Anita lives in a city with a peculiar road system: every road is a circle (not necessarily of the same radius). The rules of the system are simple: no sharp turns. That is, if you are at a transversal intersection, you continue on the circle you are on, whereas if you are at a tangential intersection, you have two options, continue on the same circle, or switch to the other circle. The rules, depicted in the figure below, ensure that the motion is always smooth.
One day, Bose, her friend, was visiting her and he wanted a tour of the entire city. Specifically, he was fascinated by the strange road system and wanted to go through all of it. Anita was more than happy to take him around. But she didn’t want to bore him by showing the same sections of the road more than once. Can she always find a way, or is there a road system where this isn’t possible?
Example:
Consider the following system of three circular roads. The blue line shows the path Anita could take through the entire city while taking no sharp turns and avoiding repetitions.
Clarifications:
- The road system is connected, that is, there is a path between any two points that avoids sharp turns.
- As I said above, the circles need not have the same radius.
- There could be more than two circles that meet at an intersection. If they all meet tangentially, then that gives more than two options for smooth motion (i.e., no sharp turns).