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J. Mukherjee, P. P. Das, M. Aswatha Kumar and B. N. Chatterji, “On Approximating Euclidean Metrics by Digital Distances in 2D and 3D,” Pattern Recognition Letters, Vol. 21, No. 6-7, 2000, pp. 573-582. doi:10.1016/S0167-8655(00)00022-2

has been cited by the following article:

• 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.