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.

Since we're talking about an infinite grid, then we have an aleph number of grid pionts.

There are also in aleph number of real numbers between 0 and 1.

It is possible, then to define the column of cells at -infinity as 100, and the column of numbers at +infinity as 100. Then each column of grid points between -infinity and +infinity can have an infinitesimally larger or smaller value than the cells on either side. The cells above and below could have the same value.

Posted by Erik O. on 2005-06-07 03:12:46