3. 3.1. Petya marked 5 points on a face of a cube, turned it and marked 6 points on an adjacent face, then turned it again and marked 7 points, and so on. He marked points on each face this way. What is the maximum number of points that can be on two opposite faces?
Answer: 18.

## Solution.

Since a cube has only 6 faces, the maximum number of points marked by Petya is 10. Then the number 9 will be on an adjacent face to the number 10 and cannot be on the opposite face. Therefore, the maximum sum will not exceed 18. The sum of 18 can be achieved, for example, as follows:

![](https://cdn.mathpix.com/cropped/2024_05_06_6a3c59b5c78061150e51g-3.jpg?height=458&width=352&top_left_y=134&top_left_x=908)