Exercise 1. 12 chairs are numbered from 1 to 12. A grasshopper can jump from chair $k$ to chair $n$ if $k-n$ is one of the four numbers $-8, -5, 5, 8$. It is known that it visited each chair exactly once. What are the possible initial positions?
Solution to Exercise 1 In the diagram below, each chair is symbolized by its number, and a line is drawn between two chairs if the grasshopper can jump from one to the other:

$$
8-3-11-6-1-9-4-12-7-2-10-5
$$

It is clear that the only possible initial positions are 5 and 8.