  The $m \times n$ king graph consists of all locations on an $m \times n$
chessboard, where edges are legal moves of a chess king. %where each vertex
represents a square on a chessboard and each edge is a legal move. Let $P_{m
\times n}(z)$ denote its domination polynomial, i.e., $\sum_{S \subseteq V}
z^{|S|}$ where the sum is over all dominating sets $S$. We prove that $P_{m
\times n}(-1) = (-1)^{\lceil m/2\rceil \lceil n/2\rceil}$. In particular, the
number of dominating sets of even size and the number of odd size differs by
$\pm 1$. %The numbers can not be equal because the total number of dominating
sets is always odd. This property does not hold for king graphs on a cylinder
or a torus, or for the grid graph. But it holds for $d$-dimensional kings,
where $P_{n_1\times n_2\times\cdots\times n_d}(-1) = (-1)^{\lceil n_1/2\rceil
\lceil n_2/2\rceil\cdots \lceil n_d/2\rceil}$.
