1 00:00:00,000 --> 00:00:01,980 So it's called the Josephus Problem. 2 00:00:01,980 --> 00:00:04,000 It's based on something from history. 3 00:00:04,000 --> 00:00:07,879 There was a group of Jewish soldiers who were surrounded by the Roman army 4 00:00:07,879 --> 00:00:12,080 and they didn't want to get captured, so they decided to come up with a system- 5 00:00:12,080 --> 00:00:14,080 - to avoid getting captured or suicide. 6 00:00:14,080 --> 00:00:18,320 So they'd sit in a circle, and the first man would kill the guy to the left of him 7 00:00:18,320 --> 00:00:22,859 The next remaining living person would kill the next remaining living person with their sword. 8 00:00:23,059 --> 00:00:26,699 And they'd go around like this, till there's only one person left, 9 00:00:26,699 --> 00:00:29,939 and the last person would have to commit suicide of course, rather than get captured. 10 00:00:29,940 --> 00:00:33,500 And the story - at least the story I was told, I'm not sure if this is historically accurate - 11 00:00:33,500 --> 00:00:37,719 Was that Josephus preferred captured than suicide, 12 00:00:37,719 --> 00:00:40,579 but he worried that if he said this, the other soldiers would turn on him. 13 00:00:40,579 --> 00:00:44,479 And so he wanted to figure out: WHERE should he sit within the circle- 14 00:00:44,500 --> 00:00:46,520 - in order to be the last man living. 15 00:00:46,520 --> 00:00:48,520 And then he would surrender, rather than kill himself. 16 00:00:48,520 --> 00:00:49,500 That was the problem 17 00:00:49,500 --> 00:00:51,500 It's a little tricky, so let's do a smaller example. 18 00:00:51,500 --> 00:00:53,020 Let's say there are seven people. 19 00:00:53,020 --> 00:00:57,420 And so just to be clear, the way it works is: person number one kills number two. 20 00:00:58,159 --> 00:01:00,159 Person number three kills four, 21 00:01:00,399 --> 00:01:01,719 Five kills six, 22 00:01:02,340 --> 00:01:05,119 But now things gets a little of harder to kind of see in advance 23 00:01:05,319 --> 00:01:09,239 Because seven, there is no eight, so seven kills one, 24 00:01:09,620 --> 00:01:11,960 Three kills five, 25 00:01:12,239 --> 00:01:14,239 Seven kills three. 26 00:01:14,480 --> 00:01:16,899 So seven's the winner; seven is the last one left over. 27 00:01:17,099 --> 00:01:20,079 For Josephus, there were 41 people. He needed to figure out where to sit. 28 00:01:20,079 --> 00:01:22,579 The mathematical problem is if there were n people. 29 00:01:22,579 --> 00:01:23,819 Where's the winning seat? 30 00:01:23,819 --> 00:01:25,719 When I learned this problem I was in high school, 31 00:01:25,859 --> 00:01:30,939 And it was one of the first problems that got me to understand how you approach- 32 00:01:30,939 --> 00:01:32,939 - difficult and- and... 33 00:01:32,939 --> 00:01:36,340 Math problems where you don't know the answer in advance, or someone hasn't shown it to you in advance. 34 00:01:36,340 --> 00:01:37,500 And it was a professor of mine 35 00:01:37,500 --> 00:01:38,879 his name was Phil Hanlon 36 00:01:38,879 --> 00:01:42,899 who played a big role in, kind of, leading me to math. 37 00:01:43,819 --> 00:01:46,779 And he suggested: what we should do is we should gather data. 38 00:01:46,939 --> 00:01:47,439 Just... 39 00:01:47,579 --> 00:01:50,859 Take various values of n and just do it by hand 40 00:01:50,859 --> 00:01:52,179 and start looking for a pattern. 41 00:01:52,180 --> 00:01:53,420 I don't know, maybe we should just do that? 42 00:01:53,420 --> 00:01:59,519 n is the number of seats, and w of n will be the winning seat. 43 00:01:59,519 --> 00:02:04,079 And what we know so far is that: n is seven, w of n it turns out is also seven. 