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Covering a Square by Small Perimeter Rectangles*
N. Alon1'** and D. J. Kleitman2"***
IMassachusetts Institute of Technology,Cambridge,MA and BellCommunicationsResearch,
Morristown, NJ 07960
2MassachusettsInstituteof Technology,Departmentof AppliedMathematics,Cambridge,MA02139
Abstract. We show that if the unit square is covered by n rectangles, then at
least one must have perimeter at least 4(2m + 1)/(n + m(m +1)), where m is
the largest integer whose square is at most n. This result is exact for n of the
form m(m +1) (or m2).
I. Introduction
In this note, we address the following problem, suggested by L. Hurwicz [1]. How
can we partition the unit square, U, into a given number, n, of rectangles (all
having edges parallel to those of the square), so as to minimize the largest of their
perimeters? The same question has been raised when the rectangles are only
required to cover the square, with overlap allowed.
These problems arose from a model of a communication problem, in which n
