5. A $5 \times 5$ square is to be cut into two types of rectangles: $1 \times 4$ and $1 \times 3$. How many rectangles can result from the cutting? Justify your answer.
5. Answer: 7.

Solution. Example:
![](https://cdn.mathpix.com/cropped/2024_05_06_8cb05b8bebe053562484g-3.jpg?height=226&width=586&top_left_y=1044&top_left_x=512)

If there are no more than 6 rectangles, then they occupy no more than $6 \cdot 4 = 24$ cells, while there are 25 cells. Therefore, there must be at least 7 rectangles.

A rectangle that is completely located in a column will be called vertical, and one that is completely located in a row will be called horizontal. Suppose we managed to cut the square into such rectangles. Consider the rectangle that contains the central cell of the $5 \times 5$ square. If necessary, rotate the $5 \times 5$ square so that this rectangle is horizontal. The cells of this rectangle are located in at least three columns, and there are no vertical rectangles in these columns. There are a total of 10 possible rectangles: 5 vertical and 5 horizontal. But 3 vertical rectangles are impossible, so there can be no more than $10 - 3 = 7$ rectangles.

The number of rectangles can only be 7.

## Grading Criteria.

Only the example: 2 points.

Proved that there are at least 7 rectangles: 1 point.

Shown that there are no more than 9 rectangles, as 9 would occupy at least 27 cells: 1 point.

Attempted to enumerate to prove that there cannot be 8 rectangles: 1 point.