Optimal comparison strategies in Ulam's searching game with two errors

AbstractSuppose x is an n-bit integer. By a comparison question we mean a question of the form “does x satisfy either condition a ⩽x ⩽b or c ⩽x ⩽d?”. We describe strategies to find x using the smallest possible number q(n) of comparison questions, and allowing up to two of the answers to be erroneous. As proved in this self-contained paper, with the exception of n = 2, q(n) is the smallest number q satisfying Berlekamp's inequality 2q⩾2nq2+ q + 1. This result would disappear if we only allowed questions of the form “does x satisfy the condition a⩽x⩽b?”. Since no strategy can find the unknown x ∈ {0,1,…,2n −1} with less than q(n) questions, our result provides extremely simple optimal searching strategies for Ulam's game with two lies—the game of Twenty Questions where up to two of the answers may be erroneous