With a dose of spherical geometry
How many surface places are there from which to start a surface trip due south for a stated distance, then due east for the same stated distance and then due north again for the same stated distance to return to the starting location.
Use a Earth that is a sphere of constant curvature, precise distances and angles. The stated distance is not a specific distance, for example a mile or a kilometer could be used. The problem could be done with 10 mega-meters as well.
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1 solution
The common answer is the North Pole. But there are an infinity of other places. They are circles around the South Pole with radii of the stated distance (whatever you chose) plus the radii of circles centered on the South Pole that circumnavigate the South Pole a positive integer number of times with a total distance of the stated distance. Altogether, that is a countable infinity of places.