One-Mile Walk Coverage

This brain teaser asks you to reason about walking on a perfectly spherical Earth in fixed compass directions, and to identify all starting points from which a particular three-segment walk brings you exactly back to where you began. At first glance it resembles the classic "North Pole puzzle," but the twist is that the planet's curvature and the periodic nature of motion along latitude circles create many less obvious possibilities. The challenge is to think globally rather than locally: you are not optimizing distance or direction errors, but classifying all geometric configurations that close the loop.

Solving it leans on spherical geometry, the structure of great circles versus parallels, and the idea that traveling east along a latitude is periodic. A strong answer uses clean geometric reasoning, maybe with some trigonometric intuition, to characterize whole families of valid starting points rather than just isolated ones. Interviewers watch for candidates who can generalize beyond the "obvious" case, reason precisely about periodic motion on curved surfaces, and clearly articulate why their classification is complete.

What it tests

This problem class is governed by the interplay between global geometry (the sphere's surface) and periodicity in movement along lines of latitude. The key is that on a sphere, moving east along a latitude circle is inherently periodic: after traveling a distance equal to the circle's circumference, you return to your starting longitude. Thus, if your eastward step is an integer multiple of a latitude's circumference, you end up where you started in longitude, regardless of where you began along that circle. The north and south movements serve to move you between these circles, so the entire path forms a closed loop if the eastward segment's periodicity matches the circle's circumference. This structure means the problem is about matching step sizes to the sphere's geometry, not just about the North Pole's uniqueness.