Question: How many points are there on the globe where, by walking one mile south, one mile east, and one mile north, you reach the place where you started?

Start by drawing a mental map: One mile south, one mile east, and one mile north covers three sides of a square. You ought to end up a mile east of where you started. The situation seems impossible, and you might think the answer is zero points.

Try again. The only way to make sense of the situation is to remember that the compass directions are relative ones applied to the surface of a sphere. At the North Pole, every horizontal direction is south. As long as you start precisely at the North Pole, you can walk a mile in any direction and that will count as walking south. Not only that, but a subsequent one-mile-east leg will curve in a circle centered on the North Pole. At any rate, it will if you interpret the puzzle to mean that you not only point yourself due east but constantly adjust your direction so that your bearing remains due east throughout the second mile. That then allows a final, straight, one-mile-north leg returning to the pole. The journey looks like a wedge of pie rather than an open square.
So the North Pole is one point where this could happen.
Notice that it couldn't happen at the South Pole. At the South Pole, every direction is north. You can't go a mile south from the South Pole.