Grids don't contain 'L'

This \(4\times 4\) grid of unit squares is tiled with L-shaped pieces of area 3, with one square blacked out.

Now, the side length \(4=2^2\) of this square grid is extended to a larger power of 2, as illustrated below, and again one of the squares is blacked out.

Can you always tile this grid with L-shaped pieces of area 3, no matter where the black square is?

The grid can have side length 8, 16, 32, or any power of 2.  One square is blacked out in each grid.

The grid can have side length 8, 16, 32, or any power of 2. One square is blacked out in each grid.

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