Where else can you start besides the North Pole?

On a perfectly spherical Earth, there are actually infinitely many other starting points. They are all located near the South Pole.

Answer: Start 1 mile north of a tiny latitude circle whose circumference is 1/n mile for some positive integer n = 1, 2, 3, .... Then your 1-mile eastward walk goes around that circle exactly n times, so you end up on the same spot and the final 1-mile north walk returns you to where you started.

Geometric idea

South Pole tiny latitude circle starting point 1 mile south Then 1 mile east goes around this circle a whole-number number of times 1 mile north
The key is making the eastward leg wrap around a tiny parallel exactly an integer number of times.

Interactive example

Choose how many complete laps the 1-mile east walk makes.

Required latitude circumference
1.000 mi
Distance of that circle from South Pole
0.159 mi
Starting distance from South Pole
1.159 mi
Using Earth radius R ≈ 3959 miles.

Why it works

  1. Pick a tiny latitude circle near the South Pole.
  2. Make its circumference exactly 1/n mile.
  3. Start exactly 1 mile north of that circle.
  4. Walk 1 mile south: you land on the circle.
  5. Walk 1 mile east: you loop around the circle exactly n times and return to the same spot.
  6. Walk 1 mile north: you return to your starting point.

General formula

If the chosen latitude circle has circumference 1/n mile, then its radius on the Earth's surface is approximately:

2πr = 1/n
so
r ≈ 1 / (2πn) miles

So the starting point is about 1 + 1/(2πn) miles from the South Pole.

More exactly, on a sphere of radius R, if dₙ is the surface distance from the South Pole to the tiny circle, then:

dₙ = R · arcsin(1 / (2πRn))

and the starting point is 1 + dₙ miles from the South Pole.

Final conclusion

Besides the North Pole, the other solutions are:

So there are infinitely many such starting points, all clustered just north of the South Pole.