Packing equal squares into a large square

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(...

On Packing Squares with Equal Squares

The following problem arises in connection with certain multidimensional stock cutting problems : How many nonoverlapping open unit squares may be packed into a large square of side a? Of course, if a is a positive integer, it is trivial to see that a2 unit squares can be successfully packed . However, if a is not an integer, the problem becomes much more complicated . Intuitively, one feels th...

A better packing of ten equal circles in a square

Let S be a square of side s in the Euclidean plane. A pucking of circles in S is nothing else but a finite family of circular disks included in S whose interiors are pairwise disjoints. A natural problem related with such packings is the description of the densest ones; in particular, what is the greatest value of the common radius r of n circles forming a packing of S? Clearly, the centres of ...

Improved Online Square-into-Square Packing

In this paper, we show an improved bound and new algorithm for the online square-into-square packing problem. This twodimensional packing problem involves packing an online sequence of squares into a unit square container without any two squares overlapping. The goal is to find the largest area α such that any set of squares with total area α can be packed. We show an algorithm that can pack an...

Translative Packing of Unit Squares into Squares