, 2015

By tiling our $8 \times 8$ table by eight $2 \times 4$ rectangles like in Tiling (1) we find that the total number of green unit squares in the table is $8b$.

By tiling our $8 \times 8$ table by four $3 \times 3$ squares, three $2 \times 4$ rectangles and one $2 \times 2$ square, like in Tiling (2), we find that the total number of green unit squares in the table is $4a + 3b + x$, where $x$ is the number of green unit squares in the $2 \times 2$ square in the left upper corner. Notice that by just rotating our tiling (2) so that the $2 \times 2$ square occupies at each time the four corners of the table, we deduce that the four $2 \times 2$ unit squares in the four corners, all have the same number of green unit squares $x$.

Finally, by tiling our $8 \times 8$ table by six $2 \times 4$ rectangles and four $2 \times 2$ squares, like in Tiling (3), we find that the total number of green unit squares in the table is $6b + 4x$

Tiling (1)

Tiling (2)

Tiling (3)

Putting together all these totals, we obtain the relations
$$8b = 4a + 3b + x = 6b + 4x$$
from which we deduce that $b = 2x$ and $9x = 4a$. Since $0 \leq a \leq 9$, either $a = 0$ and $b = 0$ or $a = 9$ and $b = 8$. In other words, either we color the whole table or we don't color at all.