 How did the messy diners finally realize that their faces needed cleaning? The \(k = 1\) case is easy, since in this case, the lone messy individual will realize he is messy immediately, since he sees that everyone else is clean. Consider the \(k = 2\) case next. At the first ring, messy individual \(i_1\) knows that one other person, \(i_2\), is messy, but does not yet know about himself. At the second ring, \(i_1\) realizes that he must be messy, since had \(i_2\) been the only messy one, \(i_2\) would have known this after the first ring when the cook made his announcement, and would have cleaned her face then. By a symmetric argument, messy diner \(i_2\) also concludes that she is messy at the second ring, and both pick up a napkin at that time.