44 00:02:04,079 --> 00:02:06,459 And so what I would do is I'd start doing some other values 45 00:02:06,459 --> 00:02:07,879 So, I don't know, why don't I do five? 46 00:02:07,879 --> 00:02:09,139 One kills two, 47 00:02:09,139 --> 00:02:10,359 three kills four, 48 00:02:10,620 --> 00:02:12,020 five kills one, 49 00:02:12,020 --> 00:02:14,360 three kills five. 50 00:02:14,360 --> 00:02:15,580 The winner is three. 51 00:02:15,580 --> 00:02:19,060 I guess there was no reason for me to skip six, so why don't we fill in six. 52 00:02:19,060 --> 00:02:22,340 One kills two, three kills four, five kills six... 53 00:02:22,919 --> 00:02:24,979 ... one kills three and five kills one. 54 00:02:24,979 --> 00:02:26,079 The winner is five. 55 00:02:26,080 --> 00:02:26,580 So 56 00:02:26,580 --> 00:02:29,300 If you're watching you might already start to notice some patterns. 57 00:02:29,300 --> 00:02:33,000 For instance, they're always odd. I mean, that's maybe the first thing you notice 58 00:02:33,000 --> 00:02:36,639 And you can start asking "why are they always odd?" And maybe you can already see that- 59 00:02:36,639 --> 00:02:39,459 - in the first loop around, all the even people get killed. 60 00:02:39,599 --> 00:02:40,099 So 61 00:02:40,099 --> 00:02:44,939 So you can - even with a tiny amount of experiment - start to see some patterns. 62 00:02:44,939 --> 00:02:48,199 - *Brady* So you DON'T want to sit in an even seat? - Don't sit in an even seat, no no no. 63 00:02:48,340 --> 00:02:50,939 And maybe we're even getting some glimmers of other patterns 64 00:02:51,139 --> 00:02:55,199 But before I really try to phrase those, I'd like to fill in the table a little bit more. 65 00:02:55,199 --> 00:02:55,799 So let's do some. 66 00:02:55,800 --> 00:02:57,280 So if there's only one person... 67 00:02:57,479 --> 00:02:57,979 well... 68 00:02:58,560 --> 00:03:00,640 That person's the winner, so that one was easy. 69 00:03:00,780 --> 00:03:04,599 If there's two people: one kills two. One is the winner 70 00:03:05,819 --> 00:03:11,079 If there's three people: one kills two, three kills one. Three is the winner 71 00:03:11,080 --> 00:03:17,500 If there's four people: one kills two, three kills four, one kills three. 72 00:03:17,500 --> 00:03:23,520 If there's eight people: one kills two, three kills four, five kills six, seven kills eight, 73 00:03:23,520 --> 00:03:27,060 one kills three, five kills seven, one kills five. 74 00:03:27,259 --> 00:03:28,739 The winner is one. 75 00:03:28,740 --> 00:03:36,060 Alright so it was one, one, three, one, three, five, seven. 76 00:03:36,259 --> 00:03:40,819 One, three, five, seven, nine. 77 00:03:40,819 --> 00:03:43,419 And you could keep going, but maybe you can see the pattern now? 78 00:03:43,419 --> 00:03:46,419 - *Brady* Is thirteen a one? - NO! Good guess actually! 79 00:03:46,419 --> 00:03:48,419 But it is not a one. So let's do thirteen. 80 00:03:48,419 --> 00:03:51,000 *Brady* This is good, I don't know then, I can't see the pattern. 81 00:03:51,000 --> 00:03:55,939 So as you see it's jumping by two each time, but then it resets at some point 82 00:03:55,939 --> 00:03:56,439 and 83 00:03:56,439 --> 00:03:58,439 And I want to go back to this, so we'll see what thirteen is quickly 84 00:04:04,219 --> 00:04:07,340 So we didn't reset that, and I claim we won't reset it the next one either. 85 00:04:07,340 --> 00:04:09,120 Fourteen will be thirteen 86 00:04:09,120 --> 00:04:11,300 and here, by the way, we're be starting to make predictions. 87 00:04:11,599 --> 00:04:12,680 It's worth noting we've- 88 00:04:12,680 --> 00:04:17,240 Even though we maybe can't say exactly what like 178 would be- 89 00:04:17,660 --> 00:04:20,560 - we're starting to see well enough so we can make a prediction. 