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

• 作者: Amanda M. Miller David L. Farnsworth
• 来源期刊: 离散数学期刊(英文)
• 学科: 数学
• 关键词: COUNTING Knight’s MOVES INFINITE CHESSBOARD Geometric ARGUMENT
• 年，卷（期）: 2013,（3）
• 页码范围: 151-154
• 页数: 4页
• 分类号: O1

摘要

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.