It is well-known that a $2^n \times 2^n$ chessboard with an arbitrary square removed is tilable by an L-shaped tromino (piece composed of three squares). The standard proof is by induction, and is constructive (gives an algorithm for producing such a tiling).
My question is: For a given chessboard with a fixed square removed, is this tiling unique?
No, it is not unique at least starting from $n=3,$ so an $8 \times 8$ board. Note that two L trominoes can combine to make a $2 \times 3$ rectangle. This rectangle can have the pieces placed two ways. Now let the removed square be a corner. Put one L next to it to make a $2 \times 2$ square. You can tile the rest of the board with five $2 \times 6$ rectangles as shown below. Each $2 \times 6$ can be tiled in four ways, so this gives $1024$ ways to tile the square. I believe the $n=2$ case is unique but have not done a careful search to prove it.
Here are two explicit tiling of the 8x8 board. I got the second board just by playing around with the tiling.