90 00:04:20,759 --> 00:04:21,259 So... 91 00:04:21,259 --> 00:04:24,740 so you could guess. Is it really true that fourteen is going to give you thirteen? 92 00:04:24,839 --> 00:04:28,919 We could do it out, and you'll see in fact that's what you'll get 93 00:04:29,300 --> 00:04:30,939 So we're getting some understanding... 94 00:04:30,959 --> 00:04:35,419 But I want to go back to this point you had, you asked if it's going to reset and go back to one there 95 00:04:35,420 --> 00:04:38,980 And the answer was no, and it's worth looking. 96 00:04:39,100 --> 00:04:41,100 Because this is the next thing I was going to do. 97 00:04:41,279 --> 00:04:42,679 Where do we get the ones? 98 00:04:42,680 --> 00:04:47,439 So if you look for a sec, there's something very special about the numbers that give you ones. 99 00:04:47,800 --> 00:04:49,180 It's the powers of two. 100 00:04:49,180 --> 00:04:54,000 And so maybe you can now guess that if i put in a sixteen, I would get a one back. 101 00:04:54,220 --> 00:04:58,680 And that'll be right, and that's actually going to be a real key in unlocking this. 102 00:04:58,879 --> 00:05:02,560 The professor I was learning this from made a really big point out of this. 103 00:05:03,379 --> 00:05:05,159 And I thought, you know, well 104 00:05:05,160 --> 00:05:07,160 I don't know what the general answer is yet! 105 00:05:07,160 --> 00:05:09,500 And he made a big point: If you know something... 106 00:05:09,759 --> 00:05:14,259 Prove it, and make sure you really understand that one thing, and that often unlocks the rest. 107 00:05:14,819 --> 00:05:15,319 So 108 00:05:15,480 --> 00:05:21,540 So he really pushed us when I was first doing this, to try to state a conjecture and then prove a theorem- 109 00:05:21,740 --> 00:05:23,079 based on what we just saw here. 110 00:05:23,279 --> 00:05:25,279 And so that's what I want to do, so here's the conjecture: 111 00:05:25,279 --> 00:05:32,519 If n is a pure power of two, then the winning seat is one. 112 00:05:32,519 --> 00:05:36,079 I want to think about this for a second and maybe see why this might be true. 113 00:05:36,079 --> 00:05:40,339 So let's do the next biggest power of two, maybe the biggest one I can bother writing down. 114 00:05:40,339 --> 00:05:41,479 So let's do sixteen 115 00:05:41,480 --> 00:05:44,180 So I want to do one pass through the circle. 116 00:05:46,160 --> 00:05:48,980 - *Brady* Take out all the evens. - Take out all the evens. 117 00:05:49,620 --> 00:05:52,519 And now, fifteen has just killed sixteen. 118 00:05:52,519 --> 00:05:55,759 So I'll put a little dot here so we'll remember it's number one's turn again. 119 00:05:55,779 --> 00:06:00,279 And it's number one's turn again, and we've removed all the evens. 120 00:06:00,279 --> 00:06:03,099 And what's left is therefore exactly half as many. 121 00:06:03,100 --> 00:06:05,480 So if i relabel at this point 122 00:06:05,839 --> 00:06:08,759 You can see that there's a power of two to start with. 123 00:06:08,980 --> 00:06:12,939 I pass through the circle once, I get rid of all the evens, I have half as many. 124 00:06:12,939 --> 00:06:14,939 And I'm back at number one's turn. 125 00:06:14,959 --> 00:06:17,519 So now if I do this again, the same thing ought to happen. 126 00:06:17,519 --> 00:06:21,159 Right? I should kill all the evens on the new labeling 127 00:06:23,800 --> 00:06:27,460 And now there are four people left, and it's STILL number one's turn 128 00:06:27,779 --> 00:06:28,979 So you go through again... 