Shapovalov A.V.

On the table, there are 9 apples, forming 10 rows with 3 apples in each (see the figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_bca77ef9e87e9e47ba11g-40.jpg?height=335&width=345&top_left_y=1193&top_left_x=868)

It is known that the weights of nine rows are the same, while the weight of the tenth row differs from them. There are electronic scales, on which you can find out the weight of any group of apples for one ruble. What is the minimum number of rubles you need to pay to determine which row's weight differs?

#
Before spending money, let's think

Let $l_{1}, l_{2}, l_{3}$ be the weights of the three highlighted (see fig. a) diagonal rows, and $v_{1}, v_{2}, v_{3}$ be the weights of the three vertical rows. Then $l_{1}+l_{2}+l_{3}=v_{1}+v_{2}+v_{3}:$ both are simply the sum of the weights of all nine apples. But at least five of these six quantities are equal to the same value $t$, so all six are equal.

![](https://cdn.mathpix.com/cropped/2024_05_06_bca77ef9e87e9e47ba11g-40.jpg?height=403&width=397&top_left_y=2017&top_left_x=836)

Similarly, we find that the weights of the other three diagonal rows are also equal to $t$. We have found nine rows, the weights of which are equal. Therefore, the weight of the remaining row, the horizontal one, is different.

## Answer

0 rubles.