Can a 12 by 9 grid be tessellated using only the L shaped tetromino?
No, it can’t. Proof: Color the grid using 9×1 stripes of 2 alternating colors. There will be 6 stripes of each color. Each L will occupy 3 squares of one color and 1 square of the opposite. As you add tetrominoes, the difference between the occupied square count of the two colors will alternate between 0 mod 4 and 2 mod 4. After 27 tetrominoes, the difference will be 2 mod 4. But for the grid to be filled, the numbers would have to be equal. This proof works for any size grid which is not a multiple of 8 (i.e. would require an odd number of tetrominoes).
