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
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