8.5. The placement of chess kings on a board is called "correct" if no king attacks another, and every square of the board is either under attack or occupied by one of the kings. What is the minimum and maximum number of kings that can be correctly placed on an $8 \times 8$ chessboard?
Answer. The minimum and maximum number of kings is 9 and 16, respectively.

Solution. Divide the board into 16 squares of size $2 \times 2$. In each of them, there can be no more than one king, otherwise they will attack each other, so their number in any correct arrangement does not exceed 16. For example, if there is one king in the lower left corner of each square in the division.

The minimum number of kings is 9. Extend the chessboard to a size of $9 \times 9$, adding an imaginary vertical line on the right and a horizontal line at the top. Divide the resulting board into 9 squares of $3 \times 3$. Place a king in the center of each of these squares. Then all the cells of the $9 \times 9$ board, and thus the original board, are under attack. It is clear that these 9 kings have also landed on the original board, so 9 kings are enough. We will prove that 8 kings are not enough. Consider the first two rows. There must be at least three kings on them (otherwise, some fields of the first row will not be attacked). Consider the seventh and eighth rows. Similarly, there must be at least three kings on them. Now consider the 4th and 5th rows. There must also be at least three kings on them, otherwise, for example, all the fields on the 4th row will not be attacked. Thus, there must be at least 9 kings.