What's added is the common knowledge that everyone else knows that Blue > 1 (including the foreigner), and that everyone else knows that everyone else knows tha

What's added is the common knowledge that everyone else knows that Blue > 1 (including the foreigner), and that everyone else knows that everyone else knows that Blue > 1, etc. 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.