A couple days ago I posted this on MSE (here) but in retrospect it might be more appropriate for this site.
This theorem is well-known (maybe it can be called Morera's theorem):
A continuous function satisfying the mean value property on balls is harmonic.
I was recently surprised to hear in a talk that the conclusion still holds if you only check the mean value property on three (I think) radii. Does anyone have a reference or name for this result? I would enjoy seeing the details and a proof.