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

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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 reachable 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 proofs.

References

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