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

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 28 k – 20 for k ≥ 5. The cumulative number of squares r each able in k or fever moves are 1, 9, 41, and 109 for k = 0, 1, 2, and 3, respectively, and 14 k 2 – 6 k + 5 for k ≥ 4. Although these formulas are known, the proofs that are presented are new and more mathematically accessible then preceding proo fs.