A chessboard has its rows and columns numbered as shown in the following figure. In each square, the number written is the sum of the numbers of the row and column of that square. For example, in the square that is in row 4 and column 5, the number $4+5=9$ is written.

![](https://cdn.mathpix.com/cropped/2024_05_01_360f8ce7ec440aed2c7ag-20.jpg?height=445&width=446&top_left_y=563&top_left_x=702)

a) What is the sum of the numbers written in all the squares of the board?

b) Let $S_{\text {black }}$ be the sum of all the numbers written in the black squares and $S_{\text {white }}$ be the sum of all the numbers written in the white squares. What is the value of the difference $S_{\text {black }}-S_{\text {white }}$?

c) What is the value of $S_{\text {black }}$?
Solution

a) See that in line 1 the number 1 is added in each of the 8 cells, in line 20 the number 2 is also added eight times, and so on. Thus, we can account for the contribution of the rows by:

$$
\begin{aligned}
8 \cdot(1+2+3+4+5+6+7+8) & =8 \cdot 36 \\
& =288
\end{aligned}
$$

The same applies to the contribution of the columns, also totaling 288. We conclude that the sum of all the cells is $2 \cdot 288=576$.

b) This difference can also be analyzed by considering the contribution of each row and each column, but in this case, for each black cell, we must add its number, while for each white cell, we must subtract it. Since each row and each column has exactly four black cells and four white cells, the number written in a row or column must be added and subtracted 4 times, that is, it contributes $4-4=0$. Therefore, we conclude that $S_{\text {black }}-S_{\text {white }}=0$.

c) Combining the two pieces of information from items $a$) and $b$), we have

$$
\left\{\begin{array}{l}
S_{\text {black }}+S_{\text {white }}=576 \\
S_{\text {black }}-S_{\text {white }}=0 .
\end{array}\right.
$$

Adding the two lines, we get $2 S_{\text {black }}=576$, that is, $S_{\text {black }}=288$.