4. A square grid consisting of 64 cells is cut along the cell edges into several pieces. All pieces must be squares, not necessarily of the same size. Indicate some ways of such cutting into 10, 11, and 12 pieces.
Solution. The ways of cutting can be different. For example, the answer is:
![](https://cdn.mathpix.com/cropped/2024_05_06_b9019ece618dbb82f1f9g-2.jpg?height=1038&width=1634&top_left_y=1046&top_left_x=286)

Comment. It is clear that there can be several ways to cut even into the same number of parts. At least one way of cutting into all three different numbers of parts is provided - 7 points. One or several ways of cutting into the same number of parts are provided - 2 points; ways of cutting into two, not three, of the specified numbers of parts are provided - 4 points.

![](https://cdn.mathpix.com/cropped/2024_05_06_b9019ece618dbb82f1f9g-2.jpg?height=188&width=1648&top_left_y=2454&top_left_x=273)
(identical letters correspond to identical digits, different letters correspond to different digits). The sixth and seventh gnomes claim that they can find not only two but even three solutions to the puzzle that will be different from those of these five gnomes. Are they right? Justify your answer.

Solution. We notice that one found solution can easily be turned into four by swapping the digits in the positions of the letters $Y$ and

![](https://cdn.mathpix.com/cropped/2024_05_06_b9019ece618dbb82f1f9g-3.jpg?height=180&width=1651&top_left_y=487&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_05_06_b9019ece618dbb82f1f9g-3.jpg?height=180&width=1648&top_left_y=664&top_left_x=273)

![](https://cdn.mathpix.com/cropped/2024_05_06_b9019ece618dbb82f1f9g-3.jpg?height=180&width=1651&top_left_y=847&top_left_x=274)

$+$| 8 | 5 | 3 | 6 |
| ---: | ---: | ---: | ---: |
| + | 7 | 1 | 0 | $\begin{array}{rl}7 & 10 \\ 924\end{array}$. Having eight different solutions to the puzzle, we can assert that the sixth and seventh gnomes are right, as from these specified solutions they can always choose three that differ from any no more than five solutions to the puzzle (even not from this list).

Answer: yes.

Comment. Eight solutions to the puzzle are provided and the answer is justified - 7 points. The number of points depends on the number of puzzle solutions indicated by the student, according to the table:

| Number of solutions | 0 | 1 | $2-3$ | 4 | 5 | $6-7$ | 8 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Number of points | 0 | 1 | 2 | 3 | 4 | 5 | 6 |