A group of people live on an island. They are all perfect logicians -- if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. If anyone has figured out the color of their own eyes, they [must] leave the island that midnight.

On this island live 100 blue-eyed people, 100 brown-eyed people, and the Guru. The Guru has green eyes, and does not know her own eye color either. Everyone on the island knows the rules and the properties stated above (except that they are not given the total numbers of each eye color) and is constantly aware of everyone else's eye color. Everyone keeps a constant count of the total number they see of each (excluding themselves). However, they cannot otherwise communicate. So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes, but that does not tell them their own eye color; it could be 101 brown and 99 blue. Or 100 brown, 99 blue, and the one could have red eyes.

The Guru speaks only once (let's say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following:

"I can see someone with blue eyes."

Who leaves the island, and on what night?


There are no mirrors or reflecting surfaces, nothing dumb, It is not a trick question, and the answer is logical. It doesn't depend on tricky wording, and it doesn't involve people doing something silly like creating a sign language or doing genetics. The Guru is not making eye contact with anyone in particular; she's simply saying "I count at least one blue-eyed person on this island who isn't me."

And lastly, the answer is not "no one leaves."

My wild guess would be, is it the Guru herself that leave the island?

Hm. I think on the 101st night, all blue eyed people would be gone, though maybe I'm missing something with the brown eyes.

I will assume that they all know that everyone else is a perfect logician.

The key is that you state that they can be make logical deductions instantly. This would imply that the fact that no one is taken away on the ferry is significant. That is, she states someone has blue eyes. By virtue of the fact that no one leaves the island on night one, the implication is that they can see someone with blue eyes. However, on night one hundred one, the implication that they can see 101 people with blue eyes is logically impossible (there are only 201 people you said, 101 are not blue-eyed, and they can't seen themselves), therefore they know their own eye color and must leave.

So it would stand that all the brown eyed people never leave, because there is the logical possibility that they could be something other than blue and brown eyed.

And my head hurts.

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You're on the right track, but that's still slightly off, because the brown eyed people have to leave as well.

Large numbers kill me

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"Who leaves the island, and on what night?"

Is it one (1) person who leave or all the blue eyed?
Would help a little to know that.

I stand by my earlier answer. All blue eyed will leave, but the brown eyed won't leave, because there remains enough untested logical possibilities (red and green) to keep them there.

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All the people will have to leave because as soon as he says that, someone will look at each blue eyed person and say or do something (such as point or just stare at) letting them know that they have blue eyes, and the brown eyed people observing this will know that they do not have blue eyes and must have brown eyes. Making all the people have to leave the island except the guru.

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No, I tried to explain that. They would know, when all the blue eyed people left, that they do not have blue eyes. However, they do not know anything beyond that. They could very well have periwinkle eyes. Thus, they stay.

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Hate to break it to you, but there is a logical answer, I've heard it, myself.

Hint: Instead of knowing that there is 100 blue eyed people, narrow it down to two.

The answer is hidden within another riddle: How many boards could the Mongols hoard if the Mongol hordes got bored?

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I don't understand what you mean by 'Board'

Board, like a wooden plank.

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Heh, I don't exactly see what you mean, but still, keep in mind there is a logical answer.

Go ahead and tell us. I get the simplification if there were 2 blue eyed people, but I don't see how that eliminates any other logical possibilities for those with non-blue eyes.

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Alrighty then.

It's quite simple. You have to start with a situation of 2 blue and 100 brown. This will happen:

Blue guy 1: Hey, I see 100 brown eyed people and 1 blue. Damn, then I can't be sure about my own eye color. I can't leave.

Blue guy 2: Hey, Blue guy 1 doesn't leave. That means he can't be sure about his own eye color. All the others have brown eyes so that must mean I have blue eyes, whoohoo!

And Blue Guy 2 leaves.

Blue Guy one however, can't know at this point what color his eyes are, since the blue guy who just left, may have been the only blue eyes person on the island. He'll have to wait for the Guru's announcement the next day. If she sees a blue eyes person again he can be sure it is him and he can leave. So 2 people leave after 2 days.

Same goes for 3 people.

Blue Guy 1 and 2: Hey, that guy over there has blue eyes but doesn't leave. That must mean at least one of us has blue eyes too!

Blue Guy 1: Damn, the other guy has blue eyes too, now I can't be sure about mine.

Blue Guy 2: Hey, the other guy doesn't know it either! That means I have blue eyes, yay!

And Blue Guy 2 leaves.

Blue Guy 1 and 3 both can't be sure about their own color now, and the next day the situation for 2 people follows.

After 3 days 3 people will have left.

Etc.. up to 100 people who will have left on the 100th day.

After that, it depends on what the Guru will say.

That's not the riddle you posted originally...
That explicitly states that she speaks exactly once in all their time on the island. Now, I agree with some of what you've said. Because blue guy 1 sees that blue guy 2 is still there after night 1, he knows that he has blue eyes. But, BG2 sees that BG1 doesn't leave, and he knows thus that he has blue eyes. This pattern will continue infinitely; for any number x of blue eyed people, all will leave on the xth night, all based on one proclamation.

So, as I stated, all 100 people will leave on the 100th day, and none before then.

You didn't explain at all why the brown eyed people would leave, since the Guru only speaks once, and that relates directly to blue eyes. After night x, all non-blue eyed people know only that they are not blue eyed. Thus, they are under no obligation to leave.

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What are you talking about? The Guru only NEEDS to talk once in order for them to to get off the island. Keeping in mind that they keep an exact memory of how many people have blue and brown eyes. So if all the blue eyes have left, the Guru only has to point to one brown eyed person and say "I see a person with brown eyes" - Therefore letting all the brown eyed folk realize that they have brown eyes. And since the Guru can keep an exact count of how many people have brown eyes, she can presume that she is the one with green eyes, and she can finally leave.

Therefore, no more people on the island.

Where did you say originally that she ever speaks again, saying "I see a person with brown eyes"? You said she only spoke once...

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I'm referring to the same time, here.