33. Let there be a square grid $n \times n$. An original set of cells is a set of $n-1$ cells, no two of which are on the same vertical or horizontal line. For which $n$ can $n+1$ non-overlapping original sets be chosen from this square?
33. Answer: for any.

Example: first, color the square in $n$ colors so that cells of the same color are located on the diagonals. Take the top left cell, it will be the first color. Repaint it to the $(n+1)$-th color, move 2 cells to the right and one down - we will land on a cell of the second color, repaint it, and so on. If $n$ is odd, reaching the right edge of the board, we will jump to the second column, again transitioning to a new color, then calmly reach the bottom row and end up in the second-to-last column. Discard the last repainted cell, the example is ready. If $n$ is even, repeating our operation near the right edge of the board, we will land on the first column, so we will move one more cell to the right. Then repeat the operation until we reach the bottom right corner, which we do not touch. After this, we get that there are no fewer than $n-1$ cells of each color, but one of the colors has $n$ cells. Discard any cell of this color.