6. (50 points) A $5 \times 5$ square of cells was cut into several pieces of different areas, each consisting of an integer number of cells. What is the maximum number of pieces that could result from such a cutting?
Answer: 6.

Solution. We will show that there cannot be more than 6 parts. Indeed, the total area of 7 parts cannot be less than $1+2+3+4+5+6+7=28$, which exceeds the area of the square.

Now we will show how a $5 \times 5$ square can be cut into 6 parts of different sizes:

![](https://cdn.mathpix.com/cropped/2024_05_06_f37bee4a288b06972b7cg-3.jpg?height=426&width=420&top_left_y=441&top_left_x=812)

Thus, the areas of the pieces will be $2, 8, 3, 7, 1,$ and 4.