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[GendoIkari]

Inspired by probability paradoxes... I'll start with my favorite:

On an island where 200 people live, there are 100 people with blue eyes, and 100 people with brown eyes. The people on this island have no mirrors, and are not allowed to discuss eye color to find out what color their own eyes are. They live by a code that says that if anyone ever discovered his own eye color, he must leave the island at dawn the morning after he makes the discovery. Also, everyone on the island is great at logic, if a conclusion can be logically deduced, he will do it.

One day, a stranger shows up, and after looking around he announces "I can see someone with blue eyes."

What effect does this have on the inhabitants? Who leaves the island when?

[Watno]

Do people on the island know that there are 100 of each eye color?
Do people on the island see each other?

[GendoIkari]

The do not know how many of each there are... they do all see each other. So a blue-eyed person will know that there are at least 100 brown-eyed people and at least 99 blue-eyed people, but he won't know if his own eyes make it 100-100 or 101-99.

[Watno]

And the stranger sees all the people?

[GendoIkari]

Yes, he makes his announcement in front of everyone.

[Davio]

Can the inhabitants discuss the number of eye colors?

Like can one say to the other "I see 100 brown eyed persons", without uniquely identifying one?

[Jimmmmm]

The stranger must leave, since he said "I can see someone with (my) blue eyes", indicating he knows that his eyes are blue?

[GendoIkari]

Can the inhabitants discuss the number of eye colors?

Like can one say to the other "I see 100 brown eyed persons", without uniquely identifying one?

No. That would make it pretty trivial.

[GendoIkari]

The stranger must leave, since he said "I can see someone with (my) blue eyes", indicating he knows that his eyes are blue?

No... the stranger doesn't live by their rules... I should have clarified that the stranger's eye color is irrelevant... assume his eyes are green.

[Jimmmmm]

The stranger must leave, since he said "I can see someone with (my) blue eyes", indicating he knows that his eyes are blue?

No... the stranger doesn't live by their rules... I should have clarified that the stranger's eye color is irrelevant... assume his eyes are green.

Yeah, I hoped that wasn't it.

[bozzball]

The blue-eyed islanders leave the island on the 100th day, with the brown-eyed ones joining them the next day.

[Jimmmmm]

The only blue-eyed guy who forgot to wear his sunnies?

Yeah, I'm stumped.

[bozzball]

The blue-eyed islanders leave the island on the 100th day, with the brown-eyed ones joining them the next day.

The second part of the sentence is only true assuming that everybody knows that their eyes are either blue or brown.

[GendoIkari]

The blue-eyed islanders leave the island on the 100th day, with the brown-eyed ones joining them the next day.

The second part of the sentence is only true assuming that everybody knows that their eyes are either blue or brown.

For those who want to spoil the answer for themselves, bozzball got it perfectly! Have you seen this before, or did you figure it out?

[Ozle]

If they are great at Logic, why cant someone just go round and count the number of brown eyed people. if he gets to 100 then he knows he has blue eyes?

No need for strangers at all....

[Reyk]

For those who want to spoil the answer for themselves, bozzball got it perfectly! Have you seen this before, or did you figure it out?

It's very beautiful and I couldn't figure it out. As a (strong) hint for others:
Assume there are 198 brown eyed people and only 2 blue eyed rather than 100/100.

[Watno]

I had a reason of thought that led me in similar direction, but I don't see how the stranger is relevant. Everyone knows he sees a blue eyed person, because they see one themselves. So, how does he provide any additional information?

[Reyk]

If they are great at Logic, why cant someone just go round and count the number of brown eyed people. if he gets to 100 then he knows he has blue eyes?

No need for strangers at all....

because ...

The do not know how many of each there are... they do all see each other. So a blue-eyed person will know that there are at least 100 brown-eyed people and at least 99 blue-eyed people, but he won't know if his own eyes make it 100-100 or 101-99.

[soulnet]

Only homosexuals (and maybe asexuals?) would remain in the Island for more than a couple of days, given that there are only males on it. Actually, I guess almost every male in the world enjoy some form of interaction with females, so probably everyone will eventually leave if they can.

On a more serious note, I've heard an equivalent problem more tha 10 years ago, non-equivalent statement and answer, but the same reasoning behind the solution, so I'm going to guess that it is an old problem, and probably is recorded in a couple of books.

About the stranger being relevant
They know there is someone with blue eyes, and they also know everyone knows that, and they also know everyone knows everyone knows that, but blue eyed people do not not (everyone knows)⁹⁹ that, because you need to see at least 100 people to be sure of that, and blue eyed people only see 99

[Watno]

I think I see it now.

[Reyk]

I had a reason of thought that led me in similar direction, but I don't see how the stranger is relevant. Everyone knows he sees a blue eyed person, because they see one themselves. So, how does he provide any additional information?

The stranger is relevant because know everybody knows that everybody knows there is a blue eyed person.

[Ozle]

http://xkcd.com/blue_eyes.html

[GendoIkari]

I had a reason of thought that led me in similar direction, but I don't see how the stranger is relevant. Everyone knows he sees a blue eyed person, because they see one themselves. So, how does he provide any additional information?

It took me a really, really long time to accept the fact that the stranger's announcement could possibly change anything / realize why it's so. Yes, everyone on the island knows that that is at least 1 blue-eyed person, because they all see a blue-eyed person. In addition, everyone knows that everyone else knows that there is a blue-eyed person, because they see at least 2 blue-eyed people, so each other person must also see at least 1 blue-eyed person as well. However, before the stranger, not every one knew that everyone knew that everyone knew that everyone knew that everyone knew ... (repeat to the 100th level)... that there's at least 1 blue-eyed person. The stranger's announcement turns it into fully common knowledge. It's easier to see if you think of the case of only 3 people with blue eyes and 3 people with brown eyes.

[Ozle]

Nope, still dont getit.

Start with 3 people with blue and 3 with green.
B1
B2
B3
G1
G2
G3

Now obviously they can all see each other, and the Blues can see 2 other blues.

So when the stranger turns up and says I can see someone with blue eyes.
If I am B1, he can possibly be talking about B2, B3 or even me. But also i know that B3 can see B2 (and vice versa), so they could assume the stranger is talking about the other one.

[Reyk]

Nope, still dont getit.

Start with 3 people with blue and 3 with green.
B1
B2
B3
G1
G2
G3

Now obviously they can all see each other, and the Blues can see 2 other blues.

So when the stranger turns up and says I can see someone with blue eyes.
If I am B1, he can possibly be talking about B2, B3 or even me. But also i know that B3 can see B2 (and vice versa), so they could assume the stranger is talking about the other one.

Yes, they can assume this in that moment, when the stranger is speaking, but only until day three.