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  • Imagine an island where 100 people,

  • all perfect logicians, are imprisoned by a mad dictator.

  • There's no escape, except for one strange rule.

  • Any prisoner can approach the guards at night and ask to leave.

  • If they have green eyes, they'll be released.

  • If not, they'll be tossed into the volcano.

  • As it happens, all 100 prisoners have green eyes,

  • but they've lived there since birth,

  • and the dictator has ensured they can't learn their own eye color.

  • There are no reflective surfaces,

  • all water is in opaque containers,

  • and most importantly,

  • they're not allowed to communicate among themselves.

  • Though they do see each other during each morning's head count.

  • Nevertheless, they all know no one would ever risk trying to leave

  • without absolute certainty of success.

  • After much pressure from human rights groups,

  • the dictator reluctantly agrees to let you visit the island

  • and speak to the prisoners under the following conditions:

  • you may only make one statement,

  • and you cannot tell them any new information.

  • What can you say to help free the prisoners

  • without incurring the dictator's wrath?

  • After thinking long and hard,

  • you tell the crowd, "At least one of you has green eyes."

  • The dictator is suspicious

  • but reassures himself that your statement couldn't have changed anything.

  • You leave, and life on the island seems to go on as before.

  • But on the hundredth morning after your visit,

  • all the prisoners are gone,

  • each having asked to leave the previous night.

  • So how did you outsmart the dictator?

  • It might help to realize that the amount of prisoners is arbitrary.

  • Let's simplify things by imagining just two, Adria and Bill.

  • Each sees one person with green eyes,

  • and for all they know, that could be the only one.

  • For the first night, each stays put.

  • But when they see each other still there in the morning,

  • they gain new information.

  • Adria realizes that if Bill had seen a non-green-eyed person next to him,

  • he would have left the first night

  • after concluding the statement could only refer to himself.

  • Bill simultaneously realizes the same thing about Adria.

  • The fact that the other person waited

  • tells each prisoner his or her own eyes must be green.

  • And on the second morning, they're both gone.

  • Now imagine a third prisoner.

  • Adria, Bill and Carl each see two green-eyed people,

  • but aren't sure if each of the others is also seeing two green-eyed people,

  • or just one.

  • They wait out the first night as before,

  • but the next morning, they still can't be sure.

  • Carl thinks, "If I have non-green eyes,

  • Adria and Bill were just watching each other,

  • and will now both leave on the second night."

  • But when he sees both of them the third morning,

  • he realizes they must have been watching him, too.

  • Adria and Bill have each been going through the same process,

  • and they all leave on the third night.

  • Using this sort of inductive reasoning,

  • we can see that the pattern will repeat no matter how many prisoners you add.

  • The key is the concept of common knowledge,

  • coined by philosopher David Lewis.

  • The new information was not contained in your statement itself,

  • but in telling it to everyone simultaneously.

  • Now, besides knowing at least one of them has green eyes,

  • each prisoner also knows that everyone else is keeping track

  • of all the green-eyed people they can see,

  • and that each of them also knows this, and so on.

  • What any given prisoner doesn't know

  • is whether they themselves are one of the green-eyed people

  • the others are keeping track of

  • until as many nights have passed as the number of prisoners on the island.

  • Of course, you could have spared the prisoners 98 days on the island

  • by telling them at least 99 of you have green eyes,

  • but when mad dictators are involved, you're best off with a good headstart.

Imagine an island where 100 people,

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A2 TED-Ed green dictator eyed prisoner island

【TED-Ed】The famously difficult green-eyed logic puzzle - Alex Gendler

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    稲葉白兎 posted on 2015/07/09
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