129 00:06:29,759 --> 00:06:33,120 Kill those guys, there's two people left, it's still number one's turn- 130 00:06:33,120 --> 00:06:37,100 - and that one kills that, and it's still number one's turn and that chair wins. 131 00:06:37,100 --> 00:06:39,720 So let's say we believe that. We know what happens to two to the n. 132 00:06:39,720 --> 00:06:43,500 So how do we explain what happens between the pure powers of two? 133 00:06:43,699 --> 00:06:48,240 If you see the pattern, you know- we know what happens at eight, and we know what happens at four... 134 00:06:48,240 --> 00:06:54,240 ... but between them it goes up by twos until at this point, if I added two to seven, I'd have a nine 135 00:06:54,379 --> 00:06:57,339 Which would be too big anyway, so that's our reset 136 00:06:57,339 --> 00:06:59,899 But we knew it was going to reset because it's a pure power of two. 137 00:06:59,899 --> 00:07:03,199 So how do we explain this jumping by two phenomena inbetween? 138 00:07:03,459 --> 00:07:05,219 So what I'm going to do- I'm gonna just mention that: 139 00:07:05,220 --> 00:07:06,680 For any number- 140 00:07:06,879 --> 00:07:12,079 - you can write it as a big power of two plus something else 141 00:07:12,079 --> 00:07:14,699 Take the biggest power of two you can subtract from a number- 142 00:07:14,699 --> 00:07:16,699 And then what's left should be smaller than that. 143 00:07:16,699 --> 00:07:18,699 When you express a number in binary notation- 144 00:07:19,160 --> 00:07:23,439 - ou choose zeroes or ones, and that gives it out as a sum of a bunch of powers of two... 145 00:07:23,439 --> 00:07:25,199 ... and you just choose the biggest one. 146 00:07:25,199 --> 00:07:29,839 So 77. The biggest power of two I can get is 64. 147 00:07:29,839 --> 00:07:32,859 And then, what is it, it's 64 + 13. 148 00:07:32,860 --> 00:07:39,560 So then for 13, the biggest power of two is 8 + 4 + 1. Did I do this right? 149 00:07:40,480 --> 00:07:43,879 72... 77, there you go. And these are all powers of two! 150 00:07:43,879 --> 00:07:50,879 And this is unique. This is the unique way to write 77 as a sum of powers of two. 151 00:07:50,879 --> 00:07:53,560 Where no powers repeated, that's a key point. 152 00:07:53,560 --> 00:07:59,800 So if you wanted to take a general number, take the biggest power of two - 64 - and then the remaining part. 153 00:08:00,100 --> 00:08:01,379 *Mumble* Twelve... Thirteen. 154 00:08:01,379 --> 00:08:06,519 I'm going to call this part two to the a, and the remaining part I'm going to call l 155 00:08:06,519 --> 00:08:12,399 And I claim that this l is going to tell us which of those odd number between are going to show up. 156 00:08:12,399 --> 00:08:14,519 So let's see how. How about we do thirteen. 157 00:08:14,519 --> 00:08:16,419 n is thirteen. 158 00:08:16,560 --> 00:08:21,220 This is the binary expansion, but using our new method the eight will be the two to the a- 159 00:08:21,560 --> 00:08:24,439 - and the five (which is the four plus one) that'll be the l. 160 00:08:24,439 --> 00:08:26,439 And here's the thing that's going to happen with the l: 161 00:08:26,439 --> 00:08:28,439 I'm going to do five steps. 162 00:08:28,439 --> 00:08:35,720 So 1 kills 2, 3 kills 4, 5 kills 6, 7 kills 8, 9 kills 10. 163 00:08:35,720 --> 00:08:37,100 So now I've dropped... 164 00:08:38,379 --> 00:08:39,860 l people. 165 00:08:40,399 --> 00:08:42,019 And it's number eleven's turn. 166 00:08:42,019 --> 00:08:46,240 Now watch what happens, how many people are left? Well what's left is a power of two. 167 00:08:46,600 --> 00:08:49,840 And we know who wins in a power of two, it's whoever starts! 168 00:08:50,360 --> 00:08:51,080 *Brady* The first killer. 169 00:08:51,080 --> 00:08:54,859 The first killer! So now, if we go from here I claim that eleven is going to win. 170 00:08:55,059 --> 00:08:55,939 And just watch, it's going to be. 171 00:08:55,940 --> 00:09:00,219 11 kills 12, 13 kills 1, 3 kills 5, 7 kills 9 172 00:09:00,419 --> 00:09:03,179 Back to eleven, and now there's only four people left. 173 00:09:03,379 --> 00:09:05,340 11 kills 13, 3 kills 7... 