7.4. What is the minimum number of kings that need to be placed on a chessboard so that they attack all unoccupied squares? (A king attacks the squares that are adjacent to its square by side or corner).

![](https://cdn.mathpix.com/cropped/2024_05_06_39d633e886f69700a0e8g-1.jpg?height=462&width=457&top_left_y=1485&top_left_x=297)
Answer: 9 kings. Solution. The required arrangement of nine kings is shown in the figure. Let's prove that it is impossible to manage with a smaller number. Indeed, consider the 9 rectangular zones outlined on the figure with bold lines. If we assume the opposite, then at least one of these zones does not contain a king. However, in this case, the cell in this zone marked with a cross will not be attacked by any king located in another zone, since this cell does not border with other zones.

## 11th grade