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 Which aleph? by Erik O.)

Note; In my first post I indicated that the average rule is for the values V(x_i,y_j), where (x_i, y_j) are the points on the grid, is a discrete aproximation to the continous equation Laplacian{V(x,y)} = 0.

Back to your example...

You can make the points in-between and outsied-of the columns follow the average rule, you only have to worry about the points on the columns;

... -100 -100 -100 -80 -60 -40 -20 0 20 40 60 80 100 100 100 ...
... -100 -100 -100 -80 -60 -40 -20 0 20 40 60 80 100 100 100 ...
... -100 -100 -100 -80 -60 -40 -20 0 20 40 60 80 100 100 100 ...


The more points you put between the columns, the finer you divide space, and the better the points on the columns follow the average rule. When you say "putting the columns at -infinity and infinity" I think of it as keeping them at the same physical distance and adding an infinite number of points in between them. That is, going from a discrete to a continious distribution.

Maybe you were already aware that you have just proposed the electromagnetic example of two parallel plates set at potendial -100 and +100. If you put an infinite number of points in between the two plates you get the exact solution for Laplaces equation; V(x) = -V_0 if x < -d; V(x) = (V_0/d)*x if -d <= x <= d; V(x) = V_0 if x > d, where V_0 = 100 and d is the physical distance between the plates (the origin is set between the two plates).

In my example (a square metalic tube), there is no analitcal solution to Laplaces equation, so I like your example better.

That being said, even in the continous case, the laplacican is not 0 for all points of space; at x = -d and +d the laplacian goes to infinity (this is related to the superficial charge on the plates). There is now way to get around this, no matter how many points exist between the plates.

In conclusion;

Laplace's equation needs a charge distribution to not have the trivial solution V = constant, and the points where there is a charge distribution do not have a 0 laplacian and cannot follow the average rule.  Therefore in the discrete case, even when the number of points goes to infinity (in other words, when the continous solution is being reached) all the numbers must be equal if they all follow the average rule.

Posted by ajosin on 2005-06-07 13:55:55