You’re a mission commander for the Riddler Space Agency, which is engaged in a space race with a competing agency. Both agencies are trying to claim regions of a newly discovered, perfectly spherical moon that possesses a magnetic field. Everywhere on the surface of this moon the magnetic field lines point from the north pole to the south pole, parallel to the surface (i.e., the magnetic field does not point into or out of the moon’s volume).
While your team will reach the moon first, the politicians in charge have entered into a rather bizarre agreement. Wherever your team lands on the moon, all the points on the surface whose magnetic field lines point in the direction of your landing site — that is, where the magnetic field points more toward your landing site than away from it — will belong to Riddler Nation. All the remaining parts of the surface will go to the competing agency’s nation.
If your team lands on a random point on this moon’s surface, then what is the expected fraction of the moon’s surface area that will be claimed by Riddler Nation?
Consider a point $L$ on the moon where your team lands and another point $M$ on the moon. If the magnetic field line through $M$ makes an acute angle with arc $LM$, then $M$ is claimed by Riddler Space Agency.
However, if the magnetic field line makes an obtuse angle with arc $LM$, then it makes an acute angle with arc $L'M$, where $L'$ is the antipode of $L$.
Ignoring the magnetic field lines that are perpendicular to $LL'$, every point on the surface points more toward than away to exactly one of $L$ and $L'$.
In other words, if the Riddler Space Agency could land two crafts, every point would be claimed exactly once.
Therfore the expected fraction is $\dfrac{1}{2}$.