This puzzle was introduced on this forum some years ago, but I think most of the current posters haven’t seen it, so here we go again:
A group of people with assorted eye colors 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. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.
On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.
The Guru is allowed to speak 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 who has 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 or anyone lying or guessing, 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.”
That was a great puzzle - didn’t get it at first, started to get it, took longer to wrap my head around it, and still not sure I can explain it succinctly.
For any natural number N of blue-eyed people, they will all leave on precisely the Nth ferry ride after the Guru’s pronouncement. Everybody else will be there forever. I don’t think it makes any difference if there are any brown-eyed people or purple-eyed people or how many there are. I’m not 100% on what would happen if the Guru had blue eyes, but probably not much would change.
They need to know they have blue eyes first, in order to leave though?
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I think I’ve got it.. I’ll reply to @Flannel_Jesus first, rather than [potentially] spoil-alert.
I did figure that only the blue-eyed people could ever leave the island. Even after using ChatGPT’s “Think longer” (“Reason”) tool and Google Gemini 2.5 Pro—and once again preferring the latter—, I still think there’s something logically flawed about the Guru’s pronouncement, though. Any scenario where N is greater than 1 relies on the then-hypothetical scenario where N = 1. And what’s ‘especially hypothetical’ about any scenario other than the actual one—in this case, the one in which N = 100—is that the Guru would then make the same pronouncement…
I meant “about” in a broader sense. But yeah, it’s illogical to say something everyone already knows everyone already knows. It only makes sense in the hypothetical scenario where there are only one or two blue-eyed people. (Not just a single one, though; I was mistaken about that). Apparently the pronouncement is meant to suggest the Guru would make the same pronouncement if there were only one or two blue-eyed people; but that’s logical, so the suggestion’s unnecessary and therefore illogical. In fact, the Guru logically shouldn’t pronounce it in the scenario where there are three! For then every blue-eyed person will immediately understand there’s three of them, not two, and leave the island that very night…
‘Originally’, I added ‘etc. etc.’ to the phrase ‘everyone already knows everyone already knows’; I just took that out again. For I’ve kept coming back to the impression that’s where the catch is. In the scenario where there are three blue-eyed people, for example, not everyone knows everyone knows everyone knows there’s at least one blue-eyed person. And with each added blue-eyed person, we can add one repetition of the phrase ‘everyone knows’.
I think it’s logically illogical to say or do truly useless things (for if saying or doing something ‘useless’ gives you release or something, it’s not truly useless). However, in the meantime I see how it’s not useless for any N.
You know it as soon as the Guru hasn’t made her once-in-a-lifetime pronouncement at noon. Or at least that would have been the case had I been right to stop at ‘everyone already knows everyone already knows’.
It’s not useless though, as it turns out. What this puzzle demonstrates is that, counterintuitively, saying something that everyone knows everyone knows can still have consequences.
I’ve been working on an explanation, I think I might have a simple way to explain it.
You’re ignoring the ‘Or at least that would have been the case [if]’ part.
I think I’ve found a way the puzzle’s still illogical, by the way. For why doesn’t the Guru say she can see someone who has brown eyes? The only logical reason would be that she found a way to choose blue over brown completely at random, which however is logically impossible: for that would contradict the rule of non-contradiction, A then being just as well able to be not-A.
To me the funniest part of the puzzle is a real person would see 100 brown eyes and 99 blue eyes and go “Well I’m probably the 100th blue-eyed, why shouldn’t the numbers be equal?”
That’s not spelling out the reasoning. That’s a completely opaque thing to quote.
If you’re saying “I, lightening, was wrong, I couldn’t figure it out”, then spelling it out is saying that explicitly. Is that what you’re saying? You were wrong?
If you’re not saying you were wrong, then what’s the explicit reasoning?
And I have no idea what you’re trying to say with your second hidden text either. Originally? Etc etc? What are you talking about? You’re speaking in incomplete thoughts. I can’t make heads or tails of what any of that means. Can’t you just start from the beginning and spell out your actual position and reasoning?
‘Had I been right’ is not tantamount to ‘Had I not been wrong’?
I edited the post where I first used the phrase ‘everyone already knows everyone already knows’. While writing that post, I initially left it at that, but then added ‘etc. etc.’ before posting it.
I think that when you take away all the riddles lightening was communicating in and just spell it out, he’s saying he was wrong - he now agrees that three blue eyed people couldn’t leave on the first day.
I agree, that’s wrong.
But if there were only three they could leave on the third day if the guru says the supposedly illogical thing she said, “I see someone with blue eyes”.
