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/(2

π) 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.