1. What is the minimum number of cells that need to be marked in a $15 \times 15$ table so that each vertical or horizontal strip $1 \times 10$ contains at least one marked cell.
Answer: 20

Solution. Let's cut the $15 \times 15$ table without the central $5 \times 5$ square into 20 rectangles of $1 \times 10$ (see the left figure). Therefore, we will need to mark at least 20 cells. An example with 20 cells: all cells of two parallel diagonals of length 10 are marked (see the right figure).
![](https://cdn.mathpix.com/cropped/2024_05_06_e0197926f6acc0f95da5g-10.jpg?height=554&width=1276&top_left_y=685&top_left_x=183)