And for an even more detailed examination of the underlying philosophical issues:
What's most interesting about this scenario is that, for k > 1, the outsider
is only telling the island citizens what they already know: that there
are blue-eyed people among them. However, before this fact is announced,
the fact is not common knowledge.
I mean, what bit of information is added here?
I think if there were an earlier all-hands-meeting without the traveler, the counts would have been synchronized then.
So, the faux pas had no effect, but he still "caused" it by accident. So, to combine the two arguments from the main link:
The foreigner's words have no effect, because his comments do not tell the tribe anything that they do not already know. On the 100th day, the blue eyed people commit suicide, unless they die/leave/disappear first.
Consider two blue-eyed people, Alice and Bob. Alice sees Bob's blue eyes and knows that Blue ≥ 1, and vice-versa. But Alice thinks, "What if I have brown eyes? In that case, Bob wouldn't know that Blue ≥ 1." So, everyone knows that Blue ≥ 1, but nobody knows that everyone knows that. Then the foreigner comes along and tells them that, between Alice and Bob, Blue ≥ 1. Now Alice knows that Bob knows that Blue ≥ 1, and realizes that, if Alice has non-blue eyes, Bob will use the new information that Blue ≥ 1 to conclude that he has blue eyes, and therefore commit suicide on the next day. When he doesn't, she concludes that she must have blue eyes, and commits suicide. Bob goes through the exact same logic as Alice.
Though the foreigner's statement did not tell Alice anything new about eye color distribution, it did tell her something about Bob's knowledge. The same goes for Bob, who learns about Alice's knowledge.
The logic is a bit more difficult to talk through with three people, so generalizing it further is left as an exercise ;P It comes down to "everyone knows that everyone knows that Blue ≥ 1", which the foreigner also contributes as common knowledge by making his announcement. For more blue-eyed people, recur on that statement as many times as necessary.