Problem 2. Consider a decomposition of the $8 \times 8$ chessboard into $p$ non-overlapping rectangles, such that each rectangle contains an integer number of squares, half of which are white, and no two rectangles have the same number of squares. Determine the maximum value of $p$.
Solution. If $p \geq 8$, the rectangles will contain, in total, at least $2+4+6+8+10+12+14+16=72$ squares, which is impossible, as the chessboard has only 64 squares.

For $p=7$ we have the possibility to decompose the board.

Here is an example: $3 \mathrm{p}$

![](https://cdn.mathpix.com/cropped/2024_06_07_a5d3ebec06670245d3a7g-3.jpg?height=697&width=746&top_left_y=1728&top_left_x=684)