I see 99 blue. These 99 blue see either 98 or 99 blue. The 100 of us are all capable of thinking and knowing that: — Michael

You've gone wrong already.You see 99 blues. The blues that you see, all see 98 or 99 blues. The 200 of you are all thinking that.

The 100 of us do not need to wait for someone to say "I see blue" for us to think and know that (1) is true. — Michael

You can know that too. but you cannot apply it to your situation because no one has said anything.

So you can only get to "if 99 days have passed and no one has left and someone had said I see blue then I would know my eyes are blue."

But no one spoke so you don't know.

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. — flannel jesus

Ok, so one person knows all the other people on the island, and knows their eye color. The only one they don't know is themselves.

On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). — flannel jesus

Ok. So I'm assuming that this is taking place when there is this number of people on the island. This is VERY important as if the elder can speak at any time and the number of people would be different, then this is a different story. In fact, if they can change from this number, this initial number never should be mentioned and this needs to be restated more carefully.

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. — flannel jesus

Ok, so no one knows what color eyes another has, and apparently we can make up colors of eyes that people don't have like purple. Are you sure this is just not limited to green, blue, or brown?

Standing before the islanders, she says the following:

"I can see someone who has blue eyes." — flannel jesus

What is "The islanders?" Because you mentioned earlier that there were 100 with blue eyes and 100 with green eyes. "Islanders" implies "All of them". Is this poorly worded? Is it "At least one islander?"

As the initial premise is written, this is the set up.

A. There are 100 blue eyes, 100 brown eyes, and one green eyed elder.
B. However, the islanders do not know that this is the limit of eye color, and their eye colors could be any color under the rainbow. They also don't know the actual number. So even if they see 100 blue eyed individuals, they're own eye color could be blue or anything else.
C. The elder is speaking to all 200 other people on the island, and we're assuming he sees all 200 people, and says, "I see someone with blue eyes".

The only uncertainty that isn't listed here is how many people the elder saw while speaking to everyone. If its more than one person, there is no one who could know their eye color, as they don't know the total of blue eyes on the island. Therefore the only logical conclusion I can think of if we know someone does figure out their eye color, is if the elder is speaking to everyone, but one person is the only one being seen, and understands they are the only one being seen. That person could figure it out then and leave that night.

it genuinely seems like you're trying to be confused

You've gone wrong already.You see 99 blues. The blues that you see, all see 98 or 99 blues. The 200 of you are all thinking that. — unenlightened

I'm not wrong because I didn't say "only the 100 of us".

You can know that too. but you cannot apply it to your situation because no one has said anything. — unenlightened

We don't need someone to say something to apply it to our current situation. We all just need to know when we will all start counting, which will be the first possible "synchronisation" point — when everyone first locks eyes. At that moment, if it helps, they can pretend that someone says "I see blue and I see brown", even though nobody does.

And if everyone starts counting from the moment they first lock eyes, the 100 blue will leave on the 100th day knowing that their eyes are blue, the 100 brown will leave on the 100th day knowing that their eyes are brown, and the green will remain knowing that they are neither blue nor brown.

So I think the initial intuitive assumption that green saying "I see blue" shouldn't make a difference is correct, even though the initial intuitive assumption that nobody will ever leave is false. 200 people will leave on the 100th day knowing their eye colour without anyone having to say anything.

↪Philosophim it genuinely seems like you're trying to be confused — flannel jesus

Out of all the replies I didn't expect this one. Any good logic problem needs to be broken down and poked at carefully. If I appear confused while breaking down your logic problem, perhaps your logic problem isn't very straight forward and needs some refinement? Feel free to correct my points if I have misread anything.

A. There are 100 blue eyes, 100 brown eyes, and one green eyed elder.
B. However, the islanders do not know that this is the limit of eye color, and their eye colors could be any color under the rainbow. They also don't know the actual number. So even if they see 100 blue eyed individuals, they're own eye color could be blue or anything else.
C. The elder is speaking to all 200 other people on the island, and we're assuming he sees all 200 people, and says, "I see someone with blue eyes".

The only uncertainty that isn't listed here is how many people the elder saw while speaking to everyone — Philosophim

Your paragraph here shows you were pretty adept at getting over most of your self-inflicted confusions. You're right about the setup.

The elder saw all of them and was looking at everyone when she said it. Not any one person. Even while saying it, she knew and could see 100 blue eyed people.

ps I'm sorry for being rude. You getting confused about what "the islanders" meant or the eye colours just seemed... concocted. But if that was all genuine confusions, then rest easy knowing you got it all right.

We don't need someone to say something to apply it to our current situation. We all just need to know when we will all start counting, which will be the first possible "synchronisation" point — when everyone first locks eyes. — Michael

Yes you do need someone to say it because the first counterfactual needs someone to say it and every iteration thereafter rests on that necessity; you cannot discharge that assumption along the way.

What you are doing is inserting 'we all know we can all see blue' in to substitute for "x says 'I see blue'"

It doesn't work, precisely because this is the counterfactual situation in which the speaking is absolutely necessary because the hypothetical solitary blue does not see blue and has to be told in order to deduce their eye colour. This produces a contradiction that the hypothetical solitary blue cannot but does see blue, and cannot but does know their own eye colour.

We all know there are multiple blues.
If there was only one blue, that blue would not know there were multiple blues or any blues.
Therefore?

