The Erdos and Campbell-Staton conjectures about square packing

Abstract

Put n open non-overlapping squares inside a unit square, and let f(n) denote the maximum possible value of the sum of the side lengths of the n squares. Campbell and Staton, building on a question of Erdos, conjectured that f(k^2+2c+1)=k+c/k, where c is any integer and k\geq |c|. We show that if this conjecture is true for one value of c, then it is true for all values of c.