Blue Eyes - The Hardest Logic Puzzle Ever

[Kodilynn]

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?

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There are no mirrors or reflecting surfaces. 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."

I've done my best to make the wording as precise and unambiguious as possible (after working through the explanation with many people), but if you're confused about anything, please let me know. 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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Borrowed from the wonderful folks of http://www.xkcd.com

Yes I know the answer, no I won't PM it to you. 30 Day Gamecard in it to whomever gets it right (and if you try googling the answer or wiki'ing it, i'll know because of the nature of the answer and the "common answers" which aren't right might I add).

You can PM me your answer or you can post it. First right (as close to right as you can be, there is a correct answer) will win!

Good luck!

[teeroyoyort]

a brown eye'd person leaves the island on the 199th night?

[teeroyoyort]

nm too tough!

[kukmuk]

It isn't fair for me to answer because my Discrete Structures teacher used an example extremely similar to this as an introduction to Mathematical Induction back in college. If you start simple and work your way up it actually isn't all that complicated.

If this comment reveals too much about the solution, feel free to remove it.

[r0xx0r]

reminds me of Search for the Holy Grail.

There the other side you see! but you must first- answer me these questions, three!



BLUE- NO GREEN AAAAIIIIIYYYYYYEEEEEEEE!!!!!!

[VyralPandemic]

Ok, While I do not posses any previous knowledge of the question, I too will refrain from posting my answer. I am just too @#$-ing smart and don't want to ruin it for everyone else.

Happy Pondering and Best Wishes to the Mathematically Challenged

With a tip of my cap, and a wave of my finger,

-Vyral

[Argo]

100 days after the guru speaks all of the blue eye'd people leave.

[Sadge]

Originally Posted by Argo

100 days after the guru speaks all of the blue eye'd people leave.

Yes, but can truly explain why?

That's the real point of this puzzle, not the answer itself.

[r0xx0r]

already sent in pm to K-man, before my reply with explanation of my answer, so i'll go ahead and let it out.



100 blue eyes leave on 100th day/night, simply because when the guru says i see one person with blue eyes, instantly (given that all the ppl know everyone's eye color, which they do as explained in the question) everyone is thinking these 99 blue eye'd ppl i see are gonna leave on the 99th day/night. And if they do, that means i'm foked because i have some other color'd eyes and i'm doomed to be on these islands with these perfectly logical mimes. But if they don't leave, that means 100 ppl have blue eyes, so that means my eyes are blue too! So I leave on the 100th day/night with all the other blue eyes. It would be non-logical for any of them to wait one day past 100 because of the simple fact that they only see 99 blue eye'd ppl.


/claps

[Argo]

Originally Posted by Sadge

Yes, but can truly explain why?

That's the real point of this puzzle, not the answer itself.

Lets say that that there is one blue eye'd person total. When the guru says this he knows he is the only blue eye'd person cause he see's all different color'd eye people. If there is two blue eye'd people. No one will know that they are blue eye'd for sure on the first day and no one will leave. On the second day they will know that there has to be at least 2 blue eye'd people because no one left on the first day. He looks and sees that there is only 1 other blue eye'd person so he can conclude that he has blue eyes. Same if there were 3 and so on all the way up to 100.