174 00:09:05,340 --> 00:09:08,440 Back to eleven, two people... Eleven wins. 175 00:09:08,440 --> 00:09:11,080 And so this is really the key to the final answer. 176 00:09:11,360 --> 00:09:12,100 Which is 177 00:09:12,340 --> 00:09:15,219 If you've written your number as two to the a plus l- 178 00:09:15,419 --> 00:09:19,919 - after l steps, whosever turn it is is going to win. 179 00:09:19,919 --> 00:09:22,120 Because it's going to be their turn and there'll be a power of two left. 180 00:09:22,120 --> 00:09:26,519 And so the winning seat will be two l plus one. 181 00:09:26,519 --> 00:09:29,919 If you write it in this way, because that's whose turn it is after l steps. 182 00:09:29,919 --> 00:09:33,939 The theorem or the claim, is that if you've written n in this way... 183 00:09:33,940 --> 00:09:37,939 So if n is two to the a plus l, where l is less than two to the a. 184 00:09:38,139 --> 00:09:39,939 It has to be strictly smaller 185 00:09:39,940 --> 00:09:43,480 So in other words: two to the a is the biggest power that sat inside of n. 186 00:09:43,539 --> 00:09:47,120 Then the winning seat is going to be 2 l plus one. 187 00:09:47,120 --> 00:09:49,379 So we've already seen it was true here, right? 188 00:09:49,519 --> 00:09:54,220 l was five and the winning seat was eleven, which is two times five plus one. 189 00:09:54,220 --> 00:09:58,279 And if you start going back through you'll see it's the same thing for all the answers. 190 00:09:58,279 --> 00:10:02,779 We've already illustrated the mechanisms, which is after l steps- 191 00:10:03,039 --> 00:10:11,079 It's the turn of person 2 l plus one, and after l steps, there are a power of two number of people left. 192 00:10:11,259 --> 00:10:11,759 and... 193 00:10:12,039 --> 00:10:16,939 Everyone knows that the power of two people, the first person who kills, that's who's going to be the winner 194 00:10:16,940 --> 00:10:18,940 *Brady* Let's see if Brady has learned! 195 00:10:18,940 --> 00:10:20,560 *chuckle* Yeah! Alright! Let's do it 196 00:10:21,340 --> 00:10:24,339 *Brady* - I think I follow. Now in the Josephus problem... 197 00:10:24,539 --> 00:10:25,039 - Yep! 198 00:10:25,039 --> 00:10:27,699 - ... There was Josephus and 40 soldiers. 199 00:10:27,700 --> 00:10:29,360 - That's right. Oh yeah! We got to go back and do that! 200 00:10:29,360 --> 00:10:31,460 - Alright so we got 41 people... 201 00:10:31,460 --> 00:10:33,080 - So n is 41 202 00:10:33,080 --> 00:10:40,139 - Alright, now I think - expressing that in the way you told me to - it's going to be 32 + 9? 203 00:10:40,139 --> 00:10:42,460 - There you go, 32 + 9 204 00:10:42,960 --> 00:10:43,599 - So... 205 00:10:43,799 --> 00:10:46,319 Two lots of nine... plus one... 206 00:10:46,419 --> 00:10:47,059 - Mhm 207 00:10:47,460 --> 00:10:49,060 - ... is nineteen 208 00:10:51,139 --> 00:10:51,819 - There we go 209 00:10:51,820 --> 00:10:53,820 - So we want to sit in position nineteen. 210 00:10:53,820 --> 00:10:56,220 - There you go. You want to sit in the nineteenth chair. 211 00:10:56,360 --> 00:10:58,539 So here we go we're going to do n = 41 212 00:10:58,799 --> 00:11:01,699 - Alright. Put them in, in the cave, waiting their fate. 213 00:11:04,299 --> 00:11:05,919 - Not a very good circle but... 214 00:11:09,519 --> 00:11:10,460 - It's getting smaller now... 215 00:11:10,460 --> 00:11:11,820 - I know. 216 00:11:12,320 --> 00:11:13,860 *Brady and Daniel chuckles* 217 00:11:14,860 --> 00:11:15,480 *Daniel chuckles* 218 00:11:15,799 --> 00:11:17,059 Should I redo this? 219 00:11:17,059 --> 00:11:18,139 - Nah you're okay. 220 00:11:18,299 --> 00:11:19,539 *Daniel mumbles* 40... 41... 