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Optimal comparison strategies in Ulam’s searching game with two errors. (English) Zbl 0902.90191
Summary: Suppose \(x\) 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 \(2^{q}\geqslant 2^{n} ((^{q}_{2})+q+1)\). 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 \(x\in \{0,1, \dots, 2^{n}-1\}\) 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.

MSC:
91A46 Combinatorial games
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References:
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