This is a solution to problem 10 on the hard interview questions node. There are infinite points where walking one mile south, one mile east and one mile north you reach the place where you started. - The North Pole. Walk a mile south (i.e. any direction), then one mile east (an arc of 1 rad around the pole), then walk back north one mile. - Exactly 1 + 1/(2π) miles from the South Pole. You walk south one mile, ending up exact 1/(2π) miles from the South Pole. You walk east one mile, around the South Pole and back (drawing a circunference of radius of 1/(2π) miles and length of 1 mile). Walk north one mile, and you're back where you started. - Exactly 1 + 1/(4π) miles from the South Pole. Just as above, except you go around the South Pole two times. - Exactly 1 + 1/(2xπ) miles from the South Pole, where x is a natural number, nonzero. You got the idea. This, of course, assuming both poles can be walked and circled around at will.
This a the solution to problem 10 on the hard interview questions node. If you have not read the question, the following will make no sense to you:

There's one point on the globe that satisfies this condition trivially: the north pole. However, consider the points 1 + 1/(2*pi) miles away from the south pole. Call point A any of these (infinite number of) points. Go a mile south to point B. When you go a mile east, you end up back at point B (you travelled once through every line of longitude). A mile north then brings you back to point A.

There are points still closer to the south pole such that going a mile east brings you through each line of longitude exactly twice, three times, or as many times as you want. Thus we have an infinite number of concentric rings of infinite numbers of points, and we can start a mile north of any of them.

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