Problem 2. In a square of grid paper containing an integer number of cells, a hole in the shape of a square, also consisting of an integer number of cells, was cut out. How many cells did the large square contain if 209 cells remained after the cutout?
Answer: 225 cells

Solution. The side of the larger square contains $n$ sides of a cell, and the side of the smaller square contains $m$ sides of a cell. Then $n^{2}-m^{2}=209 \rightarrow(n-m)(n+m)=209=11 \cdot 19$.

Case 1. $\left\{\begin{array}{c}n+m=209 \\ n-m=1\end{array} \rightarrow\left\{\begin{array}{c}n=105 \\ m=104\end{array}\right.\right.$ case does not occur due to the absence of a hole

Case 2. $\left\{\begin{array}{l}n+m=19 \\ n-m=11\end{array} \rightarrow\left\{\begin{array}{l}n=15 \\ m=4\end{array}\right.\right.$