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Kings Problem

\begin{figure}\begin{center}\BoxedEPSF{KingsMax.epsf}\end{center}\end{figure}

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

\begin{displaymath}
K(n)=\cases{
{\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
\begin{displaymath}
{1+x^2\over(1-x^2)^2(1-x)}=1+x+4x^2+4x^3+9x^4+9x^5+\ldots.
\end{displaymath} (2)


\begin{figure}\begin{center}\BoxedEPSF{KingsMin.epsf}\end{center}\end{figure}

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


References

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




© 1996-9 Eric W. Weisstein
1999-05-26