[andrew1]

OK, so we agree that if there are X <= 3 people then they're stuck and cannot kill themselves.

You claim that if X = 4 and everyone has brown eyes then on day 4 they will all kill themselves. Suppose I am one of these islanders. I can see three people with brown eyes. On days one, two and three no one kills themselves. On day four I get up and kill myself because I know that I have brown eyes.

Fine, now let's consider the case where there are three people with brown eyes, and one person with green eyes:

As I think we've agreed above, the three people with brown eyes cannot infer that they have brown eyes (i.e. the X < 3 case). So they are not able to kill themselves.

But consider the green eyed person: he can see three people with brown eyes and no one kills themselves on days one, two or three.

He is in exactly the same situation at this point as the brown eyed person we considered in the four-brown-eyed-people case. So at this point, by your logic, he must know that he has brown eyes. Which is a contradiction.

I think you are wrong that induction is not involved in establishing eye colour. (unless you can convince me in the X = 4 case that is :) ).

This isn't a problem I'm unfamiliar with it, a colleague asked me it when I was being interviewed for my current job, and it gets rediscussed periodically. I really do think you're wrong I'm afraid.

[ErrantX]

Umm, we defined X as the number of people with brown eyes. So on an island with 4 people, three of whom have brown eyes, then X=3 and, yep, there is a problem.

Where X=4 (i.e. there are 4 people with brown eyes) it works.

In your case, where the number of people is 4, but X=3, then the fourth possibly incorrectly infers that he has brown eyes and so, on his own, kills himself. On the other hand they are "highly logical" so I argue they would realise that there were too few people to know.

[andrew1]

But that's precisely the point, if they're all "highly logical" then in the island-wtih-3-brown-and-1-green case the person with green eyes can't correctly know their eye colour so can't kill themself. But they are in exactly the same situation as a person with brown eyes in the island-with-four-brown-people case. So if the green eyed person in the first case can't deduce their eye colour, then neither can the brown eyed person in the second case.

If you don't agree, please explain to me what extra piece of information the brown eyed person in the second case has which means he can kill himself on day four. As far as I can see they are both in the situation that they can see three people with brown eyes, and no one has yet killed themselves. What is the extra piece of information that allows the brown eyed person in the second case to deduce that he has brown eyes?