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