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 α? Of course, if α is a positive integer, it is trivial to see that α2 unit squares can be succesfully packed. However, if α is not an integer, the problem becomes much more complicated. Intuitively, one feels that for α = N + ( 1 100), say (where N is an integer), one should pack N2 unit squares in the obvious way and surrender the uncovered border area (which is about α 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 α 7 11, which for large α is much less than the linear waste produced by the "natural" packing above.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics