Fastest way to meet, without communication, on a sphere? I was puzzled by a question my colleague asked me, and now seeking your help.
Suppose you and your friend* end up on a big sphere. There are no visual cues on where on the sphere you both are, and the sphere is way bigger than you two. There are no means of communication. You can determine your relative position and direction by navigating the stars**. You can move anywhere, and your friend too. 
Upon inspecting the sphere, you see it is rock-solid, so you cannot create markings. To protect the environment, you are not allowed to leave other stuff, like a blood trace or breadcrumbs.
You have been put on the sphere without being able to communicate a plan. 
How would you be able to find each other (come within a certain distance $\epsilon$?) What would be the optimal strategy to move?
*Since you are here, you must be a rational person. For this puzzle we assume your friend is rational too..Which makes it odd that you end up on that sphere anyway
**While you can determine your position relatively, you are on a sphere in a galaxy so far away that you cannot determine absolute 'north', 'south' etc. by the stars.

I think completely random movement is sub optimal.
I think a better strategy is to pick a direction (any will do), stick to it and randomise your speed.
If both parties do this their paths will cross twice each orbit (unless they are on the same orbit in which case they will meet sooner due to randomised speed)
if you go full random changing direction as well as velocity you aren't guaranteed to ever cross the path of the other. (although as t gets large it becomes increasingly likely that your paths will cross at least once.)
Speed randomisation is necessary to avoid never meeting because of resonance.
Convergence will be quickest if both parties adopt the same strategy. If the other party adopts any other rational strategy that doesn't involve being stationary they will meet eventually.
