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 that for a = N + (1/100), say (where Nis an integer), one should pack N' unit squares in the obvious way and surrender the uncovered border area (which is about a/50) as unusable waste . After all, how could it help to place the unit squares at all sorts of various skew angles? In this note, we show how it helps . In particular, we prove that we can always keep the amount of uncovered area down to at most proportional to a 1111 which for large a is much less than the linear waste produced by the "natural" packing above .