8.1. From a $7 \times 7$ square grid, an equal number of $2 \times 2$ squares and $1 \times 4$ rectangles were cut out along the grid lines. What is the maximum number of these figures that could have been cut out?
Answer: 12.

Solution: Both the square and the rectangle consist of 4 cells. Therefore, the number of cut-out figures is no more than 49/4, that is, no more than 12. There are an equal number of figures of both types, so there are no more than 6 squares $2 \times 2$ and rectangles $1 \times 4$. The diagram shows how to cut out 6 squares $2 \times 2$ and 6 rectangles $1 \times 4$ from the square.

![](https://cdn.mathpix.com/cropped/2024_05_06_01eaca71e034b8dd1883g-1.jpg?height=220&width=237&top_left_y=2380&top_left_x=955)

Comment: Only proved that there are no more than twelve figures - 3 points. Only provided an example of placing 12 figures - 6 points.