1. What is the minimum number of cells that need to be marked in a $20 \times 20$ table so that each vertical or horizontal strip of $1 \times 12$ contains at least one marked cell.
Answer: 32

Solution. Let's cut the $20 \times 20$ table without the central $4 \times 4$ square into 32 rectangles of $1 \times 12$ (see the left figure). Therefore, we will need to mark at least 32 cells. An example with 32 cells: all cells of three parallel diagonals of lengths 4, 16, and 12 are marked (see the right figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_e0197926f6acc0f95da5g-15.jpg?height=731&width=736&top_left_y=717&top_left_x=183)

![](https://cdn.mathpix.com/cropped/2024_05_06_e0197926f6acc0f95da5g-15.jpg?height=723&width=734&top_left_y=718&top_left_x=1072)