Given an infinite grid of real numbers between 0 and 100, such that every number in the grid is the average of its four direct neighbours (the numbers to the left, right, above, and below it) prove that all the numbers are necessarily equal, or give a counter-example.

(In reply to Solution by Eric)

Couldn't the minimum and maximum be "at infinity" since the grid is unbounded? You really need to talk "infimum" and "supremum," also called "greatest lower bound" and "least upper bound." This problem has a difficulty rating of 4 for a reason, I think. People should be thinking of functions that are asymptotic to 0 and 100 as the infinite regions of the grid are approached. As a warmup, trying the similar thing on a two-sidely infinite line could be enlightening.

Posted by Richard on 2005-06-03 21:56:55