Bakayev E.V.

Cells of a $5 \times 7$ table are filled with numbers such that in each $2 \times 3$ rectangle (vertical or horizontal), the sum of the numbers is zero. By paying 100 rubles, you can choose any cell and find out what number is written in it. What is the smallest number of rubles needed to definitely determine the sum of all numbers in the table?
Let the central cell of the table contain the number $a$. Then the sum $S$ of all the numbers in the table is $-a$, since the sum $S+a$ can be broken down into six zero sums, each with six terms, as shown in the figure.

Therefore, it is sufficient to know one number - the central one.

![](https://cdn.mathpix.com/cropped/2024_05_06_b41405b02b6e888d68cag-34.jpg?height=394&width=552&top_left_y=1504&top_left_x=753)

On the other hand, a table filled with numbers $a$ and $-a$ in a checkerboard pattern satisfies the problem's condition. Thus, the sum $S=-a$ can be any value, and at least one number needs to be known.

## Answer

100 rubles.