This is E10.10 From Williams "Probability with Martingales" book, which I spent a lot but I could not figure out what is the relation between Gauss's theorem and this end! Here is the question:
The control system on the star-ship Enterprise has gone wonky. All that one can do is to set a distance to be travelled. The spaceship will then move that distance in a randomly chosen direction, then stop. The object is to get into the Solar System, a ball of radius r. Initially, the Enterprise is at a distance $R_o(> r)$ from the Sun. Let $R_n$ be the distance from Sun to Enterprise after n 'space-hops'. Use Gauss's theorems on potentials due to spherically-symmetric charge distributions to show that whatever strategy is adopted, $1/ R_n$ is a supermartingale, and that for any strategy which always sets a distance no greater than that from Sun to Enterprise, $1/R_n$ is a martingale.
How could it be related to something like $1/R_n$ in any way? What is a good idea?