221 00:11:19,539 --> 00:11:21,659 - Okay there we are. So we got a little tight at the end 222 00:11:21,659 --> 00:11:24,959 - Look at that, this is a lession about planning ahead, people. 223 00:11:25,080 --> 00:11:27,080 Okay so let's do it. So one kills two... 224 00:11:29,440 --> 00:11:31,440 - We're losing our even numbers. 225 00:11:31,600 --> 00:11:32,279 - Yep. 226 00:11:32,779 --> 00:11:35,159 Okay now... 41 kills 1. 227 00:11:35,980 --> 00:11:37,500 *Daniel mumbles* three kills five... 228 00:11:50,559 --> 00:11:52,679 And 19 kills 35 and there we go! 229 00:11:53,120 --> 00:11:53,620 - Alright. 230 00:11:53,779 --> 00:11:54,500 I was right! 231 00:11:54,500 --> 00:11:55,860 - There we go, nineteen! 232 00:11:55,860 --> 00:11:58,820 Oh and one other thing in this problem which is kind of fun... 233 00:11:58,820 --> 00:12:02,220 So this formula I wrote can be interpreted in binary notation. 234 00:12:02,620 --> 00:12:05,759 When we wrote a number as a sum of powers of two... 235 00:12:05,759 --> 00:12:08,100 This can be re-expressed in binary notation. 236 00:12:08,100 --> 00:12:11,460 So what was this... 32 + 8 + 1 237 00:12:11,600 --> 00:12:12,220 So that's... 238 00:12:12,220 --> 00:12:15,719 2⁵ + 2³ + 2⁰ 239 00:12:15,919 --> 00:12:17,719 And binary notation... 240 00:12:17,720 --> 00:12:23,320 The digits correspond to the various powers of two, and all the digits are either zero or one. 241 00:12:23,320 --> 00:12:27,720 So 41 in binary notation would be one copy of 2⁵, 242 00:12:27,720 --> 00:12:29,340 zero of 2⁴, 243 00:12:29,340 --> 00:12:31,340 one of 2³, 244 00:12:31,840 --> 00:12:34,960 zero 2², zero 2¹ and one 2⁰ 245 00:12:35,039 --> 00:12:37,039 Here's the trick for the Josephus problem. 246 00:12:37,039 --> 00:12:40,879 Which I won't justify but it's super cool, and you can try it own your own. 247 00:12:40,879 --> 00:12:43,620 The way to find the winning solution if this is n- 248 00:12:44,019 --> 00:12:47,759 - the winning solution in binary is you take the leading digit- 249 00:12:48,159 --> 00:12:50,159 - and you put it at the end. 250 00:12:50,159 --> 00:12:53,159 So in other words I claim that if I write 251 00:12:53,159 --> 00:12:54,459 *Mumbles* Zero... One... 252 00:12:55,059 --> 00:12:57,819 So I take this part and then I add the first digit to the end. 253 00:12:57,820 --> 00:13:01,680 That number in binary code is... Well let's see... 254 00:13:01,879 --> 00:13:10,139 It's 2⁰ + 2¹ + (I skipped 2² and I skipped 2³) and I add 2⁴ 255 00:13:10,139 --> 00:13:14,039 So it's 2⁴ + 2¹ + 2⁰ 256 00:13:14,039 --> 00:13:15,419 Which is 257 00:13:15,419 --> 00:13:18,179 16 + 2 + 1 258 00:13:18,179 --> 00:13:21,279 Which is nineteen, which is exactly what the winning seat was. 259 00:13:21,500 --> 00:13:22,000 and... 260 00:13:22,000 --> 00:13:28,559 And so there's this super efficient way in binary code to jump straight from the number n- 261 00:13:28,580 --> 00:13:30,580 - to the winning number w of n. 262 00:13:30,580 --> 00:13:35,580 So if you're writing you numbers in binary code, the pattern would've been even more quickly apparent 263 00:13:35,840 --> 00:13:36,960 Isn't that cool?! 264 00:13:36,960 --> 00:13:38,079 *Brady agrees and chuckles* 265 00:13:38,279 --> 00:13:38,899 Isn't that cool? 266 00:13:38,899 --> 00:13:39,559 - Yeah 267 00:13:40,340 --> 00:13:45,100 This video was filmed at the Mathematical Sciences Research Institute (MSRI) 268 00:13:45,100 --> 00:13:46,460 That's the building behind me. 269 00:13:46,519 --> 00:13:49,779 If you'd like to find out more about them, link's in the video description. 270 00:13:49,779 --> 00:13:55,120 They do some really important serious mathematics here, and they're also a big supporter of math outrage. 271 00:13:55,759 --> 00:14:00,759 As evidenced by this video.