Possible Duplicate:
A stronger version of discrete “Liouville’s theorem”
Let each lattice point of the plane be labeled by a positive real number . Each of these numbers is the arithmetic mean of its four neighbors ( above , below , left , right ) . Then is it true that all the labels are equal ? ( I have only been able to prove the equality of all labels when all the labels are positive integers but can not seem to get a hold when the labels are arbitrary positive reals )