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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
##### Keywords:
Ulam’s game with lies
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##### References:
  Czyzowicz, J.; Mundici, D.; Pelc, A., Ulam’s searching game with lies, J. combin. theory ser. A, 52, 62-76, (1989) · Zbl 0674.90110  Guzicki, W., Ulam’s searching game with two lies, J. combin. theory ser. A, 54, 1-19, (1990) · Zbl 0712.68027  Kleitman, D.J.; Meyer, A.R.; Rivest, R.L.; Spencer, J.; Winklmann, K., Coping with errors in binary search procedures, J. comput. system sci., 20, 396-404, (1980) · Zbl 0443.68043  Pelc, A., Solution of Ulam’s problem on searching with a Lie, J. combin. theory ser. A, 44, 129-140, (1987) · Zbl 0621.68056  Spencer, J., Balancing games, J. combin. theory ser. B, 23, 68-74, (1977) · Zbl 0374.90088  Ulam, S., Adventures of a Mathematician, (1977), Scribners New York · Zbl 0352.01009
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