Where a mile south, east, north walk closes up

Question

Treat the Earth as a perfect sphere. From how many different starting points does walking one mile south, one mile east and then one mile north bring you back to your starting point?

Short answer

Infinitely many. One is the North Pole. The rest lie near the South Pole: for each $k=1,2,\dots$ take the latitude circle of circumference $1/k$ mile; every point one mile north of such a circle works, because the eastward mile wraps around it exactly $k$ times.

Adapted from Crack, Heard on the Street, Q1.53, p.19; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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