Tiling a $23 \times 23$ square by $1 \times 1$, $2 \times 2$, and $3 \times 3$ tiles

Question

A $23 \times 23$ square is tiled by $1 \times 1$, $2 \times 2$, and $3 \times 3$ tiles. Prove that at least one 1 x 1 tile must be used. Find such a tiling with exactly one $1 \times 1$ tile.

Hint: put a number in each $1 \times 1$ square of the big square so that $2 \times 2$ and $3 \times 3$ tiles cover a total divisible by $3$.

I'm not sure how to utilize the hint in solving this problem.

Answer 1

Further hint: $$ \matrix{1 & 0 & 1 & 0 & \ldots\\ 0 & 2 & 0 & 2 & \ldots\\ 1 & 0 & 1 & 0 & \ldots\\ 0 & 2 & 0 & 2 & \ldots\\ \ldots & \ldots & \ldots & \ldots & \ldots &\cr}$$ What is the total in the $23 \times 23$ square?

Answer 2

Put all ones in the top row, twos in the second row, ones in the third row and so on, every other row ones and twos.

You can check that wherever you put a big square, it has to cover a total that is divisible by 3. You can also check that the total in the entire grid is not divisible by 3. Hence you need small squares.