Blue Eyes - "The Hardest Logic Puzzle in the World"

[Tiberius]

I managed to find this in Google's cache since the crash. For those who know the answer, please use [hide] tags, so that people who haven't heard the solution can have a go.

This is one variant of the "Blue Eyes" problem, often considered one of the hardest logic puzzles.

In the middle of the ocean there is a small island where there are a number of people living. They all are devoted to a very strange and specific religion, of which there is one simple rule:

"If you ever find out that you have blue eyes, you must kill yourself at midnight that night."

To get around this rule (and make sure all their friends don't kill themselves), nobody talks about the eye colour of anyone on the island, and nobody possesses any mirrors or other ways of accidentally finding out their own eye colour.

Every day when the sun is highest in the sky, all the residents of the island congregate in one area of the island so that every single one of them can see every other person on the island, and more importantly, what eye colour they have.

One day, whilst everyone is at this congregation, a foreigner arrives, and states to everyone (so that everyone can hear him) "I can see at least 1 person with blue eyes". He then leaves.

All the people are perfect logicians, if a conclusion can be logically deduced, they will do it instantly. Let us suppose that there are 100 blue eyed people on the island, and 100 brown eyed people. Knowing this, can you work out how many people kill themselves, and on which night they do it?

Before you answer, know that this 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 foreigner is not making eye contact with anyone in particular; he's simply saying "I count at least one blue-eyed person on this island who isn't me."

Also, even though we know that there are 100 blue eyed and 100 brown eyed people on the island, this is not common knowledge to the people themselves. For example, a blue eyed person on the island cannot deduce he is blue eyed by simply counting the number of other blue eyed people; he has no idea exactly how many blue eyed people (or brown eyed people) are on the island.

And lastly, the answer is not "no one kills themselves."

[Ashendant]

I would say "no one kills themselves.", although i know it's false

My logic is that any person that doesn't know they have blue eyes would see any other with blue eyes and would presume the foreigner would mean the other person and not himself so he doesn't know still that he has no blue eyes and hencefort commit suicide

[lilphil1989]

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I think that 100 people kill themselves on the 100th night.

Here's my reasoning:

(1) Start with a situation in which there's one blue-eyed person.
This person looks around, sees no other blue-eyed people and can therefore conclude that, since there is at least one blue-eyed person, there is exactly one: himself. He then kills himself on the first night.

(2) Now imagine a situation in which there are 2 blue-eyes. Each blue-eye will look around and see one other blue-eye. He can then follow the reasoning of (1) and conclude that if he himself has brown eyes, the other blue-eye that he sees will kill himself on the first night. When the other blue-eye doesn't kill himself on the first night (because each can see another blue-eyed person), both blue-eyes people will realise that there is one more blue-eyed person: himself. Therefore they both kill themselves on the second night.

(3) 3 blue-eyed people. Each follows the reasoning of (2), and when the other blue-eyed people don't kill themselves on the second night, all 3 kill themselves on the third.

(4)In general, for the case of N blue-eyed people, they all refer back to the reasoning of the (N-1)th case and will all kill themselves on the Nth night.

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Is this correct?

Given the title of the post, I think I may have missed some subtlety :S
Or maybe something really obvious!

[theVOID]

Nope that's it