Let f be a real-valued function on the plane such that for every square ABCD in the plane, f(A) + f(B) + f(C) + f(D) = 0.

Prove or disprove that f(P) = 0 for every point P in the plane?

Imagine a 4 by 4 grid of equally spaced points, perhaps lattice points from (0,0) to (3,3).  Let these 16 points have values a1 to a16.  There are 20 possible squares one can make from these 16 points.  So there could be 20 equations and 16 unknowns.  Take the first 16 equations:  they could be represented by a matrix 16x16 such that each row has 4 ones and the rest zeroes.  The 1x16 array this is equal to is all zeroes.  Imagine adding, subtracting, multiplying, dividing one row to/from another until the matrix has become the identity matrix.  The array is still going to be all zeroes.  QED.

Posted by Larry on 2010-11-21 19:15:29