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? 
 A: Easiest case to show what's wrong with the solution offered here and all other solutions like on wikipedia etc is to consider case of 4 blue eyed people.
Proposition 0: If there are 4 blue eyed people than everyone sees other 3 and also all of them  can conclude(even without knowing their own eye color)   that all of them know(not a typo)  that there are at least 2 blue eyed people on the island.  So if blue eyed people are A,B,C & D than A can say that B knows that C knows that D knows that A knows  that there are at least 2 blue eyed, B can make similar assumption etc.
This would be enough for a pure mathematical proof, since if all of them  know that all of them know  that there are at least 2 blue eyed people than visitors anouncement that there is one blue eyed among  them does not introduce new knowledge. 
Let's  find out  what's wrong with the proof:
General argument for any kid of proof:  If proof/solution assumes something that is impossible or contradicts to initial conditions than every conclusion based on such an assumption is useless.
All solutions I've  seen so far include some variations of : 
Suggestion A:  Suppose/if there was only person (same as suppose n=1)
Suggestion B:  Suppose/If there was only one blue eyed person
Clearly if there are 4 blue eyed person we can't suggest  that there is only one person on the island  so Suggestion A is clearly wrong and all proof based on such a suggestion are wrong. This also invalidates all recursive proofs that assume that n=1 / day one . One can abstract from the concrete example and say suppose n=1/day one,  but than you can't imply the knowledge that on day one you knew that there is one blue eyed person. 
Now what's wrong with the suggestion B? remember proposition 0. Everybody  knows that all of them know that there are at least 2 blue eyed people on the island. But suggestion B says suppose there is only one blue eyed person. This Suggestion is also wrong since we know that there are at least two of them.   So,  all profs/conclusions based on suggestion B are also wrong. 
That's it.
If you find a  proof that does not use some variation of A or B than we can reopen this discussion. 
A: 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.
A: Here's my answer as to why the outsider gives new info. I'm considering the situation of (A)lice, (B)ob and (C)athy as mentioned in a post above, where all 3 have blue eyes. For argument's sake, I'll be Alice.


*

*I know there are at least two people with blue eyes

*I know Bob and Cathy know at least one person has blue eyes

*Here's the tricky part: What do I know about what Bob knows about Cathy's knowledge?

*As far I know, I may have brown eyes, and Bob may think he has brown eyes.  So when Bob looks at Cathy, he may think she sees no one with blue eyes.  So from Bob's point of view, Cathy may not know there are blue eyed people.


Simply put, I know there are blue eyed people.  I know that Bob and Cathy know.  But I don't know that Bob knows that Cathy knows.  When the outsider announces it, I now know that everyone knows, and everyone knows that everyone knows.
It's also easier to imagine that everyone assumes they have brown eyes.  So when Alice thinks about Bob's thoughts on Cathy's thoughts, everyone down the line assumes they themselves have brown eyes.  When Alice thinks of Bob's thoughts, she's assumes he thinks they both have brown eyes, and only Cathy has blue.  When Alice thinks of Bob's thoughts on Cathy's thoughts, she imagines Bob will conclude that Cathy will not see blue eyes.
This scales up to four or more, and it's easier (for me) to think of the fourth being 'above' A (Alice), let's say Omega.  So Omega must imagine what Alice thinks that Bob is thinking about what Cathy thinks, not just what Cathy herself thinks.
So when Omega imagines Alice's thoughts, he's assuming she thinks they both have brown eyes.  When he imagines Alice's thoughts on Bob's thoughts, same thing, everyone assumes they have brown eyes in their own mind.  That's why Omega concludes that Alice may imagine Bob could think that Cathy may not know there are blue eyed people.
Very deep and difficult, but the 'everyone assumes they have brown eyes' POV helped me wrap my head around it.
A: This isn't a solution to the puzzle, but it's too long to post as a comment.  If one reads further in the post (second link), for clarification:
In response to a request for the solution shortly after the puzzle was posted, Terence Tao replied: 

I don’t want to spoil the puzzle for others, but the key to resolving the apparent contradiction is to understand the concept of “common knowledge”; see
  http://en.wikipedia.org/wiki/Common_knowledge_%28logic%29

Added much later, Terence Tao poses this question:

[An interesting moral dilemma: the traveler can save 99 lives after his faux pas, by naming a specific blue-eyed person as the one he referred to, causing that unlucky soul to commit suicide the next day and sparing everyone else. Would it be ethical to do so?]

Now that is truly a dilemma!

Added: See also this alternate version of same problem - and its solution, by Peter Winkler of Dartmouth (dedicated to Martin Gardner). See problem/solution $(10)$.
A: 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. 
A: The foreigner is only used as a point of reference for counting days. That is his role in the solution as it was thought. The puzzle merely says: look, they start their reasoning at this point in time. 
I don't agree with the solution though. If their goal was to find their eye color (such as in a different formulation of the puzzle), maybe this can be done by finding a solution which minimizes the amount of time that passes. This would allow them to agree without communicating. In this case, it is entirely possible to reach the conclusion that they all wait to see what happens in the first 99 days. If their goal is to not discover their eye color, then obviously, they will implicitly agree to do nothing and will stay alive for ever, never knowing their eye color.
