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Ulam’s searching game with a fixed number of lies. (English) Zbl 0749.90102
There are three integer parameters, \(n\), \(q\), and \(k\), in the (2-player) game of the title. Carole thinks of an integer \(x\) from \(1,2,\dots,n\). Paul may ask \(q\) questions, of the form “is \(x\) in \(A\)?”, where \(A\) is a subset of \(1,2,\dots,n\), and may use previous answers to decide his next question. Carole may lie at most \(k\) times in answering the questions. Paul wins if after \(q\) questions \(x\) is uniquely determined. If \(k=0\) it is obvious that Paul has a sure win if and only if \(n=2^ q\). The case \(k=1\) was solved by A. Pelc [J. Comb. Theory, Ser. A 44, 129-140 (1987; Zbl 0621.68056)] and several other recent papers have obtained partial results for \(k>1\). In the present paper a generalization of the game with \(n\) replaced by a sequence of nonnegative integers \(x_ 0,x_ 1,\dots,x_ k\) is analyzed. Let \(A_ 0,A_ 1,\dots,A_ k\) be disjoint sets with \(| A_ i|=x_ i\). Carole selects \(x\in A_ 0\cap\cdots\cap A_ k\). If \(x\in A_ i\) then Carole is permitted to lie at most \(k-i\) times. The original game is the case \(x_ 0=n\), \(x_ 1=\cdots=x_ k=0\). For fixed \(k\), and \(q\) sufficiently large and dependent on \(k\), necessary and sufficient conditions on \(n\) for Paul to have a sure win in the generalized game are obtained.

MSC:
91A46 Combinatorial games
91A05 2-person games
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