it's definitely a difficult, and contentious, problem. I think the official answer is correct but we've already got disagreements here. If you don't mind a spoiler, read unenlighteneds answer (which is more or less the canonical answer) and join the debate with us and Michael.

I don’t think that’s a comparable scenario. I think a minimal example requires 3 blue, 3 brown, and 1 green.

Each blue reasons: green sees blue, and so if the two blue I see don’t leave on the second day then I must be blue and green sees brown, and so if three brown I see don’t leave on the third day then I must be brown.

Therefore on the third day each blue knows they are blue and leaves

Each brown reasons: green sees brown, and so if the two brown I see don’t leave on the second day then I must be brown and green sees blue and so if the three blue I see don’t leave on the third day then I must be blue.

Therefore on the third day each brown knows they are brown and leaves.

Each person (other than green) leaves knowing their eye colour, all without anyone saying anything. The proof is in the pudding, as it were.

Or is it just a coincidence?

If you don't mind a spoiler, read unenlighteneds answer (which is more or less the canonical answer) and join the debate with us and Michael. — flannel jesus

Nope. I’m going to keep trying. Here is some more non-canonical answers.

One guy, I don’t know who, it might be @Baden, takes one of his eyes out, looks at it, and then leaves the island.

The guy who drives the ferry leaves the island every night.

I don’t think that’s a comparable scenario. I think a minimal example requires 3 blue, 3 brown, and 1 green. — Michael

If it doesn't matter what the green eyed person says, why is his presence required at all?

If it doesn't matter what the green eyed person says, why is his presence required at all? — flannel jesus

I’m not sure, but my reasoning does allow all brown and all blue to leave knowing their eye colour, so either it’s sound or it’s a very lucky coincidence.

But as a question to you, why would it require green verbally expressing what everyone already knows?

I’m not sure, but my reasoning does allow all brown and all blue to leave knowing their eye colour, so either it’s sound or it’s a very lucky coincidence — Michael

I don't think it's sound or a coincidence. I don't think it's correct. I don't think there's any reason why the green eyed person being there, not saying anything, would allow anybody to decide their eye colour, and you haven't explained why there would be.

But as a question to you, why would it require green saying what everyone already knows? — Michael

You have to follow the logic carefully one step at a time to find that out. It's very subtle and honestly strange - that what makes this such a good logic puzzle. It's completely counterintuitive, but also, once you fully grok it, undeniably true. That gives it this really unique flavour as a puzzle.

I don't think it's sound or a coincidence. I don't think it's correct. — flannel jesus

And yet every blue-eyed person leaves knowing they have blue eyes and every brown-eyed person leaves knowing they have brown eyes. So what do you mean by it “not being correct”?

You have to follow the logic carefully one step at a time to find that out. It's very subtle and honestly strange - that what makes this such a good logic puzzle. It's completely counterintuitive, but also, once you fully grok it, undeniably true. That gives it this really unique flavour as a puzzle. — flannel jesus

And I think that it goes even further: it may be counterintuitive, but one can get to the correct answer without green saying anything.

One guy, I don’t know who, it might be Baden, takes one of his eyes out, looks at it, and then leaves the island. — T Clark

Clever. Obviously everyone could do that to their own eye. That's another loophole answer though - the real answer doesn't involve a loophole

And yet every blue-eyed person leaves knowing they have blue eyes and every brown-eyed person leaves knowing they have brown eyes. So what do you mean by it “not being correct”? — Michael

You're saying "and yet" as if you've demonstrated that. You haven't

You're saying "and yet" as if you've demonstrated that. You haven't — flannel jesus

I demonstrated it in the example above.

there's many posts above. Which one is the one that demonstrates that?

The one with 3 brown, 3 blue, and 1 green

if it's this one, then the reasoning is incomplete. Why is the green person relevant? "Green sees blue" so what? So does brown. Skip the inclusion of the green person, each blue eyed person could apply the same logic but swap in a brown eyed person for the green eyed person.

So why doesn't your example just include 3 blue 3 brown?

the reasoning is incomplete — flannel jesus

How can it be incomplete it it allows all brown and blue to leave knowing their eye colour?

you haven't justified that it does, is why. You keep begging the question.

