Blue Eyes - The Hardest Logic Puzzle Ever

[VyralPandemic]

Originally Posted by Sadge

Ok -- maybe this would help ...

A lot of people like to start their answer with "imagine there were 2 people with blue eyes". While that helps, I think it's better to start with 1 blue-eyed person.

Fact: We know that there is at least 1 blue-eyed person on the island (because the guru said so, and for this case we will assume the guru never lies).

Now, if there is only 1 blue-eyed person on the island, he immediately knows it's him, right? He cannot see another blue-eyed person among the 199 other people on the island (and it is 199 people -- the guru does not count for 2 reasons: the main one being that the guru (we assume) cannot see her own eye color). So in this case, the 1 blue-eyed person leaves that night.

Now let's go to 2 blue-eyed people. Person A (the first blue-eyed person) can see 1 other blue-eyed person (Person B). But remember that neither person A or B know that their own eyes are blue. So person A can assume that person B will leave the island that night if they do not see any other blue-eyed people (because of the first example), because person B would not be able to see any other blue-eyed people. But after the first night, person B does not leave because he knows that he sees person A with blue eyes (mathematically speaking, there can only be number of people seen with blue eyes+ the person looking ... or seen+1). Since person A did not leave the first night, person B knows that A cannot logically deduce how many people have blue eyes because he sees at least 1 other person with blue eyes. Since person B cannot see any more people with blue eyes, he must have blue eyes (person A reaches the same conclusion). So on the second night, both person A&B leave being the only 2 people with blue eyes.

You can follow that logic all the way up to 100 blue-eyed people. Each person can see that there are 99 other blue-eyed people. So he simply has to wait 99 days to see if they leave and if they don't, he can assume that he himself has blue eyes and leaves on the 100th day (and of course the other 99 people are reaching the same conclusion after 99 days).

I can go further if you want.

OK, you win. That I can follow. Good explanation Sadge

[Kodilynn]

You smart people overthink too much, but it's good for a laugh. One of you is correct enough. When I get home, ill announce a winner and post the answer (which is more or less already said in someone elses post!).

[Terin01]

this was a tough one for me. I thought that i had it right away and started to post but read threw everyone elses posts and saw the same thing i thought put down. already...and a winner...so figured shit im too late again. Why do i have to watch movies with my wife?

[Sadge]

Originally Posted by Terin01

100 blue eyed people leave the island the night the guru says she sees some one with blue eyes.

How'd ya figure that?

[Kodilynn]

Originally Posted by Sadge

How'd ya figure that?

Don't worry, he's wrong.

[Sadge]

Originally Posted by Kodilynn

Don't worry, he's wrong.

I know that =)

I'm just wondering how he came to that conclusion. Like I stated before with a puzzle like this, it's not about the answer -- it's how you come to the answer.

[r0xx0r]

he's thinking, they all were like fok it. /zone AnywhereButHere
after they heard what the guru said.

[Kodilynn]

VP kinda ruined this by doing what I said not to do, which was using the net to get the answer and posting the direct link. As such, nevermind.

Technically there are two solutions. VP and Argo more or less posted one.

What I was looking for was:

Let us consider the special case of 1 blue-eyed person. That person will observe that there are no blue eyed people and thus deduce that they are the blue-eyed person. Since they now know their eye color, they leave that night.

In the case where there are two blue-eyed people, they will each observe someone with blue eyes - but that's not enough information to determine their eye color, so they both wait to see if the other one leaves the island that night. If they don't, then they each know their own eyes are blue as well. So they leave a night later.

Generalizing, in the case of n blue-eyed people (for n>2), each person will observe n-1 blue-eyed people, but this is not enough information to tell them their eye color. Each of the n blue-eyed people waits. Unfortunately, waiting for a night does not provide sufficient information - as far as each person is concerned there could be n-1 people with blue eyes waiting to see if the other one leaves. Thus, they all must wait n-1 days before leaving.

Thus, in the case of n=100, they all wait for 99 days and leave on the 100th night.

[Dragonthorne]

Logical people know how to talk and communicate

Illogical people do not use all resources that are available

[Hostagecs]

what happens if they are color blind?