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Imagine an island where 100 people,

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all perfect logicians,
are imprisoned by a mad dictator.

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

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

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

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

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As it happens, 
all 100 prisoners have green eyes,

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but they've lived there since birth,

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

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There are no reflective surfaces,

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all water is in opaque containers,

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and most importantly,

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they're not allowed 
to communicate among themselves.

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

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

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without absolute certainty of success.

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After much pressure 
from human rights groups,

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the dictator reluctantly agrees
to let you visit the island

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and speak to the prisoners
under the following conditions:

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you may only make one statement,

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and you cannot tell them 
any new information.

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What can you say 
to help free the prisoners

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without incurring the dictator's wrath?

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After thinking long and hard,

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

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The dictator is suspicious

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

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

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But on the hundredth morning 
after your visit,

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all the prisoners are gone,

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each having asked to leave
the previous night.

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So how did you outsmart the dictator?

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

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

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Each sees one person with green eyes,

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

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For the first night, each stays put.

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

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they gain new information.

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

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he would have left the first night

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after concluding the statement 
could only refer to himself.

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Bill simultaneously realizes 
the same thing about Adria.

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The fact that the other person waited

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

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

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Now imagine a third prisoner.

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

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

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or just one.

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They wait out the first night as before,

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

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

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Adria and Bill were just 
watching each other,

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and will now both leave 
on the second night."

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But when he sees both 
of them the third morning,

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he realizes they must 
have been watching him, too.

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

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and they all leave on the third night.

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Using this sort of inductive reasoning,

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

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The key is the concept 
of common knowledge,

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coined by philosopher David Lewis.

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The new information was not contained
in your statement itself,

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but in telling it to everyone 
simultaneously.

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

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each prisoner also knows 
that everyone else is keeping track

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of all the green-eyed people they can see,

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

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What any given prisoner doesn't know

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

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the others are keeping track of

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

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

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

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