A $5\times 5$ board is covered by eight hooks (a three unit square figure, shown in the picture) so that one unit square remains free. Determine all squares of the board that can remain free after such covering.
Color the board by painting the corners black, the square in the middle of the edges and the square in the center of the board. When we arrange a piece in the shape of a hook it only covers a house painted black. So we would need nine pieces and the statement only allows us $8$. Soon these boxes can be filled with the monomino and we have $9$ ways to accomplish this task.
