Question: An $8$ X $8$ chess board is missing its corners. Can you completely cover the chess board with L-shapes resembling the way a knight can move, i.e., shapes that look like this:$\begin{matrix}X&X&X\\X \end{matrix} $ in any orientation ?
I don't even really know where to begin with discrete math problems like this. I would appreciate any tips, hints, or answers to this question. Thanks.
Color the board this way. Note that wherever you place an L tetromino it covers three white squares and one orange, or one white and three orange. If you cut off the four corners, you have $30$ of each color. To match the area you need $15$ tetrominos. They cannot cover an equal number of squares of each color, so the tiling is impossible.