Can you solve the famously difficult green-eyed logic puzzle? – Alex Gendler

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.


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