2. In a $11 \times 11$ square grid, we sequentially wrote the numbers $1,2, \ldots, 121$ from left to right and from top to bottom. Using a $3 \times 3$ square tile, we covered exactly nine cells in all possible ways. In how many cases was the sum of the nine covered numbers a perfect square of an integer?
2. Let $n$ be the number covered by the middle cell of the square tile. Then the first row of this tile covers the numbers $n-12, n-11, n-10$, its second row covers the numbers $n-1, n$ and $n+1$, and finally the third row covers the numbers $n+10, n+11$ and $n+12$. The sum of all numbers covered by the tile is thus $9n$.

The sum of the numbers covered by the square tile will be a perfect square of an integer if and only if the number $n$ covered by its center is itself a perfect square of an integer. Among the numbers $1, 2, \ldots, 121$, only the numbers in the set $\{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121\}$ have this property, but the numbers $1, 4, 9, 100$, and 121 lie on the edge row or edge column of the $11 \times 11$ grid and cannot be covered by the center of the $3 \times 3$ square tile.

Of all possible coverings of the grid, 6 coverings have the desired property, where the center of the tile covers one of the numbers $16, 25, 36, 49, 64$, and 81.

For a complete solution, award 6 points, of which up to 2 points for expressing the sum of the numbers under the square tile in terms of some (covered) cell, up to 2 more points for a correct discussion of when this sum is a perfect square of an integer. Award one point for finding at least three valid coverings of the grid, deduct 1 point if the solver incorrectly specifies any position of the tile, and deduct 2 points if the solver incorrectly specifies at least three positions of the tile.