Yes you do need someone to say it because the first counterfactual needs someone to say it and every iteration thereafter rests on that necessity; you cannot discharge that assumption along the way. — unenlightened

That someone speaking is required in the counterfactual scenario isn't that someone speaking is required in the actual scenario.

You're making a false logical step.

Edit
See here which I think explains it best.

Original
It is a fact that if 100 browns, 100 blues, and 1 green lock eyes then if everyone immediately starts applying the counterfactuals and says "if the 98/99/100 blues I see don't leave on the 98/99/100th day then I am blue" and "if the 98/99/100 browns I see don't leave on the 98/99/100th day then I am brown" then the 100 browns and 100 blues will leave on the 100th day knowing their eye colour, and the 1 green will remain knowing that they are neither blue nor brown.

You might think that they shouldn't reason this way, but nonetheless if they do reason this way then (other than green) they will leave knowing their eye colour — and each of them knows from the start that reasoning this way will allow either 199 or 200 people to leave by the 101st day knowing their eye colour.

Not a worry, that's just how I break down things to make sure I understand the situation correctly.

The elder saw all of them and was looking at everyone when she said it. Not any one person. — flannel jesus

And when you say, "All of them" do you mean the 100 blue eyes and 100 brown eyed individuals on the island? Or could this be a variable number like there only being 1 besides the elder at the point the elder speaks?

all 200 people.

This puzzle isn't trying to trick you with wording. The most natural interpretation of that bit of the text is that ALL of the islanders are there, and that's how you should interpret it.

"Standing before the islanders" - no need to try to think of clever alternate ways of interpreting it, at face value "the islanders" means all of them.

↪Philosophim all 200 people. — flannel jesus

Thank you. Then logically taking only the information given, no one would be able to leave the island. All the elder has confirmed is that blue is a color of eye that at least someone has. Of course, everyone already knew that. It didn't need to be the elder that stated it, it could be anyone. "Someone" in the logical sense means "at least one".

At least, this is assuming there is no other outside information that is needed to know about eye color etc. Taking the problem verbatim with no outside knowledge needed, its impossible for anyone to determine the necessary logical conclusion about their own eye color. If you're concerned about posting the answer here, feel free to give me a direct message. I promise I won't reveal the answer.

It doesn't work, precisely because this is the counterfactual situation in which the speaking is absolutely necessary because the hypothetical solitary blue does not see blue and has to be told in order to deduce their eye colour. This produces a contradiction that the hypothetical solitary blue cannot but does see blue, and cannot but does know their own eye colour. — unenlightened

I'm not sure about this.
If we take as a premise that "everyone sees at least one blue", then the counterfactual still works: If there is one blue, he would leave on day one. As you pointed out, that the counterfactual is false is irrelevant.

What if the sage had said instead, "I see at least two blues"?

I'll work my way up to the answer.

Imagine instead that of the 200 people the guru was speaking to, 199 of them had brown eyes and 1 had blue eyes. The guru says "I see someone with blue eyes". What happens next? Can anybody leave then?

If we take as a premise that "everyone sees at least one blue", then the counterfactual still works: If there is one blue, he would leave on day one. As you pointed out, that the counterfactual is false is irrelevant. — hypercin

If there were only one blue, then it WOULDN'T be true that everyone sees at least one blue.

↪Philosophim I'll work my way up to the answer.

Imagine instead that of the 200 people the guru was speaking to, 199 of them had brown eyes and 1 had blue eyes. The guru says "I see someone with blue eyes". What happens next? Can anybody leave then? — flannel jesus

You agreed with me earlier here:

A. There are 100 blue eyes, 100 brown eyes, and one green eyed elder. — Philosophim

Are you now saying this was incorrect and that the number of people with different eye color could be different when the elder finally speaks?

I'm asking you to imagine something. Can you do that?

If there were only one blue, then it WOULDN'T be true that everyone sees at least one blue. — flannel jesus

A = Only one Blue
B = Everyone sees one blue
C = Blue leaves on first night

B
A -> ~B
A -> C

Still valid.

except for the fact that premise B wouldn't be true in that scenario. Are you just refusing to acknowledge that I said that?

If there wer eonly one blue, it wouldn't be true that everyone sees one blue. Right? Do you understand why that is?

↪Philosophim I'm asking you to imagine something. Can you do that? — flannel jesus

I'm asking you to clarify the rules. Imagining something that isn't in the rules is pointless if I'm unsure of the rules. Please clarify the rules as I noted, then I will gladly imagine it.

I acknowledge it. But not that it is relevant to the counterfactual logic.

I don't even understand what you're asking.

Your logic relies on the statement "everyone sees one blue". That's not true if there's one blue eyed person.

↪Philosophim I don't even understand what you're asking. — flannel jesus

I said in my summary that there are at the time of the elder speaking, 100 blue eyed, 100 brown eyed, and 1 green eyed elder. You said that summary was correct. You are now presenting a scenario in which there could be a different balance of eye color. Are you saying that the balance of eye color could be any variety at the time the elder speaks?

But there is NOT one blue eyed person. The logic just says, IF there is one blue eyed person, he would leave. He did not, therefore there is not one blue eyed person.

I'm asking you to imagine something. That's it. Either you can, or you can't. If you cannot imagine any different scenario than the one presented, then you will be incapable of understanding the logic of the solution.

Do you think that if there were only 2 blue eyed people, and the guru didn't say anything, they could leave on the second day?

no

But you think if there's 100 blue and 100 brown, and the guru says nothing, they can leave on the 100th day?