7. Variant 1.

In a $101 \times 101$ square, an $88 \times 88$ corner square is painted red. What is the maximum number of non-attacking queens that can be placed on the board without placing the figures on the red cells? A queen attacks along the horizontal, vertical, and diagonals of the square. Attacking through the painted cells is allowed.
Answer: 26.

Solution. Consider 13 horizontal and 13 vertical strips of size $1 \times 101$, not containing any red cells. Each of the queens stands on at least one of these strips, and no strip can contain more than one queen, so there are no more than 26 queens. To construct an example, consider two rectangles of size $13 \times 25$ (see the figure) and place the queens in each of them according to the scheme shown in the figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_0d3dbdd740b551c86dd8g-6.jpg?height=580&width=986&top_left_y=179&top_left_x=514)