Is there no solution to the blue-eyed islander puzzle? Text below copied from here 


  The Blue-Eyed Islander problem is one of my favorites. You can read
  about it here on Terry Tao's website, along with some discussion.
  I'll copy the problem here as well.
  
  There is an island upon which a tribe resides. The tribe consists of
  1000 people, with various eye colours. Yet, their religion forbids
  them to know their own eye color, or even to discuss the topic; thus,
  each resident can (and does) see the eye colors of all other
  residents, but has no way of discovering his or her own (there are no
  reflective surfaces). If a tribesperson does discover his or her own
  eye color, then their religion compels them to commit ritual suicide
  at noon the following day in the village square for all to witness.
  All the tribespeople are highly logical and devout, and they all know
  that each other is also highly logical and devout (and they all know
  that they all know that each other is highly logical and devout, and
  so forth).
  
  [For the purposes of this logic puzzle, "highly logical" means that
  any conclusion that can logically deduced from the information and
  observations available to an islander, will automatically be known to
  that islander.]
  
  Of the 1000 islanders, it turns out that 100 of them have blue eyes
  and 900 of them have brown eyes, although the islanders are not
  initially aware of these statistics (each of them can of course only
  see 999 of the 1000 tribespeople).
  
  One day, a blue-eyed foreigner visits to the island and wins the
  complete trust of the tribe.
  
  One evening, he addresses the entire tribe to thank them for their
  hospitality.
  
  However, not knowing the customs, the foreigner makes the mistake of
  mentioning eye color in his address, remarking “how unusual it is to
  see another blue-eyed person like myself in this region of the world”.
  
  What effect, if anything, does this faux pas have on the tribe?


The possible options are 

Argument 1. The foreigner has no effect, because his comments do not tell the tribe anything that they do not already know (everyone in the tribe can already see that there are several blue-eyed people in their tribe). 

Argument 2. 100 days after the address, all the blue eyed people commit suicide. This is proven as a special case of

Proposition. Suppose that the tribe had $n$ blue-eyed people for some positive integer $n$. Then $n$ days after the traveller’s address, all $n$ blue-eyed people commit suicide.

Proof: We induct on $n$. When $n=1$, the single blue-eyed person realizes that the traveler is referring to him or her, and thus commits suicide on the next day. Now suppose inductively that $n$ is larger than $1$. Each blue-eyed person will reason as follows: “If I am not blue-eyed, then there will only be $n-1$ blue-eyed people on this island, and so they will all commit suicide $n-1$ days after the traveler’s address”. But when $n-1$ days pass, none of the blue-eyed people do so (because at that stage they have no evidence that they themselves are blue-eyed). After nobody commits suicide on the $(n-1)^{st}$ day, each of the blue eyed people then realizes that they themselves must have blue eyes, and will then commit suicide on the $n^{th}$ day.   

It seems like no-one has found a suitable answer to this puzzle, which seems to be, "which argument is valid?" 

My question is... 
Is there no solution to this puzzle?
Argument 1 is clearly wrong.

Consider the island with only two blue-eyed people.  The foreigner arrives and announces "how unusual it is to see another blue-eyed person like myself in this region of the world." The induction argument is now simple, and proceeds for only two steps; on the second day both islanders commit suicide. (I leave this as a crucial exercise for the reader.)

Now, what did the foreigner tell the islanders that they did not already know? Say that the blue-eyed islanders are $A$ and $B$. Each already knows that there are blue-eyed islanders, so this is not what they have learned from the foreigner. Each knows that there are blue-eyed islanders, but neither one knows that the other knows this.  But when $A$ hears the foreigner announce the existence of blue-eyed islanders, he gains new knowledge: he now knows that $B$ knows that there are blue-eyed islanders. This is new; $A$ did not know this before the announcement.  The information learned by $B$ is the same, but mutatis mutandis.

Analogously, in the case that there are three blue-eyed  islanders, none learns from the foreigner that there are blue-eyed islanders; all three already knew this.  And none learns from the foreigner that other islanders knew there were blue-eyed islanders; all three knew this as well.  But each of the three does learn something new, namely that all the islanders now know that (all the islanders know that there are blue-eyed islanders).  They did not know this before, and this new information makes the difference.

Apply this process 100 times and you will understand  what new knowledge was gained by the hundred blue-eyed islanders in the puzzle.
it looks like an administrator here deleted my answer. i just want to say that if you consider it important in any way to be any sort of ambassador to the layperson, then you need to seriously reconsider your attitude. theres a post on here that literally is written by a salesman, talking about the psychological reasons of why the 100 people would know. that one didnt get deleted. i wrote a clearly stated, reasoned response to the best of my ability, and it was snarkily ridiculed with no explanation, then deleted. terrible attitude dude. i am extremely disappointed.

if there is a problem with my logic that i am unaware of, then let me know. i am interested in discussion. i just dont think it is helpful to simply tell someone they havent answered the question without any explanation other than: its a math problem. it might be listed in the math section of this site, but as it is stated, it isnt a math problem. its a logic problem. the whole point of the problem is to determine what can and cannot be known with what information exists. if there is a flaw in my logic, please point it out.

i have re-answered the question, with a few clarifications. i expect you will delete it again mr. hardmath. but i just wanted you to know that if you do, it is very petty and uncool, especially considering you left up the answer from the salesman.

ANSWER:

the correct answer is option one: there would be no change. everyone already knows that everyone knows there are people with blue eyes. as long as there are at least 2 people amongst the islanders with blue eyes, everyone would see at least one other person with blue eyes, but not know if they themselves have blue eyes or not. in this case there are 100 people with blue eyes, and 900 without. but it doesnt matter how many people there are with blue eyes as long as there are more than 2 - they would have no way of knowing their own eye color, EVER. since there is no new information, why would anything change?

the problem that i see with the reasoning in the answers supporting option 2 is that it claims that the new knowledge is that the islanders now know that all of the other islanders know. this is not logical! all of the islanders must know that all the others know already! what one person sees, the others can see. they already understand that what they see the others can see, so therefore they already know that everyone already knows they already know. or whatever. 

to take the above example of bob, cathy, and alice: all three must know that someone has blue eyes, and they all must know that they all know, because they all have blue eyes, therefore bob knows cathy knows and cathy knows alice knows bob knows, etc.

therefore, there is no new information. therefore, behavior does not change.

if someone disagrees with this, please explain. i have read through all of the answers and i really cannot understand why anyone thinks that the new information is that now all the islanders know that all the islanders know.