The sequence gets started by the initial statement, and although in the case of n > 1 it seems as if it doesn't impart any new information, it does. The distin

The sequence gets started by the initial statement, and although in the case of n > 1 it seems as if it doesn't impart any new information, it does. The distinction that matters to get the chain started is the difference between "I personally know there is at least one blue-eyed islander" and "everybody knows there is at least one blue-eyed islander and I know everybody knows it and they know I know it and I know they know I know it... (so on)". Prior to that you can't know that other islanders could deduce all the things you can deduce from available information, and they can't know that you could deduce all the things they could. Afterward, you know exactly what they can deduce from the available information, they know what you can deduce, and you know that it produces a guaranteed conclusion, by day n, that n is the number of blue-eyed islanders.