How can line segments in a closed path intersect only one other segment?

Question

In the 'parity' chapter of Dmitri Fonin's Mathematical Circles, problem 7, he asks if we can draw a closed path made up of nine line segments, each of which intersects exactly one of the other segments. But is this possible in any closed path? Shouldn't each segment intersect two other segments?

Answer 1

It seems that the $9$ vertices of the path as such are not counted as "crossings". On the other hand I would count the passage of a third edge through a vertex as two "crossings". Given that, each crossing uses up the allowance of exactly two edges. Since there is an odd number of edges the task can not be fulfilled.