Tutoring math, you encounter patterns. The tutor brings up one about sums of squares.

If two whole numbers sum to a constant even one, then their minimum sum of squares will come from each being half. The reason:

Let the numbers sum to 2n, and let the numbers then be n-k and n+k. Then the sum of their squares is

(n-k)² + (n+k)²

Expanding each square, we get

n²-2nk+k² + n²+2nk+k²=2n²+2k²

Therefore, the minimum sum of squares will be achieved when k=0, that is, when n-k=n+k=n.

Source:

Travers, Kenneth J et al. Using Advanced Algebra. Toronto: Doubleday Canada Limited, 1977.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.