8. (5 points) From a $12 \times 12$ grid, a $4 \times 4$ square was cut out, lying at the intersection of the fourth to seventh horizontal and vertical lines. What is the maximum number of non-attacking rooks that can be placed on this field, if the rooks do not attack through the cut-out cells?

## Answer: 15

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# Solution:

The available part of the board is divided into the following four areas, in each of which no more than the specified number of rooks can stand (see the left figure), as in each rectangle, no more rooks can stand than the length of its shorter side. Therefore, the total number of rooks does not exceed 15.

An example with 15 rooks is shown in the right figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_7411c515b7731f5e6328g-6.jpg?height=642&width=1788&top_left_y=667&top_left_x=160)