Every square of an infinite chessboard contains a positive number. Given that the number in each square is equal to the arithmetic mean of the numbers in the four neighboring squares, is it true that all numbers are equal to each other?
I would expect that this is a well-known problem, but I was unable to find it on the Web.
Notice that the question has an immediate solution if we drop the positivity assumption, and also if we insist on all numbers to be integers.
The problem can also be stated in terms of the adjacency operator of the infinite grid graph: we want to show that it does not have eigenvectors with all components positive.
(Please, feel free to re-tag this question.)
