A simple solution to Ulam's liar game with one lie

Abstract

We consider the following game between a questioner and a responder, first proposed by Ulam [9]. (A variation of this game was independently proposed by Renyi, see [5].) Both Renyi and Ulam were motivated by questions arising from communication over noisy channels. The responder thinks of an integer x ∈ {1, . . . , n} and the questioner must determine x by asking questions whose answer is ‘Yes’ or ‘No’. The responder is allowed to lie at most k times during the game. Let qk(n) be the maximum number of questions needed by the questioner, under an optimal strategy, to determine x under these rules. In particular, Ulam asked for the value of q1(10) (as this is related to the well-known ‘twenty questions’ game). It follows from an observation of Berlekamp [1] that q1(10) ≥ 25 and