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.

Laplace's equation can not be used to prove this, and the reason is that in complex number theory there are functions called doubly periodic (or elliptic) functions. these functions are not necesserily constand and have a preiod in both x and y directions (this about something like sin(x)*sin(y)). these functions both hold the laplace equation, and are not constand, and taking discrete values of them will just give you constand values.

i suggest you try to read the proof of lioville theorem in complex analysis, which states that an analytic complex function with no poles which is bounded is necessarily constant.

 

Posted by ronen on 2005-06-26 21:13:49