9.1. What is the maximum number of different rectangles that an 8 by 8 chessboard can be cut into? All cuts must follow the grid lines. Rectangles are considered different if they are not equal as geometric figures.
Answer: 12.

Solution. Let's list the possible sizes of different integer rectangles of minimal areas that can fit along the grid lines on an 8 by 8 board in ascending order of these areas: 1 by 1, 1 by 2, 1 by 3, 1 by 4, 2 by 2, 1 by 5, 1 by 6, 2 by 3, 1 by 7, 1 by 8, 2 by 4, 3 by 3, 2 by 5. There are already 13 rectangles, and the sum of their areas is 73, which is greater than the area of the board. Therefore, it is impossible to cut the board into more than 12 rectangles in the required manner.

On the other hand, the sum of the areas of all of them except the 3 by 3 is exactly 64, and it is possible to provide an example of such a partition into all 12 rectangles except the 3 by 3: the first four verticals of the board can be cut into strips of width 1 and lengths 1 and 7, 2 and 6, 3 and 5, and 8, respectively. The remaining 4 verticals can be divided into two vertical rectangles 2 by 7, composed of rectangles 2 by 5 and 2 by 2, and 2 by 4 and 2 by 3. Above them, add a horizontal strip 1 by 4.

Other examples of such a partition are also possible.

Grading criteria. It is proven that the number of rectangles in the partition does not exceed 12: 4 points. A correct example for 12 rectangles is provided: 3 points.