Packing equal squares into a large square

Let s(x) denote the maximum number of non-overlapping unit squares which can be packed into a large square of side length x. Let W (x) = x − s(x) denote the “wasted” area, i.e., the area not covered by the unit squares. In this note we prove that W (x) = O ( x √ 2)/7 log x ) . This improves earlier results of Erdős-Graham and Montgomery in which the upper bounds of W (x) = O(x) and W (x) = O(x(3− √ 3)/2 log x), respectively, were obtained. A complementary problem is to determine s′(x) the minimum number of unit squares needed to cover a large square of side length x. We show that s(x) = x + O ( x √ 2)/7 log x ) , improving an earlier bound of x + O(x).