Tetrominos

Label the lower left square $(1,1)$ and the upper right square $(9,6)$ .

Consider the coloring where we color every odd row blue. There are 24 cells that are blue. Notice that no matter where we place a L-tetromino, it must cover an odd number of blue cells. Since we use 11 L-tetrominos, this implies that they will only cover an odd number of blue cells. Hence, the monomino must cover a blue cell.

Consider the coloring where we color every even column red. There are 18 cells that are red. Similar to the argument above, the monomino must cover a red cell.

We can check that if the monomino is at $(2, 1)$ , then there is no way to fill up the first three columns. Hence $(2,1)$ and $(8,1)$ are not valid.

We can check that if the monomino is at $(2, 3)$ , then there is no way to fill up the first three columns. Hence $(2,3)$ and $(8,3)$ are not valid.

We can check that the remaining 6 positions are valid in the following way:

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