Knights, spies, games and ballot sequences

In this paper we solve the Knights and Spies Problem: In a room there are n people, each labelled with a unique number between 1 and n. A person may either be a knight or a spy. Knights always tell the truth, while spies may either lie or tell the truth, as they see fit. Each person in the room knows the identity of everyone else. Apart from this, all that is known is that strictly more knights than spies are present. Asking only questions of the form: ‘Person i, what is the identity of person j?’, what is the least number of questions that will guarantee to find the true identities of all n people? Despite its apparently recreational character, the Knights and Spies Problem is surprisingly deep; it is unusual to find such an easily stated problem that can challenge and be enjoyed by professionals and amateurs alike. The following remarks introduce some basic ideas and should clarify its statement.