The green-eyed/ blue-eyed puzzle/ conundrum
Today I want to share a puzzle that my friend Aaron Abrams told me a few days ago. I’m sure some of you have heard it before, but it’s confusing me, so I’m asking for your help.
Here’s the setup. There’s an island of people, all of whom have either blue eyes or green eyes. By social convention they never discuss eye color, because there’s a tragic rule that states that, if you ever figure out your eye color, you have to leave the island within 24 hours. Oh, and there are no mirrors.
OK, get it? So think of the island as pretty small, maybe 100 people, so you know everyone else’s eye color but not your own.
Here’s what happens next. Some castaway arrives by swimming onto the island, stays for a few days and hangs out with the folks there eating island food and having island parties, and then after building himself a boat he prepares to leave. Not being trained in the social customs of the island, on the day he leaves he says, “hey, it’s good to see some people with green eyes here!”.
So the puzzle is, what happens next?
Here’s what’s obvious. If you are a person who only sees blue eyes, you know by his statement that you must have green eyes. So you have to leave the next day.
But actually he said “some people.” So even if you only see one other person with green eyes, then you have to leave, with that other green-eyed person, after one day.
With me so far?
But hey, what if you see two other people with green eyes? Well, you might think you’re safe, and you’d wait to see them leave together the next day. But what if they don’t leave after one day? That must mean that you also have green eyes. Then all three of you have to leave, after two days. Get it?
Then you work by induction. If you see N other people with green eyes, they should all leave after N-1 days, or else you have green eyes too and all (N+1) of you leave after N days.
OK, so here’s the conundrum. The guy who started this whole mess really didn’t do much. He just stated what was obvious to everyone already on the island, namely that some people had green eyes. I mean, yes, if there were really only 2 people with green eyes, then he clearly added real information, because both those people had thought only 1 person had green eyes.
But just for the fun of it, let’s assume there were 17 people with green eyes. Then they guy really didn’t add information. And yet, 16 days after the guy left, so do all the green-eyed islanders. So really the guy just started a count-down more than anything.
So, is that it? Is that what happened? Or was the original set-up inconsistent? Is it not an equilibrium at all? Or is it an unstable equilibrium?
In any case, Aaron and his friend Jamie have developed a saying, it’s a green-eyed/ blue-eyed thing, which means it’s an apparently information-free fact which changes everything. I think I’ll use that.