4. Alexei drew 7 lines on a plane, which divided it into several parts. Then he chose two adjacent parts (adjacent parts are those that share a common side), counted how many sides each of them contains, and added these two numbers. What is the largest number he could have obtained? Explain your answer.

![](https://cdn.mathpix.com/cropped/2024_05_06_ab9ddc946a4ab378ccc3g-1.jpg?height=525&width=357&top_left_y=457&top_left_x=1566)
4. Answer: 10.

Solution. Consider any two adjacent parts. Temporarily remove the line containing their boundary. Then these two parts will become one large part. It has a maximum of 6 sides (each line can contain no more than one side of this part). Return the line we removed. It will be a side for both small parts (+2),

![](https://cdn.mathpix.com/cropped/2024_05_06_64b656e4ffe249f631e0g-1.jpg?height=540&width=372&top_left_y=518&top_left_x=1590)
and it intersects a maximum of two sides of the large part, turning one large side into two small sides (+2).

Example. Lines containing the sides of a regular hexagon and a line passing through the midpoints of two of its adjacent sides. The two parts forming the hexagon have a total of 10 sides.