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 3×3 to 10×10, plus 12×12 and 13×y3 have been known since the 19th century. The best solution for 11×11 was found in 1973 by Bernard Lemaire, and Davis found the best solution for 14×14 in 1977.
       On this site we present solutions for boards as large as 50×50. The solutions for boards 20×20 through 26×26 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 10×10 proving optimality is a much more difficult problem than finding a good cover.
       As of August 2005, the solutions for boards up through 20×20 have been verified as optimal by Lee Morgenstern using exact methods. He has also developed optimality proofs for square boards up to 13×13, and for several rectangular boards.
       To view the solutions click on any of the links in the box below.

3×3 through 10×10 boards
11×11 through 14×14 boards
15×15 board
16×16 board
17×17 board
18×18 board
19×19 board
20×20 board
21×21 board
22×22 board
23×23 board
24×24 board
25×25 board
26×26 board
27×27 board
28×28 board
29×29 board
30×30 board
31×32 board
35×35 board
40×40 board
45×45 board
50×50 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 N×N Knights Covering Problem," J. Recr. Math. 23(1991), pp 255-267.
David C. Fisher, "On the N×N Knight Cover Problem," Ars Combinatoria 69(2003), pp 255-274.
Bernard Lemaire, "Knights Covers on N×N 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 50×50 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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