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 re: Solution by Richard)

The 'two-sidely infinite line' problem reduces to:
 if two adjacent line points differ by e, then just by stepping 50/e boxes to the right and left will take us outside the range one way or the other.

Similarly, in our 2-dimensional grid we can look at rings around a pair of non-equal grid points.  If they differ by e, then amoungst the six adjacent neighbors, at least two of these will differ by 5e/3. Within the next ring of 10, there will be two points which differ by at least 11e/3. This pattern continues (although is admittedly much more labor-intesive to work out), but clearly within 100/e rings we will have exceeded the range of the grid. Unless, of course, e=0.

Posted by Eric on 2005-06-04 01:25:12