KNIGHT COVERINGS

KNIGHT COVERINGS
FOR LARGE CHESSBOARDS
Frank Rubin
November 9, 2000

       The problem of how to cover a chessboard with the smallest number of knights has fascinated both mathematicians and chess players for years. The problem is to place the smallest number of knights so that every square on the board is covered. That is, every square on the board is either occupied by a knight or attacked by a knight.
       The optimal solutions for boards of sizes 3x3 to 10x10, plus 12x12 and 13x13 have been known since the 19th century. The best solution for 11x11 was found in 1973 by Bernard Lemaire, and Davis found the best solution for 14x14 in 1977.
       On this site we present solutions for boards as large as 50x50. The solutions for boards 20x20 through 26x26 were first discovered in Oct. and Nov. 2000 by Frank Rubin (the webmaster of this site). It is likely that some or all of these coverings are optimal, but for boards larger than 10x10 proving optimality is a much more difficult problem than finding a good cover.
       As of August 2005, the solutions for boards up through 20x20 have been verified as optimal by Lee Morgenstern using exact methods. He has also developed optimality proofs for square boards up to 13x13, and for several rectangular boards.
       To view the solutions click on any of the links in the box below.

3x3 through 10x10 boards
11x11 through 14x14 boards
15x15 board
16x16 board
17x17 board
18x18 board
19x19 board
20x20 board
21x21 board
22x22 board
23x23 board
24x24 board
25x25 board
26x26 board
27x27 board
28x28 board
29x29 board
30x30 board
31x32 board
35x35 board
40x40 board
45x45 board
50x50 board

Optimality Proofs

You might also enjoy the Contest Center's
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Bishop Covering webpage
Mathematical Puzzles webpages
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REFERENCES

Anderson H. Jackson, Roy P. Pargas, "Solutions to the NxN Knights Covering Problem," J. Recr. Math. 23(1991), pp 255-267.
David C. Fisher, "On the NxN Knight Cover Problem," Ars Combinatoria 69(2003), pp 255-274.
Bernard Lemaire, "Knights Covers on NxN Chessboards," J. Recr. Math. 31(2003), pp 87-99.
Frank Rubin, "Improved Knight Coverings," Ars Combinatoria 69(2003), pp 185-196.
Frank Rubin, "Knight Covers for the 50x50 Chessboard," Mathfest 2004, Providence RI.
Frank Rubin, "A Family of Efficient Knight Covering Patterns," J. Recr. Math. 33(2004), pp 165-175.

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