1. (7 points) In an $8 \times 8$ frame with a width of 2 cells (see figure), there are a total of 48 cells. How many cells are in a $254 \times 254$ frame with a width of 2 cells?

![](https://cdn.mathpix.com/cropped/2024_05_06_af60849ded79da38d4cfg-1.jpg?height=288&width=280&top_left_y=661&top_left_x=891)
Answer: 2016.

Solution. First method. Cut the frame into four identical rectangles as shown in the figure. The width of the rectangles is equal to the width of the frame, i.e., 2 cells. The length of each rectangle is 2 less than the side of the frame: $254-2=252$ cells. Then the area of one rectangle is $2 \cdot 252=504$. Therefore, the total number of cells in the frame is $4 \cdot 504=2016$.

![](https://cdn.mathpix.com/cropped/2024_05_06_af60849ded79da38d4cfg-1.jpg?height=283&width=280&top_left_y=1355&top_left_x=888)

Second method. The area of the frame can be obtained by subtracting the area of the inner square from the area of the large square $254 \times 254$. The side of the inner square is 4 cells shorter than the side of the large square. Therefore, the area of the frame is

$$
254^{2}-250^{2}=(254-250)(254+250)=4 \cdot 504=2016.
$$

Remark. If the side of the frame is denoted by $n$, it can be proven (for example, by the methods described above) that its area will be $8 n-16$ cells.

## Grading Criteria.

- Any complete correct solution - 7 points.
- Correct approach, but an arithmetic error was made - 3 points.
- Correct reasoning, but an error was made in estimating the dimensions (for example, in the second method, it is incorrectly assumed that the side of the inner square is 2 cells shorter than the large one) - 2 points.
- Only the correct answer - 1 point.