No. There is nothing wrong with the induction and the second argument is the correct one. This is a well known puzzle that became very popular on Terrence Tao'

No. There is nothing wrong with the induction and the second argument is the correct one. This is a well known puzzle that became very popular on Terrence Tao's blog (http://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/ http://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islan...) - where you can also find the solution (and various wrong attempts) in the comments. With 3 blue-eyed people: Everybody knows that there are other people with blue eyes. Everybody knows that everybody knows that there are other people with blue eyes. However, prior to the visitor's statement, nobody knows that everybody knows that everybody knows that there are other people with blue eyes. Because, consider the case where there are 3 blue eyed people named A,B,C. A knows that B knows that C has blue eyes. However A cannot possibly know that B knows that C knows that somebody has blue-eyes, because that somebody can either be B - in which case B has no way of knowing that, and therefore no way of knowing that C knows that, A - in which case A has no way of knowing that , and therefore no way of knowing that B knows that C knows that, or C, in which case C has no way of knowing that, and therefore B has no way of knowing that C knows that. Following the visitor's comment - A does know that, and eventually everybody kills himself. Formally this new information is knows as "common knowledge": http://en.wikipedia.org/wiki/Common_knowledge_(logic) http://en.wikipedia.org/wiki/Common_knowledge_(logic)