 The general case follows by induction. Suppose that if \(k = j\), then each of the \(j\) messy diners can determine that he is messy after \(j\) rings. Then if \(k = j + 1\), then at the \(j + 1\)st ring, each of the \(j + 1\) individuals will realize that he is messy. For if he were not messy, then the other \(j\) messy ones would have all realized their messiness at the \(j\)th ring and cleaned themselves then. Since no one cleaned herself after the \(j\)th ring, at the \(j + 1\)st ring each messy person will conclude that someone besides the other \(j\) messy people must also be messy, namely, himself.