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.
All the numbers are necessarily equal. Here is the proof;
To describe the plane with an inifinite number of points, we can use the continous version of the average rule; laplace's equation
(see equation 6).
Since ALL the points in the continuum must follow laplace's equation,
there is only one possible solution; a constant field.
Conversly, the only way to not get a constant field is by having at
least one point with a laplacian different than 0 (such a point does
not follow the average rule in neither the rule's discrete or
continious form). Such points carry the charge distribution of
the field and are forbidden by the problem's premise.
Unfortunatly this is not a "fundamental" proof. It relies on the well
known results of laplace's and poison's equations used in
electromagnetism, and on the axiomatic assumption that an infinite 2-D
grid of points can be described by a continous plane.
Posted by ajosin
on 2005-06-08 19:00:53