I’ve got your number: a multiple choice guessing game

We consider the following guessing game: fix positive integers k, m, and n. Player A (“Ann”) chooses a (uniformly) random integer α from the set {1,2,3, . . . ,n}, but does not reveal α to Player B (“Gus”). Gus then presents Ann with a k-option multiple choice question about which number she chose, to which Ann responds truthfully. After m such questions have been asked, Gus must attempt to guess the number chosen by Ann. Gus wins if he guesses α. The purpose of this note is to find all “canonical” m-question algorithms which maximize the probability of Gus winning the game. An analysis of a natural extension of this game is also presented.