The paths are not arbitrary. From what I could see, here are the rules governing them:

- Paths are composed of a series of straight line segments.
- Each line segment connects a pair of circles.
- If the preceding line segment arrived at a given circle, then the next line segment must leave that same circle.
- A given circle may be visited only once.
- A line segment may pass over a given circle without visiting it only if that circle has already been visited. (Attempting to pass over a not-yet-visited circle will instead visit that circle.)

How many unique paths are there?

## Comments

(Anonymous)You missed a condition. There's also a minimum sequence length.

paulmckpaulmck