No, the key is that the foreigner's statement establishes common knowledge at some point in time. What happened before that point in time is irrelevant.
> the blue-eyed islanders all know that there are 99 other islanders with blue eyes
That's true, but what they don't know (until the 2nd) day) is that all the other blue-eyed islanders know that all the other blue-eyed islanders know that there are 99 other blue-eyed islanders. On the 3rd day they will realize that all the other blue-eyed islanders know that all the other blue eyed islanders know that all the other blue eyed islanders know that... and so on. Then on the 100th day there are 100 iterations of recursive knowledge, and all the blue-eyed islanders realize they themselves must have blue eyes.
UPDATE: note that it is crucial that all the islanders are together when the foreigner makes his statement. If he goes to each islander individually and says "some of you have blue eyes" then it doesn't work. What matters is not the statement, but that all the islanders witness all the other islanders hearing the statement.
Strictly, all blue eyed people need to hear the statement, right? If someone is missing and everyone (but them) knows the missing person has brown eyes, that doesn't change the logic of those who heard.
1. What happens if one blue-eyed person is somewhere else on the island when the foreigner makes his statement (and his absence is known to everyone)?
2. What happens if the next day a blue-eyed stranger wanders into the village, thereby establishing common knowledge that the day before there was in fact an additional blue-eyed person on the island (though no one in the village knew it at the time)?
3. What happens if the next day a blue-eyed baby is born in the village?
1. Suppose the foreigner makes his statement to a group of islanders C ("contaminated"), and the rest of the islanders P ("pure") do not hear it, and it is known to all that they didn't hear it. Call the group of blue-eyed people B. Then the intersection of C with B will kill themselves after a number of days equal to the size of that group.
2. Nothing. (I interpreted this as being without a statement by a foreigner.)
3. Nothing. (I also interpreted this one as being without a statement by a foreigner. With such a statement, it's the same problem as case 1; everyone will recognize that the baby, having not existed on Foreigner Day, can't know about nor have been mentioned in the statement.)
EDIT:
I should point out that I've assumed the foreigner's statement refers to the group he's addressing, not to the population of the island. ("At least one of you who I see before me has blue eyes".)
With a better interpretation of your problem 2:
2a. On some day, the foreigner addresses a village, saying "at least one person on the island has blue eyes". A blue-eyed stranger wanders into the village shortly after he leaves, allowing the villagers to believe that he was referring to the stranger.
In this case, there is no synchronization point, and "nothing" will still occur.
2b. A blue-eyed stranger wanders into the village the day after the foreigner leaves, allowing the villagers to believe that he was referring to the stranger.
As far as I can see, this has gone back to case 1 again. The foreigner's statement provoked a first day of blue-counting, and while it is revealed to have possibly not meant what they thought it meant, day 1 of blue-counting is sufficient for day 2. The blue-eyed villagers should kill themselves after a number of days equal to the size of their group. (The stranger, even if he settles into the village, will be unaffected.)
"what they don't know (until the 2nd) day) is that all the other blue-eyed islanders know that all the other blue-eyed islanders know that there are 99 other blue-eyed islanders."
I don't see how they know that on the second day, either, though. Before the foreigner's statement, every blue-eyes islanders knows a) that there are at least 99 blue-eyed islanders b) that all the blue-eyed islanders know that there are at least 98 blue-eyed islanders. I don't see what the foreigner's statement adds to this knowledge.
You're right, my explanation is actually wrong. The idea is right, but the recursive knowledge is not gained incrementally the way I described but rather all at once at the end when they wake up on the nth day and see that everyone is still alive.
Each one reasons (recursively) that if their eyes are brown then, given that common knowledge of the existence of blue-eyed people has been established, the remaining n-1 blue-eyed people will realize their eyes are blue (and kill themselves) on the n-1'th day. Since this doesn't happen, their eyes must be blue.
> the blue-eyed islanders all know that there are 99 other islanders with blue eyes
That's true, but what they don't know (until the 2nd) day) is that all the other blue-eyed islanders know that all the other blue-eyed islanders know that there are 99 other blue-eyed islanders. On the 3rd day they will realize that all the other blue-eyed islanders know that all the other blue eyed islanders know that all the other blue eyed islanders know that... and so on. Then on the 100th day there are 100 iterations of recursive knowledge, and all the blue-eyed islanders realize they themselves must have blue eyes.
UPDATE: note that it is crucial that all the islanders are together when the foreigner makes his statement. If he goes to each islander individually and says "some of you have blue eyes" then it doesn't work. What matters is not the statement, but that all the islanders witness all the other islanders hearing the statement.