The original problem was walk a mile south walk a mile west (was it east -- no matter) walk a mile north. The east-west walk is less than the length of line of latitude reached by the southword leg so that the return trip is along a different line of longitude. Let me give an example. The coordinates of the north geographic pole are (90, lon) where lon can be any angle between 0 and 360. The north geographic pole and the south geographic pole are the only two points on the earth sphere that do not have unique coordinate. Now let me widen the problem out Start at a point, walk to the equator in a southerly direction, walk east along the equator the same number of steps that one took to reach the equator then march in a northerly direction the same number of step.
Case 1 the starting point is the north pole. Assume the first leg is south along the Greenwich meridian, that is to say 0 longitude. This gets us down to (0, 0) on the equator. Walk west the same distance and we get to (0, 90). Now walk north the same distance and we get to (90,90) which is the same point as (90, 0) the north pole.
Case 2. The starting point is (x-lat, x-long) where x-lat is greater than 0 and less than 90. Assume x-long = 0 without loss of generality. Now leg 1: (x-lat, 0) to (x1, 0) where x1 < x-lat and greater or equal to 0. Leg 2 (x1,0) (x1, y1) where y1 > 0 but < 360. That means leg2 moved us to a different point with the same latitude. Now leg 3 northward by the same distance. This gets is to (x2, y1) because going north means following a meridian of longitude. Notice that x2 not = x1. The final destination is (x2, y1) which is different from (x-lat, 0). So we do not end up at the same place if we started out from a point that was not the pole.
Forget drawings. The proof is abstract and mathematical. Drawings are crutches for the logically feeble.