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Usinggeometric techniques, formulas for the number of squares that require k moves in order to be reached by asole knight from its initial position on an infinitechessboard are derived. The number of squares reachable in exactly k moves are 1, 8, 32, 68, and 96 for k = 0, 1, 2, 3, and 4, respectively, and28k – 20 for k ≥ 5. The cumulative number of squares reachable in k or fever moves are 1, 9, 41, and 109 for k = 0, 1, 2, and 3, respectively, and 14k2 – 6k + 5 for k ≥ 4. Although these formulas are known, the proofs that are presented are new and moremathematically accessible then preceding proofs.

KEYWORDS

Counting; Knight’s Moves; Infinite Chessboard; Geometric Argument

Cite this paper

A. Miller and D. Farnsworth -Counting the Number of Squares Reachable in k Knight’s Moves,- Open Journal of Discrete Mathematics, Vol. 3 No. 3, 2013, pp. 151-154. doi: 10.4236-ojdm.2013.33027.





Autor: Amanda M. Miller, David L. Farnsworth

Fuente: http://www.scirp.org/



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