Knights Problem

The problem of determining how many nonattacking knights can be placed on an chessboard. For , the solution is 32 (illustrated above). In general, the solutions are

$K(n)={1/2n^2 n>2 even; 1/2(n^2+1) n>1 odd,$

(1)

giving the sequence 1, 4, 5, 8, 13, 18, 25, ... (OEIS A030978, Dudeney 1970, p. 96; Madachy 1979).

The minimal number of knights needed to occupy or attack every square on an chessboard (i.e., domination numbers for the knight graphs) are given for , 2, ... by 1, 4, 4, 4, 5, 8, 10, 12, 14, ... (OEIS A006075), with corresponding numbers of such solutions given by 1, 1, 2, 3, 8, 22, 3, ... (OEIS A006076).

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