## The Blue-Eyed Islanders: is it a Paradox?

There’s a logic puzzle that’s been around for a while.

It is set on an island, populated by logicians, and they have a terrible common belief that anyone with blue eyes must be exiled.   Yet, they are empathic and believe even more strongly in self-determination.   So, they won’t tell anyone the terrible fact that their eyes are blue.

These logicians have been living on the island, when a mysterious stranger swims ashore and over cookies and tea at the next week’s research colloquium (attended by all, of course) he mentions that there is at least one blue-eyed person on the island.   (Xkcd.com describes this problem, and the problem even has its own Wikipedia page.)

Disaster ensues, of course.  If there is one blue-eyed person, that person must know immediately that they must be blue eyed, because they don’t see anyone else with blue eyes.  Thus, they sneak into exile at midnight.

If there are two blue-eyed people X and Y, nothing happens the first night, because both of them expect the other to go into exile.

But the second morning, they wake up and each finds that the other is still there.  X realizes that the only reason that Y could still be on the island is if Y saw someone with blue eyes.  And the only possible person that Y could see who might have blue eyes is X.  So X realizes, uh oh, I must have blue eyes.  (As does Y.)  So, at midnight, both sneak off to shameful exile.

And, one can recursively prove that if there are N blue-eyed people, they (morning by morning) make deductions, and that all the blue-eyed people leave the island on the Nth midnight.  (Xkcd provides the solution in fairly simple form.)

The paradox is that for N > 2, the mysterious stranger didn’t actually bring any new information onto the island.  Because for N > 2, all the islanders can already see two people with blue eyes, so they already know that everyone can see at least one person with blue eyes.  So, everyone already knows that there is at least one blue-eyed person.

So, what does the mysterious stranger actually do?  He seems essential as a starting point, and people who write about this problem talk about him bringing the fact of blue-eyes into “common knowledge”.  He seems paradoxically both necessary and useless.

But the correct answer is that this situation appears to be logically impossible, so a paradox is both expected and unimportant.  As suggested by Cathy O’Neill, the problem is actually logically inconsistent in the sense that (under the problem’s assumptions) you could never set it up.

Obviously, if you have an island and you parachute N>2 blue-eyed people in on January 1, everyone can look around and see blue eyed-people as soon as they land.  N days later, they sneak off to exile, and when the mysterious stranger arrives, he only sees brown eyes.  (Tragedy would then follow if he lies that he sees blue eyes, but that’s another story.)

If the mysterious stranger arrives before January N, he brings no new information, and the logical machinery has already begun its inexorable motion.  So, when he stands up at the seminar and mentions that there are blue eyed people, everyone knows that everyone knows, so he is just quietly told that it’s none of his business.

Alternatively, you could try to form the population of N blue-eyed people by bringing them in one-by-one.  But, as soon as you have N=3 on the island, they all know that they all know, and on the third midnight, they head for exile.  So, you can never have more than two blue-eyed logicians with this belief structure on an island for more than a short time.

If you have N=1 or N=2, the mysterious stranger is necessary, but it’s not a paradox because he brings in information that islanders didn’t already have.

The only hope for setting the problem up is by introducing some uncertainty about who knew what when.   But, then the mysterious stranger is bringing in useful information again: it’s no longer paradoxical.

So, no, it isn’t a paradox.  But it’s still a beautiful chain of logic.  Keep this kind of thing in mind, just in case you start meeting too many logicians…