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.