Exercise 4. Consider an $8 \times 8$ chessboard whose squares are alternately colored white and black. An infernal rook is a piece that can attack squares of its own color on its row, as well as squares of the opposite color on its column. What is the maximum number of infernal rooks that can be placed on the chessboard such that no two infernal rooks can ever attack each other?
## Solution to Exercise 4

To show that a certain integer $c$ is the largest integer satisfying a certain property, there are necessarily two distinct parts: the analysis, in which we establish that any integer $n$ satisfying the stated property satisfies $n \leqslant c$, and the construction, in which we provide an example of $c$ towers that meet the conditions of the statement.

## Analysis:

First, we note that if two towers belong to the same row without attacking each other, they must be on different colored squares. Thus, there can be at most two towers per row. Since there are 8 rows, there can be at most $2 \times 8 = 16$ towers on the chessboard.

## Construction:

Conversely, we show that it is possible to place 16 towers on the chessboard.

|  | $\bigcirc$ |  |  |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  |  | $\bigcirc$ | $\bigcirc$ |  |  |  |  |
|  | $\bigcirc$ |  |  |  |  |  |  |
|  |  | $\bigcirc$ | $\bigcirc$ |  |  |  |  |
|  | $\bigcirc$ |  |  |  |  |  |  |
|  |  | $\bigcirc$ | $\bigcirc$ |  |  |  |  |
|  | $\bigcirc$ |  |  |  |  |  |  |
|  |  | $\bigcirc$ | $\bigcirc$ |  |  |  |  |

In the configuration below, where each red dot represents a tower, we verify that no two towers attack each other. Moreover, there are indeed 16 towers on the chessboard.

Thus, the maximum number of towers that can be placed is 16.