A 5*5 chessboard is given.A square is cut from the board.Show that it is not possible to cover the rest of the board by 3*1 triminoes if the square is not cut out from the center.
Can the result be generalised for any (2n+1)*(2n+1) chessboard?
Colour each square of the 5x5 board Red, Green or Blue as indicated in the diagram. There are 8 red, 8 blue and 9 green squares. Each triomino covers one square of each colour, so the square that is cut out has to be a green one. By symmetry (or by considering the mirror image of the above colouring), only the centre square is eligible.
Hopefully that idea will help you to investigate the (2n+1)x(2n+1) case also.
Let's see if I understand this right. The mirror image (vertical or horizontal) maps a trimino-filled board into another trimino-filled board. However, the image of a green square is not green unless it is the central square. Therefore, if one has a filled board where a non-central green square is cut out, then one gets a similar board where a non-green square is cut out, which is impossible.By symmetry (or by considering the mirror image of the above colouring), only the centre square is eligible.