Trominos

Consider the L shaped tile, and 2x2 and 4x4 grids shown below. We wish to ``tile'' these grids with L shaped tiles so that only one square in the grid is not covered.

1.

Show how to ``tile'' the 2x2 grid so one square is not covered - easy.

2.

Show how to ``tile'' the 4x4 grid so one square is not covered - little harder.

3.

Can you ``tile'' a 5x5 grid so one square is not covered? If so, show such a tiling. If not, give a convincing argument.

4.

Can you ``tile'' a 6x6 grid so one square is not covered? If so, show such a tiling. If not, give a convincing argument.

5.

Look for a pattern. In general, which N x N size grids can always be tiled? Give an argument supporting your claim.