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Further analysis on the “king and rook vs. king on a quarter-infinite board” problem. (English) Zbl 1194.91060

Summary: In [“Unsolved problems in combinatorial games”, in: More games of no chance. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 42, 457–473 (2002; Zbl 1047.91008)] by R. K. Guy and R. J. Nowakowski posed the following open problem: Played on a quarter-infinite board, with initial position WKa1, WRb2 and BKc3. Can White win? R. M. Low and M. Stamp [Integers 6, Paper G03, 8 p. (2006; Zbl 1157.91323)] showed that White has a winning strategy that can be implemented within a \(9 \times 11\) region. In this short note, we extend this result and show the following: With initial position WK\((1,1)\), WR\((x,y)\) and BK\((a,b)\), where \(1 < x < a\), White has a winning strategy that can be implemented within an \((a+b+3) \times (a+b+5)\) region.

MSC:

91A46 Combinatorial games
00A08 Recreational mathematics
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