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.

Quite simply there can be no minimum value which is not surrounded by four equal squares, which in turn would be similarly surrounded by four equal squares ad infinitum covering the entire grid with the same minimum value. The same is of course true for any 'maximum' value. 

Posted by Eric on 2005-06-03 21:09:33