The king is a chess piece that can move only one square vertically, one horizontally, or one diagonally. We say that a king attacks a square if it can occupy it with a single move. For example, a king placed in the central squares of a $6 \times 6$ board attacks 8 squares, a king placed on the side squares attacks 5 squares, and a king positioned in one of the four corners of the board attacks only 3 squares.

![](https://cdn.mathpix.com/cropped/2024_05_01_80f7f1185898b39e7876g-23.jpg?height=582&width=580&top_left_y=1071&top_left_x=841)

a) Consider a $6 \times 6$ board. What is the minimum number of kings that can be placed on the board so that all the squares of the board are either occupied or attacked by one of the kings?

b) Still considering the $6 \times 6$ board, what is the maximum number of kings that can be placed on the board so that they do not attack each other?

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Solution

a) Divide the $6 \times 6$ board into 4 $3 \times 3$ boards. If one of these four regions does not have a king, the central cell of that region will not be occupied or attacked by any king. Therefore, at least 4 kings are needed. If we place a king in the central cell of each $3 \times 3$ board, then all cells of the board will be attacked. Thus, the minimum number of kings is 4.

b) Divide the $6 \times 6$ board into 9 $2 \times 2$ boards. If two kings are in the same $2 \times 2$, then they will be attacking each other. Therefore, we can have at most 9 kings. If we place a king in the top-left corner of each of these $2 \times 2$ boards, we will have 9 kings that do not attack each other.