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.

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    $\begingroup$ There is so much hidden human context in this problem, that I cannot imagine any solution to it which did not involve querying a gigantic database of human knowledge. Like a human brain. $\endgroup$ – Lee Mosher Feb 27 '16 at 14:29
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    $\begingroup$ There is no mathematical solution meeting your requirement. However, it can be verified (referring to the other solution near the South Pole). Use solid geometry method to find the latitude of the largest equi-latitudal circle that has the said effect. Then, reverse the whole question. The next question is for the second largest and so on. $\endgroup$ – Mick Feb 27 '16 at 18:11

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