Here is the answer to the Blue Eyes Logic Puzzle I posted.

This answer comes from mathematician Terence Tao, and has to do with common knowledge.
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100 days after the Guru’s comment, all the blue eyed people will leave. This is proven as a special case of

Proposition. Suppose that the tribe had n blue-eyed people for some positive integer n. Then n days after the traveller’s address, all n blue-eyed people leave the island.

Proof: We induct on n. When n=1, the single blue-eyed person realizes that the traveler is referring to him or her, and thus leaves on the next day. Now suppose inductively that n is larger than 1. Each blue-eyed person will reason as follows: “If I am not blue-eyed, then there will only be n-1 blue-eyed people on this island, and so they will all leave n-1 days after the traveler’s address”. But when n-1 days pass, none of the blue-eyed people do so (because at that stage they have no evidence that they themselves are blue-eyed). After nobody leaves on the (n-1)st day, each of the blue eyed people then realizes that they themselves must have blue eyes, and will then leave on the nth day.

If you need any explanation of this, let me know.

As you probably know, I love logic puzzles. I came across a particularly difficult one today, so I thought I would share it with you. I first came across it on mathematician Terence Tao’s blog, but I saw another formulation by xkcd creator Randall Munroe, and I like his formulation better. It is his formulation which is reproduced below. This puzzle is not of my own thinking. It has been around for a long, long time.

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Blue Eyes:
The Hardest Logic Puzzle in the World

A group of people with assorted eye colors live on an island. They are all perfect logicians — if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. 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. Everyone on the island knows all the rules in this paragraph.

On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), 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.

The Guru is allowed to speak once (let’s say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following:

“I can see someone who has blue eyes.”

Who leaves the island, and on what night?

There are no mirrors or reflecting surfaces, nothing dumb. It is not a trick question, and the answer is logical. It doesn’t depend on tricky wording or anyone lying or guessing, and it doesn’t involve people doing something silly like creating a sign language or doing genetics. The Guru is not making eye contact with anyone in particular; she’s simply saying “I count at least one blue-eyed person on this island who isn’t me.”

And lastly, the answer is not “no one leaves.”

A word of warning: The answer is not simple. This is an exercise in serious logic, not a lateral thinking riddle. There is not a quick-and-easy answer, and really understanding it takes some effort.
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I am willing to talk this over with anyone who is struggling with it. Puzzles like this are fascinating to me.

The ship wherein Theseus and the youth of Athens returned had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place, insomuch that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.
–Life of Theseus by Plutarch

Richard, my parents, and I were discussing the classic paradox of Theseus’ ship (stated above from Plutarch.) We each came up with different examples and related them to a number of different situations. Afterwards, I looked online to other possible solutions to the paradox, and found that my answer was a combination of two older ones: Aristotle’s causes, and the concept of four dimensions.

I am interested in hearing what other people think about this. If you have a little time to think about this, please do so and either email me or leave a comment with your thoughts