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An average table (Posted on 2005-06-03) Difficulty: 4 of 5
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.

No Solution Yet Submitted by ronen    
Rating: 4.2500 (4 votes)

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Solution Solution in Brief | Comment 9 of 10 |
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
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