On packing squares with equal squares.

[摘要] 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.

[学科分类] 计算机科学（综合）