The fewest kings controlling the board

Lower bound. Number rows and columns $1,\dots,8$ and look at the $9$ squares with both coordinates in $\{1,4,7\}$. Two distinct such squares differ by at least $3$ in some coordinate. A king controls squares whose coordinates differ from its own by at most $1$ in each, so two squares it controls differ by at most $2$ in each coordinate. Hence no king controls two of the $9$ squares, and at least $9$ kings are needed.

Upper bound. Place kings on the $9$ squares with row and column in $\{2,5,8\}$. A king in row $r$ controls rows $r-1,r,r+1$ (restricted to the board), so the kings in rows $2,5,8$ control rows $\{1,2,3\},\{4,5,6\},\{7,8\}$, i.e. all rows; the same holds for columns. Every square is therefore controlled.

Answer. $9$.

Remark. For an $n\times n$ board the answer is $\lceil n/3\rceil^2$ by the same two arguments.

Lesson. To prove minimality, exhibit a set of "far apart" objects that each need their own resource.