# The famously difficult green-eyed logic puzzle

"One hundred green-eyed logicians have been imprisoned on an island by a mad dictator. Their only hope for freedom lies in the answer to one famously difficult logic puzzle. Can you solve it? Alex Gendler walks us through this green-eyed riddle."

## Notable Replies

1. I never got why this was suddenly the hardest puzzle. I think it was just Randall Monroe couldn't work it out, and so called it "the hardest logic puzzle in the world", and everyone knows Monroe is smart so just followed along...

I guess it helps if you've seen a proof by induction before, or think like a programmer.

2. There's a philosophy joke:

Three logicians walk into a bar. The bar tender says "Do you three want a drink?" The first logician says "I don't know," the second logician says "I don't know," the third logician says "Yes."

This has the same logic as green-eyes problem.

3. "Everyone you see has green eyes."
So much simpler.

4. Both your solution and the one in the video offer new information, otherwise they couldn't lead to the perfect logicians in the puzzle drawing a new conclusion and leaving.

The new information being that what they themselves can observe is equally true of everyone else.

5. Everyone has always known that at least one prisoner is green-eyed.

The facts:

• No prisoners are not green-eyed.
• Everyone has always known that at most one prisoner is not green-eyed.
• Everyone has always known that everyone has always known that at most two prisoners are not green-eyed.
• Everyone has always known that everyone has always known that everyone has always known that at most three prisoners are not green-eyed.
• For all i, (Everyone has always known)^i that at most i prisoners are not green eyed.

What is the information contained in hearing the statement "at least one person is green-eyed" addressed to everyone?

• At most n-1 prisoners are not green-eyed
• Everyone knows that at most n-1 prisoners are not green-eyed
• For all i, (Everyone knows)^i that at most n-1 prisoners are not green-eyed
• Specifically, a new piece of information: (Everyone knows)^n that at most n-1 prisoners are not green-eyed

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