Problem 7.1. In the picture, nine small squares are drawn, with arrows on eight of them. The numbers 1 and 9 are already placed. Replace the letters in the remaining squares with numbers from 2 to 8 so that the arrows from the square with the number 1 point in the direction of the square with the number 2 (the number 2 can be in square $A$ or $B$), the arrows from the square with the number 2 point in the direction of the square with the number 3, and so on, the arrows from the square with the number 8 point in the direction of the square with the number 9.

![](https://cdn.mathpix.com/cropped/2024_05_06_43be4e09ee3721039b48g-18.jpg?height=389&width=379&top_left_y=96&top_left_x=529)

Construct the correspondence.

- In square $A$
- In square $B$
- In square $C$
- In square $D$
- In square $E$
- In square $F$
- In square $G$
- stands the number 2.
- stands the number 3.
- stands the number 4.
- stands the number 5.
- stands the number 6.
- stands the number 7.
- stands the number 8.
Answer: In square $A$ there is the number 6, in $B-2$, in $C-4$, in $D-5$, in $E-3$, in $F-8$, in $G-7$.

Solution. Let's order all the squares by the numbers in them. This "increasing chain" contains all nine squares.

Notice that in this chain, immediately before $C$ can only be $E$ (only the arrows from $E$ point to $C$). Similarly, immediately before $E$ can only be $B$ (not $C$, because $C$ is after $E$). Immediately before $B$ can only be 1. Therefore, the numbers in squares $B, E, C$ are $2,3,4$ respectively. Then 5 is definitely in $D$, 6 is definitely in $A$, 7 is definitely in $G$, and 8 is definitely in $F$.