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P. P. Das, “An Algorithm for Computing the Number of the Minimal Paths in Digital Images,” Pattern Recognition Letters, Vol. 9, No. 2, 1989, pp. 107-116. doi:10.1016/0167-8655(89)90043-3

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• TITLE: Counting the Number of Squares Reachable in k Knight’s Moves

  AUTHORS: Amanda M. Miller, David L. Farnsworth

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

  JOURNAL NAME: Open Journal of Discrete Mathematics, Vol.3 No.3, July 12, 2013

  ABSTRACT: Using geometric techniques, formulas for the number of squares that require k moves in order to be reached by a sole knight from its initial position on an infinite chessboard 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, and 28k – 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 more mathematically accessible then preceding proofs.