8.5. On an $8 \times 8$ chessboard, $k$ rooks and $k$ knights are placed such that no figure attacks any other. What is the largest $k$ for which this is possible?
Answer: 5.

Solution: From the condition, it follows that in one row (column) with a rook, no other figure can stand.

Suppose 6 rooks were placed on the board. Then they stand in 6 rows and 6 columns. Therefore, only 4 unpicked cells will remain (located at the intersection of two empty rows and two empty columns). It is impossible to place 6 knights in these cells. Therefore, $k$ is no more than 5.

In Fig. 3, it is shown how to place 5 rooks and 5 knights on the board so that they do not attack each other.

![](https://cdn.mathpix.com/cropped/2024_05_06_3c28e9e0b8098c58331eg-2.jpg?height=411&width=417&top_left_y=1482&top_left_x=822)

Fig. 3

Remark. There are other examples of placement.

Comment. Only proved that $k$ is no more than $5-4$ points.

Only provided an example of placing 5 knights and 5 rooks - 3 points.