On October 4, Hongyuan Zhan posted a brain teaser. Here is one possible solution. Scott gave an argument that explains why it is always possible to construct these path even if we had more than 12 hours in our clocks. The white path in the center can be ignored. I made a mistake along the way and Paint is not very forgiving.

Path to return the hours to their position.

Originally I thought spiral-like paths would do it, but some curve quite a bit in other shapes.