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 Cheating....
I just realized that in my setup, once you arrive at the converging solution, the points ON the perimeter do not following the average rule.
That being said, you can make the points on the perimeter follow the
average rule "as much as you want". That is, given a number delta >
0, One can allways find a finer subdivision with more points, such that
the difference between the fixed value of 100 V for the points on the
perimeter and the value gotten by doing the average around them, is
less than delta.
Since you can get as close as you want, but never really get there, now I'm not sure if my solution is valid or not.
Edited on June 6, 2005, 1:56 pm
Posted by ajosin
on 2005-06-06 13:55:22