Interesting logical puzzle (see if you can solve it).

There is an island of 100 people, in which 90 have blues and 10 have green eyes.  Each person can see every other person's eye color, but discussion of eye color is forbidden and no one knows the color of his/her own eyes (there are no reflective surfaces on the island).    If anyone were to discover that he/she has blues eyes, he/she would have to leave the island at dawn the next day.   One day, an outsider comes to the island and makes the following announcement that everyone heres:

"At least one of you has blues."

Assuming that every islander is completely logical, what effect, if any, does this have?

 

How do you mean "completely logical?"

Don't think too much into that.  Just means that they are very logical.  

That doesn't really help. Perhaps they'd "kill the messenger" for discussing eye color.

Eventually they'd figure out exactly how many blue eyed people there were.

If there were, say, three people with blue eyes then no one would leave the first day bc they would expect the other two to leave. Next day same scenario. But on the third day theyd figure out they had to be the third person (or 4th/5th/6th etc). 

This has no effect. Everyone has already seen multiple people with blue eyes so this adds no new information.

While my heart is with this response:

ThatOldGuy:

That doesn't really help. Perhaps they'd "kill the messenger" for discussing eye color.

I think this response is best:

Prime:

This has no effect. Everyone has already seen multiple people with blue eyes so this adds no new information.

Friedmanite:

There is an island of 100 people, in which 90 have blues and 10 have green eyes.  Each person can see every other person's eye color, but discussion of eye color is forbidden and no one knows the color of his/her own eyes (there are no reflective surfaces on the island).    If anyone were to discover that they have blues eyes, they'd have to leave the island at dawn the next day.   One day, an outsider comes to the island and makes the following announcement that everyone heres:

"At least one of you has blues."

Assuming that every islander is completely logical, what effect, if any, does this have?

Answer: all of them leave the island. Am I correct?

^^^
I think so too. He said "they'd have to leave," so perhaps this means the collective "they." Assuming the OP's grammar is correct, this must be the case as "they" is improper to describe a single person.

gotlucky:

While my heart is with this response:

ThatOldGuy:

That doesn't really help. Perhaps they'd "kill the messenger" for discussing eye color.

I think this response is best:

Prime:

This has no effect. Everyone has already seen multiple people with blue eyes so this adds no new information.

QFT.

I'd also be curious to know what kind of sorcery is preventing them from discussing eye color and forcing them to leave this eyeland.

 

The point of the puzzle is to figure out why the "Nothing, because it adds no new information" reasoning is wrong.  Mtn Dew is pretty much correct.

Everyone's mind constructs a huge logical chain giving every possibility of what would happen if he/she had blue eyes and what would happen if he/she had green eyes, given the fact that everyone is uncertain of their own eye color, and that everyone knows that everyone else is uncertain of their own eye color, and that everyone has perfect deductive abilities, and that everyone can deduce what they would be thinking if they were anyone else, etc.  And it extends on and on to the point that they are able to eliminate scenarios that have been proven wrong from other peoples' actions, until the blues figure out that they are blues and leave.

But I think that they would have to know that every islander was completely logical as well.

Also, if they really are completely logical, and if they wanted to stay on the island, they would just not think about it in the first place.

Sat. 12/05/26 16:21 EDT
.post #153

Let's take the simplest case: Only one blue-eyed person on the island.

Everyone except the blue-eyed person already knows "there is at least one blue-eyed person on the island." After the announcement, the blue-eyed person now also possesses this knowledge. Seeing nothing but green-eyed persons, he correctly deduces that he is the blue-eyed person, then leaves.

Now let's consider the case of two blue-eyed persons:

Everyone already knows that there is at least one blue-eyed person on the island, so the announcement "there is at least one blue-eyed person on the island" adds no new knowledge. Since no new knowledge has been added, nothing changes.

The same is true for any number greater than one of blue-eyed persons.

I know this isn't the "officially correct" answer, but I think the officially correct answer is wrong (or perhaps I just don't understand it).

Friedmanite:

The point of the puzzle is to figure out why the "Nothing, because it adds no new information" reasoning is wrong.

The assertion "it adds no new information" is correct (except in the case of only one blue-eyed person). The reasoning "if it adds no new information, then nothing will change" seems perfectly sound to me. If you disagree, please elaborate.

"If there were, say, three people with blue eyes then no one would leave the first day bc they would expect the other two to leave."

I reject this statement. Why would one expect the other two to leave if he knows that the other 2 do not know what color their eyes are? In fact, he would expect them to stay, not leave. From what I can tell, nobody ever has to leave so long as they don't know what color their own eyes are. It is possible that all stay on the island as before.

Again, the statement from the new guy on the island changes nothing. If there was a reason for the islanders to leave, they would have already done so prior to the foreigner arriving.

It's absolutely true that what the outsider tells the island adds no new information (each islander knows there are blue-eyed people among them).  However, the fact that the announcement was made in public, changes the dynamics of what each islander thinks the other islanders know.  Using an inductive, argument, we can show that the all the blue-eyed people, leave on the 90th dawn after the announcement.  In fact, the number of green or blue eyed people is irrelevant.  On any island with n blue-eyed people, the same result will follow.

