an:01177768
Zbl 0902.90191
Mundici, Daniele; Trombetta, Alberto
Optimal comparison strategies in Ulam's searching game with two errors
EN
Theor. Comput. Sci. 182, No. 1-2, 217-232 (1997).
00042334
1997
j
91A46
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.