Let *n* be a positive integer that is not a multiple of 3. Which single squares can be removed from an
*n*×*n* checkerboard so that the remaining squares can be covered by straight trominoes
(1×3 rectangles) without overlap? For example, in a 4×4 board, one can cover the region obtained
by removing a corner square, but one cannot cover the region obtained by removing a square on the diagonal
next to a corner square. A successful tiling is shown on the left and a forced non-tiling is shown on the right
(the number in a tromino indicates the order in which it is laid down).