 The “paradox” of this argument is that for \(k \gt 1\), like the case of the clumsy waiter of Example 1.1, the cook’s announcement told the diners something that each already knew. Yet apparently the cook’s announcement also gave the diners useful information. How could this be? By announcing a fact already known to every diner, the cook made this fact common knowledge among them, enabling each of them to eventually deduce the condition of his own face after sufficiently many rings of the  bell.[4]