Island of the Blue-Eyed People

[hugme]

Ok, this one is going to take a little bit of thought to understand, believe me it DOES make sense. you have to really think about what's going on to figure out what would happen... I may give you a little bit more time on this one as it is a bit harder than the other ones.

remember N is any number

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There is an isolated island in the middle of the sea... On this island, people who have blue eyes kill themselves.

Because of this, no one talks about eye color, and there are no reflective surfaces on the island.

The island inhabitants are inherently logical geniuses and trust everything everyone says.

When someone decides to kill him/herself, he sleeps on it first, and then commits suicide the next morning.

The meet every morning after the death time and look at each other.

Suppose an evil demon arrives on the island and says, "At least one person on this island has blue eyes".

What happens if there are N blue-eyed people on the island?

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rember that when they all meet in the morning, they all look around and see the eye color of all the other villagers, so every villager knows the color eyes of every one else. even if they don't talk about it.

Comments

[dwivian]

Hee! Perfect case for Proof By Induction!

[nicker.livejournal.com]

They all yell "Happy Birthday!"

[hugs.livejournal.com]

you rock!

**SMOOCH**

[chirt.livejournal.com]

one thing . . . if there are N blue eyed people, does everyone know there are N blue eyed people? like is there a billboard saying "there are currently 34 blue eyed people?

if not, which is what i figure, then there seems to be two distinct cases, 1 and N>1.

If there is one person, then they will obviously die the bext morning knowing that someone has to have blue eyes and they don't see any.

In case two, N>1, If people know that there is at least one person with blue eyes, but the blue eyed people each see another blue eyed person, then they have no reason to think that they have blue eyes.

If no one talks about it . . . no harm done.

also, if a blue eyed person does kill him/herself, the other villagers have no reason to believe that there might be another. so again . . . if they keep their mouths shut, no harm done.

am i close or am i missing something?

[hugs.livejournal.com]

you have gotten further than anyone else so I will give you the hint...

you are missing the point of this line:

"The island inhabitants are inherently logical geniuses and trust everything everyone says."

[cor256.livejournal.com]

You are welcome to say no to answering these questions, but I have a few:

Do the islanders actually know what the color blue is? since they dont talk about it, woudl that be a factor?

That line says they believe what everyone says, but does the evil demon count as a person? If not, they wouldnt believe it and therefor noone kills themselves (that goes in general also since noone talks about it).

I do have a way they tell for all but 1 person though. It plays off the logical part of this, but I dont think that is the answer you are looking for.

[hugs.livejournal.com]

yes, they know what the color blue is, there is more that is blue than just eyes... like um.. the sky...

yes, the evil demon counts they do believe him

What would you like to answer?

[cor256.livejournal.com]

One at a time they line up. They look at the color of the eyes if the first two. IF there is a blue-eyed person (now to be known as B) and a non blue-eyed person (now to be known as O). so if we have a BO lineup, then the 3rd person who has O will go between them so the lineup is now BOO. the next person has O as well, so not is BOOO. the next has B so now its BBOOO and so on, the person always going inbetween the B and the O.
that will tell everyone but the B in the middle of the line if they should kill themselves.

But that aside, they would find out as children and would the kids kill themselves?

and another side note, if they meet after death time, then what does it matter? the blue eyes already killed themselves so everyoen left is non blue eyed.

Ok, I'll shutup now. :)

[hugs.livejournal.com]

I have no idea what you just answered.

[t-rex.livejournal.com]

Well, if *no one* has blue eyes, they will all kill themselves.

If the number is anything other than zero, I got no clue.

[hugs.livejournal.com]

well, the demon wasn't lying... there is at least one.

[lovelylotus.livejournal.com]

Ok... since this has gone on for a while...

Here we go... Day 1, everyone looks at everyone else's eyes.

Scenario 1: If there is only 1 person with blue eyes, and that person looks at everyone else and doesn't see anyone else with blue eyes, they will assume they are the one with the blue eyes and kill themselves the the morning of Day 2.

