# Task № 3.1

## Condition:

On a sheet of graph paper, there are three $5 \times 5$ squares, as shown in the figure. How many cells are covered by exactly two squares?

![](https://cdn.mathpix.com/cropped/2024_05_06_4cfa7cdc7a8a51b9e752g-09.jpg?height=596&width=671&top_left_y=593&top_left_x=698)
Answer: 15

Exact match of the answer -1 point

## Solution 1

Direct calculation. We will draw the squares and shade the cells that are covered exactly twice.

![](https://cdn.mathpix.com/cropped/2024_05_06_4cfa7cdc7a8a51b9e752g-09.jpg?height=457&width=506&top_left_y=1636&top_left_x=775)

Solution 2.

The total area covered by the squares is $13+25+18=56$. It is easy to understand that all three squares cover exactly two cells. The total area of all squares is 75. The cells covered once (56 cells) are counted once, the cells covered twice (x cells) are counted twice, and the cells covered three times (2 cells) are counted three times. Thus, when adding the areas of all squares, the number $x$ is counted twice, and the number 2 is counted three times. Therefore, we need to subtract one extra $x$ and two times the number 2. We get the equation $56=75-x-2 \cdot 2$, from which $x=15$.

![](https://cdn.mathpix.com/cropped/2024_05_06_4cfa7cdc7a8a51b9e752g-10.jpg?height=417&width=899&top_left_y=234&top_left_x=584)