8. (5 points) From a $12 \times 12$ grid, a $4 \times 4$ square was cut out, lying at the intersection of the third to sixth horizontal lines and the same 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: 14

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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 be placed (see the left figure), as in each rectangle, no more rooks can be placed than the length of its shorter side. Therefore, the total number of rooks does not exceed 14.

An example for 14 rooks is shown in the right figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_7411c515b7731f5e6328g-3.jpg?height=636&width=1772&top_left_y=778&top_left_x=176)

## Open Mathematical Olympiad for Schoolchildren

March 11, 2018

## 7th Grade  2nd Variant  Solutions