Scenario 2: If there are more than 1 person with blue eyes, and someone doesn't kill themselves the morning of Day 2, because they know that there is at least one other person on the island with blue eyes, you wait until the number of days that are the number of people that you know have blue eyes. If at that point, someone still hasn't killed themselves, then you can only assume that you also have blue eyes. By this time, everyone else on the island that has blue eyes would also figure out that they have blue eyes and everyone would kill themselves on that same morning.

Conclusion: Everyone that has blue eyes on the island will kill themselves the number of days that equals the number of people with blue eyes.

So, to the question "What happens if there are N blue-eyed people on the island?"
The answer is: That N blue-eyed people will all kill themselves N days after being told that at least one person on the island has blue eyes.

(I really hope that makes sense...I don't think I explained it all that well...)

[dwivian]

You did very well. You missed a special case, though. I had it at the "proof by induction" comment at the first, but I'd solved enough already to not take over Hugs' posts.

The trivial base case is 1, and it is easy -- one death, all is well. If you trust that someone has blue eyes, and you don't see it, you're it. Off you go.

After that, we have two interesting divergences. If you see a single set of blue eyes (base case = 2) and nobody dies that first night, you know that you have blue eyes, too, as they are expecting you to die. So, you kill yourself, and they kill themselves, on day 2 (for the same reason, since they see your eyes, and they didn't die). Nobody else should die, unless they have a different assumption, based on the fact that they see two sets of blue eyes.

The induction says that if you see N sets of blue eyes, you wait N-1 days to see if others kill themselves. If they don't, you assume you have blue eyes, and you off yourself. The problem here is that, past 2, there is no way to achieve the base case. Everyone waits together, and when N days pass, they all die without regard to their eye color.

So, 1 = 1 death. 2 = 2 deaths. 2+ = everyone dies on day N.

[lovelylotus.livejournal.com]

Ah... I see the typo... Sorry... Thanks clarifying!

:-)

[dwivian]

Alas, I made an error -- I allowed an N+1 induction in the N case. When I went over it again, I found that I was wrong. You have it correct.

[hugs.livejournal.com]

I don't see it... on day 3 if you see 2 people still alive with blue eyes you would kill yourself the next morning BEFORE the meeting. (like the story says)

everyone is 100% trustworthy and 100% logical.

so still... all the blue eyed people would die on day N

[dwivian]

Case 1 is trivial. So, Case 2 with blue eyes:

Day 1, everyone sees 2 with blue, except the blue eyed people, who see one. Nobody knows if there are two, or three blue eyed people. Blue eyes know there is at least one, but there could be two. Everyone else sees two, but considers that there could be three.
Day 2, everyone sees 2 with blue, except the blue eyed people, who now KNOW they have blue eyes, as a single set would have died already (trivial base case). Those that see two people consider that there may be 3 with blue eyes, but the only way to know is to wait one more day.
Day 3, two dead blue eyed people. The incredibly logical people know that they couldn't confirm 2 or 3 until today, and now that two are dead, and they see no blue eyes, they're fine. The possibility is that there were three blue eyed people, but until someone spills the beans again that single blue-eyed person survives. Remember, they don't kill themselves UNLESS THEY KNOW THEIR EYE COLOR. No knowledge, so no death.

Now.... 3 blue eyed people.
Day one -- Blue eyed people see two blue eyes. The rest see three. So, there are 3 or 4 blue eyed people, across the board. Incredibly logical people realize that they cannot determine the difference yet.
Day two -- Everyone that sees two blue eyes, and no deaths, KNOWS there are three, per the induction above. They kill themselves that night.
Day three -- Nobody sees blue eyes. Three dead blue eyed people.

SO, I was wrong. There is no mass death!

[dwivian]

(I carried an incorrect value -- I allowed for the N+1 case in the N induction.)