2. A rectangular grid with sides 629 and 630 is cut into several squares (all cuts follow the grid lines). What is the smallest number of squares with an odd side that can be in such a partition? Don't forget to explain why there cannot be a smaller number of squares with an odd side in the partition.
Answer: Two. Solution. An example when there are exactly two squares: two squares with a side of 315 adjoin the side of the rectangle with a length of 630, and the remaining rectangle $630 \times 314$ is cut into squares $2 \times 2$. A smaller number of squares with an odd side cannot be: to each of the sides of length 629, at least one square with an odd side adjoins, and this cannot be the same square, as then its side would be at least 630, which is impossible. Note. One square with an odd side cannot exist also because the area of the rectangle is even, while the total area of all squares in this case would be odd.