the maximum number of Skew-Tetrominos that can be placed on an $8 \times 9$ rectangle without overlapping?

Note: The tetrominos can be rotated and reflected.
Solution: The maximum number of Skew-Tetrominos is 16.

We have already shown that you can place 16. You prepare the rectangle in 4 plus small rectangles of 2 lines and 9 columns. On each of these small rectangles, you can place 4 Skew-Tetrominos (by placing them all in the orientation on the exam sheet), so you can place 16 on the entire rectangle.

Let's now show that you cannot place more than 16.

First solution: Color all the columns in black and all the paired columns in white. We then notice, on the one hand, that each Skew-Tetromino must cover two white squares and two black squares, and on the other hand, that there are 40 black squares and 32 white squares. Therefore, there are at least 8 black squares that will not be covered by a Tetromino, which means that the Tetrominos cover a maximum of 64 squares and therefore you cannot place more than \(16 = 64 / 4\).

Second solution: Color black all the squares located in an even column and an even row, and white all the others. We then notice, on the one hand, that there are 16 black squares, and on the other hand, that each Tetromino must cover exactly one black square, so you can place a maximum of 16 Tetrominos.