Knights, spies, games and ballot sequences

This paper solves the Knights and Spies Problem: In a room there are nn people, each labelled with a unique number between 11 and nn. A person may either be a knight or a spy. Knights always tell the truth, while spies may 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 ii, what is the identity of person jj?’, what is the least number of questions that will guarantee to find the true identities of all nn people? We present a questioning strategy that uses slightly less than 3n/23n/2 questions, and prove that it is optimal by solving a related two-player game. The performance of this strategy is analysed using methods from the famous ballot-counting problem. We end by discussing two questions suggested by generalisations of the original problem.