Packing equal squares into a large square

Let s(x)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)=x2−s(x)W(x)=x2−s(x) denote the “wasted” area, i.e., the area not covered by the unit squares. In this note we prove thatW(x)=O(x(3+2)/7logx). This improves earlier results of Erdős–Graham and Montgomery in which the upper bounds of W(x)=O(x7/11)W(x)=O(x7/11) and W(x)=O(x(3−3)/2logx), respectively, were obtained. A complementary problem is to determine s′(x)s′(x) the minimum number of unit squares needed to cover a large square of side length x. We show thats′(x)=x2+O(x(3+2)/7logx), improving an earlier bound of x2+O(x7/11)x2+O(x7/11).