Proof:  For n = 1, the the sole blue-eyed person will realize that he/she is the blue-eyed person the outsider was referring to and so would leave the island next dawn.  Now assume for an arbitrary n (number of blue eyes), the blue-eyed people will leave on the n'th dawn after the announcement. Then if there were n+1 blue eyes, each of them would reason: "If I am not blue-eyed, those n blue-eyed people will leave on the n'th dawn."  So on the n'th day, seeing that they haven't left, each of the n+1 blues eyes would realize that he/she is blue-eyed, and so would leave the next dawn.

 

Prime, start with 1, then 2 follows, so does 3.....continue ad infinitum.  

Friedmanite:

Proof:  For n = 1, the the sole blue-eyed person will realize that he/she is the blue-eyed person the outsider was referring to and so would leave the island next dawn.

This seems to require an additional assumption, namely that the sole blue-eyed person knows what everyone else's eye color is.

^Reread the problem.

Sat. 12/05/26 18:13 EDT
.post #154

Friedmanite:

However, the fact that the announcement was made in public, changes the dynamics of what each islander thinks the other islanders know.

I'm not trying to give you a hard time, but I don't know what this sentence means. What are "the dynamics of what each islander thinks the other islanders know"? What does this mean?

The way I see it, the (public) announcement either adds new information, or it doesn't. In the case of one blue-eyed islander, it does. In the case of two or more, it doesn't. In the case of two or more blue-eyed islanders, every islander knows:

1. That there is at least one blue-eyed islander;
2. That every islander knows this.

Now assume for an arbitrary n (number of blue eyes), the blue-eyed people will leave on the n'th dawn after the announcement.

But, as I've pointed out, we shouldn't make this assumption, because for any number of blue-eyed islanders greater than one, it is clearly false.

I repeat what I said in my previous post:

The assertion "it adds no new information" is correct (except in the case of only one blue-eyed person). The reasoning "if it adds no new information, then nothing will change" seems perfectly sound to me. If you disagree, please elaborate.

I can't see any flaw in what I've said, yet I know what I've said disagrees with the officially correct answer. So, this is a bit weird. I think I'm right and the official answer is wrong, but I'd love to be proved wrong.

MMMark, 

I was using an inductive argument, which has the form:

1.) Prove it's for n = 1

2.) Prove that if it's true for n, it's true for n+1

If you can do this it's true for every n.

Now back to your question.   Let's look at the case where n = 2.   Each of the blue-eyed people reason as follows:

"Hey, if that guy with blue eyes is the only blue-eyed person on the island he/she must leave tomorrow at dawn."  

Upon seeing that other hasn't left, each of the two blue-eyed people thinks to him/herself:

"So that guy didn't leave?  This must mean that I am also blue-eyed!"

Hence, they both leave the next day.

You can use similar reasoning to generalize this to n = 3, 4, 5 ......     

With the solution you gave,

are you sure you set the story up right?

If there are more than 2 people, what is the point of using the outsider as some type of trigger to set off the event ?

The information would already be a given in the people's head and the process of leaving would have already taken place, right?

That's precisely why the answer is so counterintuitive.   The outsider comes and doesn't tell the islanders anything new, but the process of telling the islanders this in public does something.   The case for n = 3 is no harder.  By the reasoning I just gave, each of the 3 blue eyes, sees other two and reasons that they must leave the 2nd dawn after the announcement.  After seeing that they don't, all three come to the realization that they're blue eyed

 

 

 

So If I'm thinking about this right, it's about setting off an event / rule on knowing who must know - it's not about the knowledge about the community - but what happens when the outsider actually sets off a chain by verbalizing?

Because you are asking what effect it will have, and nothing can be effected until something is said?

 

EDIT

Friedmanite, this is only valid for N=1 or N=2. Once N=3 or more, it changes the game. I am out of time right now, but will post a scenario later this evening...

 

1B|10G scenario:

Blue realizes he only sees green eyes and leaves on first night.

2B|10G scenario:

Blues individually realize that if they had green eyes, the other blue would have left on the first night.  The blues leave on the second night.

3B|10G scenario:

Blues individually realize that if they had green eyes, the 2B|11G scenario would be the same as the 2B|10G scenario, and the blues would have left on the second night.  The blues leave on the third night.

And it continues in this pattern, so the blues will leave on the ninetieth night.

Sat. 12/05/26 19:30 EDT
.post #155

Oh wait...I'm wrong. Assertion #2 in my previous post is wrong.

In the case of two (or more, or even one) blue-eyed islanders, it is not true that "every islander knows that every islander knows that there is at least one blue-eyed islander."

Using the case of two blue-eyed islanders again, then there are two islanders, namely the blue-eyed ones, who don't know that "every islander knows that there is at least one blue-eyed islander."

One blue-eyed islander doesn't know the other knows "that there is at least one blue-eyed islander."

But, after the announcement, both know that the other knows. So, the announcement does add new information.


There's more to this than I originally noticed. I'll think about what you've said a bit more before I post again.

MMMark,

I think you misunderstand my problem/solution.   The outsider comes and says in public,

"At least one of you has blue eyes."

This is not the same as the outsider telling every person in private, "At least one of you has blue eyes."    The latter would have no effect at all (unless n = 1)  as intuition would suggest.

Vive,

What's the effect?  The blue-eyed people leave.

 

More explanations on the puzzle:

http://en.wikipedia.org/wiki/Common_knowledge_(logic)

 

http://plato.stanford.edu/entries/common-knowledge/#1.2