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Rook endgame problems in \(m\) by \(n\) chess. (English) Zbl 1378.91039

Summary: We consider Chess played on an \(m \times n\) board (with \(m\) and \(n\) arbitrary positive integers), with only the two Kings and the White Rook remaining, but placed at arbitrary positions. Using the symbolic finite state method, developed by Thanatipanonda and Zeilberger, we prove that on a \(3 \times n\) board, for almost all initial positions, White can checkmate Black in \(\leq n + 2\) moves, and that this upper bound is sharp. We also conjecture that for an arbitrary \(m \times n\) board, with \(m, n \geq 4\) (except for \((m, n) = (4, 4)\) when it equals 7), the number of needed moves is \(\leq m + n\), and that this bound is also sharp.

MSC:

91A46 Combinatorial games

Software:

ToadsAndFrogs
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References:

[1] Elkies, N. D., On numbers and endgames: combinatorial game theory in chess endgames, (Games of No Chance, Math. Sci. Res. Inst. Publ., vol. 29, (1996)), 135-150 · Zbl 0873.90142
[2] Thanatipanonda, T. A., How to beat capablanca, Adv. in Appl. Math., 40, 2, 266-270, (2008) · Zbl 1213.91046
[3] Thanatipanonda, T. A.; Zeilberger, D., A symbolic finite-state approach for automated proving of theorems in combinatorial game theory, J. Difference Equ. Appl., 15, 111-118, (2009) · Zbl 1232.91071
[4] Wikipedia, Eight queens puzzle
[5] Wikipedia, Rook polynomial
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