## Task A-4.2.

The upper right quarter of the chessboard (dimension $8 \times 8$) is covered with paper. How many rooks can we place on the remaining part of the chessboard so that they do not attack each other? In how many ways can this be done?

(Two rooks attack each other if they are in the same row or in the same column.)
## Solution.

Obviously, it is not possible to place more than 8 rooks since each row can have at most one. 1 point The example in the image shows that it is possible to arrange eight rooks in the desired way.

![](https://cdn.mathpix.com/cropped/2024_05_30_abcf4bcd4243d7db1a46g-19.jpg?height=651&width=645&top_left_y=1970&top_left_x=631)

Let's consider all the ways this can be done.

It is clear that in the lower half of the board, there cannot be more than four rooks, so in the upper part of the board, that is, in the upper left part of the board, there must be four rooks, each in its own row and column.

In the lower left quarter of the board, there cannot be any rooks, as all those columns are already occupied, so the remaining four rooks are in the lower right quarter.

Four rooks on a $4 \times 4$ board can be arranged in $4!=24$ ways.

Indeed, in the first row, a rook can be in any of the four positions. In the second row, a rook cannot be in the same column as the first rook, so there are three possibilities. In the third row, there are only two unattacked positions left, and finally, in the fourth row, only one position remains.

Therefore, the total number of possible arrangements is $24^{2}=576$.