Packing equal squares into a large square

Abstract

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 2 − 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 ( 3 + 2 ) / 7 log x ) . This improves earlier results of Erdős–Graham and Montgomery in which the upper bounds of W ( x ) = O ( x 7 / 11 ) 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 2 + O ( x ( 3 + 2 ) / 7 log x ) , improving an earlier bound of x 2 + O ( x 7 / 11 ) .