Report Number: CS-TR-75-483
Institution: Stanford University, Department of Computer Science
Title: On packing squares with equal squares.
Author: Erdoes, Paul
Author: Graham, Ronald L.
Date: March 1975
Abstract: The following problem arises in connection with certain multi-dimensional stock cutting problems: How many non-overlapping open unit squares may be packed into a large square of side $\alpha$? Of course, if $\alpha$ is a positive integer, it is trivial to see that unit squares ean be successfully packed. However, if $\alpha$ is not an integer, the problem beeomes much more complicated. Intuitively, one feels that for $\alpha$ = N + 1/100, say, (where N is an integer), one should pack $N^2$ unit squares in the obvious way and surrender the uncovered border area (which is about $\alpha$/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 ${\alpha}^{7/11}$, which for large $\alpha$ is much less than the linear waste produced by the "natural" packing above.
http://i.stanford.edu/pub/cstr/reports/cs/tr/75/483/CS-TR-75-483.pdf