Which places on the earth can you walk one kilometer south, then one kilometer west, then one kilometer north and end up in the same spot?
Assume the earth is a perfect sphere and oceans or mountains don’t get in your way.
North pole
The simplest place with this property is the north pole. After you walk south, any amount of distance walking west will leave the north pole one kilometer north of you.
Other solutions
However, there are also an infinite number of places near the south pole with this property. Consider the circle centered on the south pole, with a circumference of one kilometer. Walking west for one kilometer from anywhere on this line will return you to the same position. Thus if we start one kilometer north of this circle, then we also satisfy the problem.
Walking once around the south pole is not the only way to stay in the same place. We could choose a smaller circle such that walking west one kilometer would circle the south pole \(n\) times, for any positive integer \(n\).
The circumference of a circle on a sphere is given by \(c = 2 \pi R \sin\left(\frac{r}{R}\right)\) where \(R\) is the radius of the sphere (the Earth). Since the circumference must be \(\frac{1}{n}\) we have:
\[r = R \arcsin\left(\frac{1}{2 \pi R n}\right)\]Since we need to start one kilometer north of any of these circles, that means that we need to start at any point:
\(\left(R \arcsin\left(\frac{1}{2 \pi R n}\right) + 1\right)\) kilometers north of the south pole for any positive integer \(n\).