So, I think many people have heard the riddle:
"A hunter is hunting a bear. He goes 1 mile south, 1 mile west and 1 mile north and ends up back where he started. What colour is the bear?"
The answer is supposed to be white, as the hunter is at the North Pole. Of course, there are the other solutions at $1 + \frac{1}{2*n*\pi}$ miles away from the South Pole, where $n$ is a positive whole number.
My question is: it's easy enough to solve this by thinking about the sphere and considering the edge cases. Is there some systematic equational method for working out these solutions using e.g., geometry / topology / etc? I'm looking for something that offers a more general idea on how to solve such problems in other non-standard geometries.