## Task A-3.5.

A chessboard is a board with 8 rows and 8 columns, whose squares are alternately colored black and white, with the square in the first row and first column being black. Each square on the chessboard has a single integer written on it. It is known that the sum of all numbers on the white squares is 26, and the sum of all numbers in the odd-numbered columns is 43.

If we change the signs of all numbers on the white squares, what will be the sum of all numbers in the odd-numbered rows after this change?
## Solution.

Let $C_{s}$ and $B_{s}$ be the sum of the numbers on the black and white squares in the odd columns, and $C_{r}$ and $B_{r}$ be the sum of the numbers on the black and white squares in the odd rows.

Notice that the odd rows and odd columns contain the same black squares, so $C_{r}=C_{s}$.

Notice that the odd rows and odd columns do not share any white squares, but together they cover all the white squares. Therefore, $B_{r}+B_{s}=26$.

From the problem, we know that $C_{s}+B_{s}=43$.

After changing the sign on the white squares, the sum in the odd rows is

$$
C_{r}-B_{r}=C_{s}-\left(26-B_{s}\right)=C_{s}+B_{s}-26=43-26=17
$$

Note: It is sufficient for the student to explain the solution in words. However, if the student only writes that the solution is the difference between the sum of the numbers in the odd columns and the sum of the numbers on the white squares, without an explanation, they should receive 2 points.