After traveling from a point X on earth 10 km South, 10 km East and 10 km North you arrive at the same location X.  Where is X situated?  How many such points X exist?

The obvious solution is X being the North Pole. But there are other positions on earth that respect the conditions.

Consider for example going on a meridian toward the South Pole until the length of the parallel you are situated on is exactly the same as the distance traveled. Moving to East in this case would mean making a complete rotation on that parallel, and moving to North is then performed in the reverse direction on the first segment of the trajectory.

More concrete, from any point situated 10 km North from the parallel having the length of 10 km (\frac{10}{\pi} diameter) we can travel as described above so there are infinite points X.

Even more, from a point situated at 10 km from the parallel of length 5km (that would be quite close to the South Pole), you can go to South until reaching the parallel, make two complete rotations then back in North direction.

Or more generally, move 10 km to South until reaching the parallel of length \frac {10}{n}, make n rotations, and then move back 10 Km North.

Source:  Martin Gardner, “The Colossal Book of Short Puzzles and Problems”

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