10.3. Vova placed several (possibly 0) chess pieces on an $8 \times 8$ board. Lena noticed that in each $2 \times 2$ square, the same number of pieces is placed. And Vlad noticed that in each $3 \times 1$ (or $1 \times 3$) rectangle, the same number of pieces is placed. How many pieces were placed on the board? (List all possible answers and prove that there are no others.)
Answer: 0 or 64.

Solution. Suppose that in each $2 \times 2$ square there are $m$ figures, and in each $1 \times 3$ rectangle there are $n$ figures. Let's select a $2 \times 6$ rectangle from the board. On one hand, this rectangle can be divided into three $2 \times 2$ squares, and thus it contains $3 m$ figures. On the other hand, it can be divided into four $1 \times 3$ rectangles, and then it contains $4 n$ figures. We get the relation $3 m = 4 n$, from which it follows that $n$ is divisible by 3. But $n$ can take the values $0, 1, 2, 3$. Thus, $n = 0$ or $n = 3$. In other words, either all $1 \times 3$ rectangles are empty, and then there are 0 figures on the board, or all $1 \times 3$ rectangles are completely filled with figures, and in this case there are 64 figures on the board.

Note: Any correct solution: 7 points.

The case of zero figures is missing, but the solution is otherwise correct: 6 points.

It is proven that all $1 \times 3$ rectangles (or all $2 \times 2$ squares) are either empty or filled, but the solution is not complete: 5 points.

Only the correct answer is provided: 1 point.