South, East, North, and Home from Uncountably Many Places

Walk one mile south, then one mile east, then one mile north, and find yourself exactly where you started. From how many points on Earth can you do that?

Almost everyone says one, the North Pole, and the first half of that answer is right. The North Pole is a solution. What is wrong is the second half, the silent "and nowhere else". There is an entire family of other starting points, huddled near the South Pole, and the family is large enough that the count is not a number at all.

Why the North Pole works

Walk a mile south from the pole and you are on the parallel one mile out, on some meridian. Walk east along that parallel and you finish on a different meridian, having gone some fraction of the way round. Then walk a mile north. Every meridian runs to the pole, so whichever one you happen to be standing on, a mile north takes you back to the pole you left. The eastward leg is irrelevant here. It could be one mile or five thousand.

That degeneracy is exactly what makes the North Pole feel like the whole answer. It works for a reason so blunt that it seems to leave nothing to look for. But look at what the argument actually needed: the eastward leg had to return you to a meridian that leads back to your start. At the pole, all of them do. Elsewhere, exactly one does, which is the one you left, and that is a condition rather than an impossibility.

Walking on a sphere, written down

Take the Earth as a sphere of radius $R$, using the mean radius $R = 3958.76$ miles when a number is wanted. Walking south or north means following a meridian, so the distance to a pole changes by exactly the distance walked. Walking east means following a parallel, which is a circle of constant latitude.

The parallel at geodesic distance $s$ from a pole is a circle whose radius is not $s$ but

because the parallel is a slice of the sphere and its radius is measured in the plane of the slice, not along the surface. Walking a distance $d$ east along a parallel of radius $r$ changes your longitude by

which is just arc length over radius. This is the one step where a reader can lose the thread, so it is worth saying plainly: the eastward leg is not a geodesic. Apart from the equator, a parallel is not the shortest route between its own points, and a walker who keeps a compass bearing of due east is turning constantly. Equation (2) is right precisely because it treats the path as a circle of radius $r$ rather than as a great circle.

The condition for coming home

Suppose you start at distance $s + 1$ from the South Pole, so the mile south lands you on the parallel at distance $s$. The mile north afterwards will retrace your outbound meridian only if the eastward mile put you back on the longitude you started the eastward leg from. Any other longitude leaves you on a different meridian, and a mile north along that meridian ends a positive distance away from your start.

One mile is $n$ laps of a parallel exactly when that parallel has circumference $1/n$ miles. Setting $C(s) = 1/n$ in equation (1) and solving:

The starting circle for $n$ laps is one mile north of that, so it sits at distance

from the South Pole. In miles, $d_1 = 1.159$, $d_2 = 1.080$, $d_3 = 1.053$, and so on. The circles are distinct, they all sit strictly beyond one mile from the pole, and they crowd in towards the parallel exactly one mile out without ever reaching it. So the walk never crosses the pole and no two lap counts describe the same circle.

Why the numbers look so ordinary

Equation (3) contains an arcsine and a planetary radius, yet the answers come out as tidy fractions of $2\pi$. That is the sphere behaving locally like a plane. As $R$ grows the arcsine linearises:

On a flat sheet, a circle of circumference $1/n$ has radius $1/(2\pi n)$, and that is what equation (5) recovers. For a single lap the flat figure $1/(2\pi) = 0.159154\ldots$ miles agrees with the exact spherical value to seven decimal places. A mile is small enough compared with the Earth that the curvature contributes almost nothing, which is why an answer that sounds like spherical geometry can be computed with elementary plane geometry and still be right to the eighth digit.

How many points, said carefully

The usual phrasing of the answer is "infinitely many", which is true and weaker than what the geometry gives. Nothing in the construction picked out a longitude. Rotate the whole picture about the polar axis and the walk rotates with it, so if one point of a starting circle works then every point of that circle works. A circle has uncountably many points.

So the solution set is the North Pole together with a countable family of circles, each uncountable. The set is uncountable, and its structure is worth stating rather than its size alone: one isolated point at the north, plus a sequence of circles near the south accumulating on the parallel one mile from the South Pole.

What the puzzle assumes without saying

Four things are being taken for granted, and each one changes the answer if read differently. "Walk east" has to mean follow the parallel. A walker who instead sets off on the great circle whose initial bearing is east drifts away from constant latitude immediately, and the closing condition becomes a different problem.

The South Pole itself is not a solution, since there is no south to walk. Nor is any point within one mile of it, because the first leg would run past the pole and up the far side, at which point the tidy accounting in equation (4) no longer describes the path. The bound $d_n > 1$ in the construction is doing real work.

The Earth is not a sphere. On an oblate ellipsoid the parallels have slightly different circumferences at the same distance from the pole, so the exact latitudes in equation (3) shift a little. The structure of the answer survives untouched, since it depends only on parallel circumferences shrinking continuously to zero at the pole, which happens on any surface of revolution with a smooth pole.

Finally, the family really has to be constructed rather than guessed at. There is no useful rule of thumb like "somewhere a mile or two from the South Pole". Start one and a half miles out and the walk ends 2.524413 miles from where it began. Only the discrete distances in equation (4) close the loop, and everything between them fails.

Sources and further reading

• The circles of constant latitude in equation (1) — Circle of latitude
• The north and south legs — Meridian
• Why the eastward leg is not a shortest path — Great circle
• The measured radius used for $R$ — Earth radius
• The sense in which a circle of points is uncountable — Uncountable set

Everything above was checked twice before publication, once symbolically and once by walking the sphere in three-dimensional Cartesian coordinates and measuring the closing error as a Euclidean distance in miles. For each lap count from one to ten the loop closes to under $3 \times 10^{-13}$ miles and the North Pole walk closes to under $10^{-9}$, while the control start one and a half miles from the South Pole misses by 2.524413 miles.