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Kings Problem


The problem of determining how many nonattacking kings can be placed on an $n\times n$ Chessboard. For $n=8$, the solution is 16, as illustrated above (Madachy 1979). In general, the solutions are

{\textstyle{1\over 4}}n^2 & $n$\ even\cr
{\textstyle{1\over 4}}(n+1)^2 & $n$\ odd\cr}
\end{displaymath} (1)

(Madachy 1979), giving the sequence of doubled squares 1, 1, 4, 4, 9, 9, 16, 16, ... (Sloane's A008794). This sequence has Generating Function
\end{displaymath} (2)


The minimum number of kings needed to attack or occupy all squares on an $8\times 8$ Chessboard is nine, illustrated above (Madachy 1979).

See also Bishops Problem, Chess, Hard Hexagon Entropy Constant, Knights Problem, Queens Problem, Rooks Problem


Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 39, 1979.

© 1996-9 Eric W. Weisstein