# The famously difficult green-eyed logic puzzle

I never got why this was suddenly the hardest puzzle. I think it was just Randall Monroe couldnâ€™t work it out, and so called it â€śthe hardest logic puzzle in the worldâ€ť, and everyone knows Monroe is smart so just followed alongâ€¦

I guess it helps if youâ€™ve seen a proof by induction before, or think like a programmer.

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Thereâ€™s a philosophy joke:

Three logicians walk into a bar. The bar tender says â€śDo you three want a drink?â€ť The first logician says â€śI donâ€™t know,â€ť the second logician says â€śI donâ€™t know,â€ť the third logician says â€śYes.â€ť

This has the same logic as green-eyes problem.

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â€śEveryone you see has green eyes.â€ť
So much simpler.

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Not sure what you mean. If youâ€™re saying that the person who comes to the island should say to all of them at once â€śeveryone you see has green eyes,â€ť then yes, that does make the problem much simpler â€” everyone would leave instantly. But thatâ€™s a different puzzle.

EDIT: Strike that. I kind of skipped through the video assuming it was the regular olâ€™ puzzle. Listening to it from the start, I hedgingly say youâ€™re right, the new version messed it up. If the puzzle is â€śwhat can you say to help free the prisoners, without giving them new informationâ€ť you could certainly argue that â€śeveryone you see has green eyesâ€ť doesnâ€™t give them new information, and is much simpler.

Of course, it does give them new information. But the regular olâ€™ solution does as well. (The new information being that everyone has now heard the information.) This version of the puzzle depends on you â€śoutsmartingâ€ť the dictator, who is just not smart enough to realize you gave new information, and thatâ€™s a very fuzzy question.

In trying to make this puzzle different, I think they messed it up. Should have stuck with the original: someone new comes in and for whatever-reason says â€śthereâ€™s at least one person with green eyes.â€ť Who leaves, and on what day?

Edit #2: Indeed, there are a number of solutions to this new version of the puzzle. How about just lying? â€śThe guards have promised me that tonight-only they will free anyone, regardless of eye color.â€ť Since itâ€™s a lie, it canâ€™t possible constitute new information. But it would have the exact same effect of getting everyone out safely that night.

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That gives them a new piece of information since any given prisoner doesnâ€™t themselves know if they themselves have green eyes. You could say that at least prisoners-1 has green eyes because any given prisoner knows that everyone else has green eyes already.

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No, â€śeveryone you see has green eyesâ€ť does not give new info. Everyone each person sees has green eyes. They know this already, it is not new info. If the person hearing this ascertains that every other person is hearing the same thing, and thus each person must have green eyes, that is a deduction not new information. This statement provides no new information any more than the suggested solution in the video.

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Come for the puzzle, stay for the commenters who say they are puzzled as to why anyone would be puzzled by this puzzle.

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I totally read that as â€śeveryone has green eyes.â€ť Youâ€™re right.

Both your solution and the one in the video offer new information, otherwise they couldnâ€™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.

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You are incorrect. The statement itself provides no new information.

Letâ€™s say there are two people - you and another. You can see their eyes, they can see yours. You donâ€™t know the color of your own eyes, only theirs. Somebody new shows up and says â€śboth of you have blue eyes.â€ť And you argue with him, saying he is not providing new information, because you knew all along, you just didnâ€™t want to tell anybody. The next morning you wake up alone on the island, everybody else having left in the dead of night, leaving naught but a note saying â€śactually we were all wearing colored contacts.â€ť

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Pffft. Like that would ever happen in real life.

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Try as I might I canâ€™t figure out how in the hell the solution as originally presented could possibly work the way they say. Your solution does it though.

Why wait 100 days? They could have all left the first night. Figure that one out

I was defeated by the puzzle of when to stop the video if I wanted to actually try to solve it myself.

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Why would it take 100 days? If you can see 99 people with green eyes, then you know everyone can see at least 98 people with green eyes. In that case, since they are logicians everyone knows that everyone can see at least 97 people with green eyes, a far stronger piece of information than knowing that everyone knows there is at least one, they should all be gone in four days and since they have been there since birth they all would have left years ago. So either the induction logic is wrong or the puzzle is invalid.

The simplest form that is truly analogous to the 100 case is four, since with four everyone would know everyone else can see at least one person with green eyes. To be persuasive the video really should have dealt with the case of four people since two and three are special cases rather than purely inductive examples.

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Albert leaves on Cherylâ€™s birthday.

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Sure it does. Both solutions provide more information. The â€śoddityâ€ť of this version of the puzzle is that you have to debate whether the new information is obvious enough to pass the dictator.

The new information is that you now know that other people have the information.

When the person shouts â€śeveryone you see has green eyes,â€ť you say to yourself â€śwait, but if that applies to everyone, then I have green eyes.â€ť Thatâ€™s new information â€” you obtained it by deduction, but deduction based on the statement.

Itâ€™s also the case with the standard version. Use the two person version as an example: youâ€™ve been sitting around with this green-eyed person for years, thinking â€śpoor sap, if only he had someone of knowing he had green eyes he could leave.â€ť Now someone comes and says â€śat least one person has green eyes.â€ť You now have new information: the poor sap heard this statement, and therefore does have a way of knowing he has green eyes!

If there was no possible way of deducing something before statement A, and there is a way of deducing it after statement A, statement A must have given you new information.

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Four people is exactly analogous to the three-person situation.

Proof:

1. You accept that, with three green-eyed people, they would all leave on the third day (it sounds from above like you accept this â€” if not, we can start further back).
2. Suppose there were three green-eyed people and one brown-eyed person. Would that change the puzzle? No, all three green-eyed people would leave on the third day.
3. You are on the island with three other people. They all have green eyes. If there were exactly three green-eyed people, they would all leave on the third day. You, however, couldnâ€™t leave on the third day because you donâ€™t know what color your eyes are.
4. Everyone else is having the same thought-process. They canâ€™t leave on the third day.
5. No one leaves on the third day
6. On the fourth day, you conclude that you must have green eyes and leave
7. Everyone else is having the same thought-process. They all leave on the fourth day.

The key is that up until the third day, everything that you, as a prisoner, know is perfectly consistent with the possibility that there are three green-eyed people and you, a brown-eyed person. If it were the case that you had brown eyes, nothing would have changed until the third day.

In is only with the lack of people leaving on the third day that you can conclude you have green eyes. (If you do.)