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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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