7. In a $100 \times 100$ square, 10000 cuts were made along the grid lines (each cut one cell long), and it split into 2500 four-cell figures. How many of them are $2 \times 2$ squares?
Solution:

Answer: 2300.

Solution. Note that the perimeter of a $2 \times 2$ square is 8, while for the other four-cell figures (rectangle $1 \times 4$, T-shape, L-shape, or S-shape) it is 10. Let the number of squares be $x$, and the number of other figures be $2500-x$. Then their total perimeter is $8 x + 10 \cdot (2500 - x) = 25000 - 2 x$. On the other hand, this is the perimeter of the square plus twice the length of the cut, i.e., 20400 cells. Therefore, $2 x = 25000 - 20400 = 4600$, and as a result, $x = 2300$.