The Puzzle

You’re standing on a perfectly spherical Earth. You:

  1. Travel 1 mile south
  2. Then 1 mile east
  3. Then 1 mile north

…and you end up exactly where you started.

If you’re not at the North Pole, where else could you possibly be?

Solution 1: The Obvious Answer — The North Pole

If you start at the North Pole:

  • Going 1 mile south puts you on a circle of latitude exactly 1 mile from the pole.
  • Going 1 mile east moves you along that circle (your distance from the pole remains 1 mile).
  • Going 1 mile north takes you straight back, because all paths heading north converge at the North Pole.

That’s the well-known solution.

But You’re Not at the North Pole?

There are infinitely many such points near the South Pole!

Generalized Solution

Suppose there exists a parallel circle of latitude near the South Pole where the eastward circumference is:

\[ \frac{1}{k} \text{ miles for some integer } k \ge 1 \]

Then going 1 mile east will take you around the circle exactly \( k \) times, returning you to your original east-point.

If you’re 1 mile north of that circle, then:

  1. Going south puts you on that circle
  2. Going east wraps around \( k \) times and returns you
  3. Going north brings you back to the start

These circles lie increasingly close to the South Pole, and there are infinitely many such latitudes.

Final Answer

Besides the North Pole, your starting point could be 1 mile north of any circle of latitude near the South Pole where traveling 1 mile east makes a full \( k \)-loop around the Earth for any integer \( k \ge 1 \).

Reference