0:00:06.629,0:00:09.070 Imagine an island where 100 people, 0:00:09.070,0:00:14.150 all perfect logicians,[br]are imprisoned by a mad dictator. 0:00:14.150,0:00:18.291 There's no escape,[br]except for one strange rule. 0:00:18.291,0:00:23.051 Any prisoner can approach the guards[br]at night and ask to leave. 0:00:23.051,0:00:26.392 If they have green eyes, [br]they'll be released. 0:00:26.392,0:00:30.342 If not, they'll be tossed [br]into the volcano. 0:00:30.342,0:00:34.103 As it happens, [br]all 100 prisoners have green eyes, 0:00:34.103,0:00:36.676 but they've lived there since birth, 0:00:36.676,0:00:41.491 and the dictator has ensured[br]they can't learn their own eye color. 0:00:41.491,0:00:43.446 There are no reflective surfaces, 0:00:43.446,0:00:46.155 all water is in opaque containers, 0:00:46.155,0:00:47.663 and most importantly, 0:00:47.663,0:00:50.780 they're not allowed [br]to communicate among themselves. 0:00:50.780,0:00:54.754 Though they do see each other[br]during each morning's head count. 0:00:54.754,0:00:59.196 Nevertheless, they all know no one would[br]ever risk trying to leave 0:00:59.196,0:01:02.607 without absolute certainty of success. 0:01:02.607,0:01:05.008 After much pressure [br]from human rights groups, 0:01:05.008,0:01:09.048 the dictator reluctantly agrees[br]to let you visit the island 0:01:09.048,0:01:12.465 and speak to the prisoners[br]under the following conditions: 0:01:12.465,0:01:14.549 you may only make one statement, 0:01:14.549,0:01:17.858 and you cannot tell them [br]any new information. 0:01:17.858,0:01:20.582 What can you say [br]to help free the prisoners 0:01:20.582,0:01:24.246 without incurring the dictator's wrath? 0:01:24.246,0:01:25.979 After thinking long and hard, 0:01:25.979,0:01:31.381 you tell the crowd, [br]"At least one of you has green eyes." 0:01:31.381,0:01:33.269 The dictator is suspicious 0:01:33.269,0:01:37.652 but reassures himself that your statement[br]couldn't have changed anything. 0:01:37.652,0:01:42.401 You leave, and life on the island[br]seems to go on as before. 0:01:42.401,0:01:44.656 But on the hundredth morning [br]after your visit, 0:01:44.656,0:01:47.183 all the prisoners are gone, 0:01:47.183,0:01:50.881 each having asked to leave[br]the previous night. 0:01:50.881,0:01:53.933 So how did you outsmart the dictator? 0:01:53.933,0:01:59.108 It might help to realize that the amount[br]of prisoners is arbitrary. 0:01:59.108,0:02:04.178 Let's simplify things[br]by imagining just two, Adria and Bill. 0:02:04.178,0:02:06.498 Each sees one person with green eyes, 0:02:06.498,0:02:09.744 and for all they know, [br]that could be the only one. 0:02:09.744,0:02:12.310 For the first night, each stays put. 0:02:12.310,0:02:15.231 But when they see each other [br]still there in the morning, 0:02:15.231,0:02:17.641 they gain new information. 0:02:17.641,0:02:22.268 Adria realizes that if Bill had seen[br]a non-green-eyed person next to him, 0:02:22.268,0:02:24.230 he would have left the first night 0:02:24.230,0:02:28.209 after concluding the statement [br]could only refer to himself. 0:02:28.209,0:02:32.321 Bill simultaneously realizes [br]the same thing about Adria. 0:02:32.321,0:02:34.340 The fact that the other person waited 0:02:34.340,0:02:39.124 tells each prisoner his[br]or her own eyes must be green. 0:02:39.124,0:02:42.282 And on the second morning, [br]they're both gone. 0:02:42.282,0:02:44.526 Now imagine a third prisoner. 0:02:44.526,0:02:48.795 Adria, Bill and Carl each see [br]two green-eyed people, 0:02:48.795,0:02:53.905 but aren't sure if each of the others[br]is also seeing two green-eyed people, 0:02:53.905,0:02:55.512 or just one. 0:02:55.512,0:02:57.821 They wait out the first night as before, 0:02:57.821,0:03:01.072 but the next morning, [br]they still can't be sure. 0:03:01.072,0:03:03.801 Carl thinks, "If I have non-green eyes, 0:03:03.801,0:03:06.977 Adria and Bill were just [br]watching each other, 0:03:06.977,0:03:09.703 and will now both leave [br]on the second night." 0:03:09.703,0:03:12.468 But when he sees both [br]of them the third morning, 0:03:12.468,0:03:16.226 he realizes they must [br]have been watching him, too. 0:03:16.226,0:03:19.284 Adria and Bill have each [br]been going through the same process, 0:03:19.284,0:03:22.295 and they all leave on the third night. 0:03:22.295,0:03:24.255 Using this sort of inductive reasoning, 0:03:24.255,0:03:28.883 we can see that the pattern will repeat[br]no matter how many prisoners you add. 0:03:28.883,0:03:31.900 The key is the concept [br]of common knowledge, 0:03:31.900,0:03:34.539 coined by philosopher David Lewis. 0:03:34.539,0:03:38.935 The new information was not contained[br]in your statement itself, 0:03:38.935,0:03:42.953 but in telling it to everyone [br]simultaneously. 0:03:42.953,0:03:46.680 Now, besides knowing at least one[br]of them has green eyes, 0:03:46.680,0:03:51.090 each prisoner also knows [br]that everyone else is keeping track 0:03:51.090,0:03:54.321 of all the green-eyed people they can see, 0:03:54.321,0:03:58.787 and that each of them [br]also knows this, and so on. 0:03:58.787,0:04:00.839 What any given prisoner doesn't know 0:04:00.839,0:04:04.316 is whether they themselves are one[br]of the green-eyed people 0:04:04.316,0:04:06.431 the others are keeping track of 0:04:06.431,0:04:12.571 until as many nights have passed[br]as the number of prisoners on the island. 0:04:12.571,0:04:17.098 Of course, you could have spared[br]the prisoners 98 days on the island 0:04:17.098,0:04:21.491 by telling them at least 99 of you[br]have green eyes, 0:04:21.491,0:04:26.251 but when mad dictators are involved,[br]you're best off with a good headstart.