess that when you have one 35 00:03:42,756 --> 00:03:50,739 blue and a bunch of red on top, you end up with a 1 over a power of 2. 36 00:03:50,739 --> 00:04:00,766 Now, once you add these all up, so you can take any of these and add them up, what 37 00:04:00,766 --> 00:04:10,616 you can get by adding up these games is anything of the form p over two to the n. 38 00:04:10,616 --> 00:04:19,781 P is any integer, n is in the integer and these are called the dyadic, dyadic 39 00:04:19,781 --> 00:04:25,986 rationals. So all the dyadic rationals turn out to be 40 00:04:25,986 --> 00:04:34,229 games that, they all turn out to be expressible in terms of Hackenbush. 41 00:04:34,230 --> 00:04:47,757 So, let's look at a problem. You will contrive. 42 00:04:47,757 --> 00:04:57,329 What number is this? These two games. 43 00:04:57,329 --> 00:05:03,652 This is one game that's the sum of two games. 44 00:05:03,652 --> 00:05:07,127 What number is this? Okay. 45 00:05:07,127 --> 00:05:12,350 Here is the solution. This is one fourth. 46 00:05:12,350 --> 00:05:18,840 This is two. And 1 4th plus 2 is 2 and 1 4th, also 47 00:05:18,840 --> 00:05:31,217 known as, what C, two is eight, no, 9 4ths so either answer would be correct. 48 00:05:31,217 --> 00:05:35,575 And that's the end of Module 1.