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A characterization on \(n\)-critical economical generalized tic-tac-toe games. (English) Zbl 0781.90097

Summary: There is a so-called generalized tic-tac-toe game playing on a finite set \(X\) with winning sets \(A_ 1,A_ 2,\dots,A_ m\). Two players, \(F\) and \(S\), take in turn a previous untaken vertex of \(X\), with \(F\) going first. The one who takes all the vertices of some winning set first wins the game. P. Erdős and J. L. Selfridge [J. Combinat. Theory, Ser. A 14, 298-301 (1973; Zbl 0293.05004)] proved that if \(| A_ 1|=| A_ 2|=\cdots=| A_ m|= n\) and \(m< 2^{n-1}\), then the game is a draw. This result is best possible in the sense that once \(M=2^{n-1}\), then there is a family \(A_ 1,A_ 2,\dots,A_ m\) so that \(F\) can win. In this paper we characterize all those sets \(A_ 1,\dots,A_{2^{n-1}}\) so that \(F\) can win in exactly \(n\) moves. We also get similar results in the biased games.

MSC:

91A46 Combinatorial games

Citations:

Zbl 0293.05004
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Full Text: DOI

References:

[1] Beck, J., Remarks on positional games, Acta Math. Acad. Sci. Hungar., 40, 65-71 (1982) · Zbl 0515.90100
[2] Beck, J.; Csirmaz, L., Variations on a game, J. Combin. Theory Ser. A, 33, 297-315 (1982) · Zbl 0505.90100
[3] Erdo&#x030B;s, P.; Selfridge, J., On a combinatorial game, J. Combin. Theory, 14, 298-301 (1973)
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