I do not believe it allows that. Do you understand that? I don't think it does. I don't think it actually allows anyone to leave.

you haven't justified that it does, is why — flannel jesus

I explained the reasoning that each person performs and the conclusion they draw from it; a conclusion that is correct.

I don’t understand what else you’re looking for.

well I'm trying to talk to you about it but you have to actually engage lol.

What does the green eyed person have to do with it? Why not just have 3 blue 3 brown?

As I said, I’m not sure. But it appears to be a fact that if the blue-eyed people reason in such a way then they correctly deduce that they have blue eyes and that if the brown-eyed people reason in such a way then they correctly deduce that they have brown eyes.

Therefore either the reasoning is sound or it’s a coincidence.

Just as you don’t appear to be sure why green must verbally express what everyone already knows to be true.

your reasoning is still based on nothing other than unenlighteneds reasoning, and he's already told you his reasoning is based on the guru saying something. When are you going to apply your own reasoning?

"I don't know" isn't reasoning. Why is the green eyed person relevant in your scenario? If he's not saying anything? For every blue eyed person who relies on the green eyed person, couldn't they just as easily rely on a brown eyed person? For every brown eyed person who relies on the green eyed person, can't they just as easily rely on a blue eyed person? I don't think the green eyed person is doing anything.

Clever. Obviously everyone could do that to their own eye. That's another loophole answer though - the real answer doesn't involve a loophole — flannel jesus

I’ve given up on being correct. I’m working on being amusing.

simplify the question as asked, instead of 100 blue eyes 100 brown eyes, think about the same scenario but 2 blue eyes 2 brown eyes

your reasoning is still based on nothing other than unenlighteneds reasoning, and he's already told you his reasoning is based on the guru saying something. — flannel jesus

And as I have repeatedly explained, it doesn’t actually require the Guru to say anything. It’s a red herring. It might appear to be necessary, but counterintuitively it isn’t.

As evidenced by the fact that my reasoning allows all blues and browns to correctly deduce their eye colour and leave on the 100th day, answering the question in the OP.

so are you going to answer why the green eyed person, who doesn't say anything, is relevant?

I already answered. I don’t know. But the reasoning nonetheless allows all blues and browns to correctly deduce their eye colour and leave on the 100th day, answering the question in the OP.

it doesn't though. You're just insisting it does, but you aren't starting from a reasonable place. There's no reason in your scenario that anybody could figure out their own eye colour.

I could justify unenlighteneds reasoning with actual complete syllogisms. I don't think you can do so for yours.

There's no reason in your scenario that anybody could figure out their own eye colour. — flannel jesus

It’s explained in the post.

just because you think you've explained something doesn't make it correct. I've had a lot of wrong ideas explained to me in my life.

You're still basing your reasoning off the words of a guy who explicitly has different premises from you. Until you stand on your own two feet with your reasoning, your explanations are on shaky ground

Each blue reasons: green sees blue, and so if the two blue I see don’t leave on the second day then I must be blue — Michael

Like this for example. Why do you think this is true? If green eyed person says nothing, what reason would the two blue have to leave on the second day? Without resting on the coattails of unenlightened, why would this be the case?

If green eyed person says nothing, what reason would the two blue have to leave on the second day? Without resting on the coattails of unenlightened — flannel jesus

It's the same reasoning.

Just as we can stipulate some hypothetical in which I don't see anyone with blue eyes, even though "in reality" I do, we can stipulate some hypothetical in which green says "I see blue" (and so I can know that she sees blue), even though "in reality" she doesn't.

And we can do this because we know "in reality" that green sees blue even if she doesn't say so.

If you want it as a step-by-step argument:

P1. Green sees blue
P2. Therefore, if I don't see blue then I must be blue
P3. Therefore, if I see one blue and he leaves on the first day then I must not be blue
P4. Therefore, if I see one blue and he doesn't leave on the first day then I must be blue
etc.

I know that P1 is true because I see 100 blue and I know that green can see them too, I know that the antecedent of P2 is false because I see 100 blue, I know that the antecedent of P3 is false because I see 100 blue, etc.

This reasoning doesn't require green to actually say "I see blue" and will allow me to correctly deduce my eye colour.

Therefore either the reasoning is sound or the correct "deduction" is a coincidence.