The Open Problems Project

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Problem 78: Rectangling a Rectangle


Do there exist rectangles that may be partitioned into a finite number n of rectangular pieces of equal area but with all perimeters different?


Posed by R. Nandakumar, Feb. 2012: The phrase “rectangling a rectangle” was introduced by Michael Brand at


A partial solution for rectangular pieces with real edge lengths is known—a spiral layout of 7 rectangular pieces forming a larger rectangle. See Brand’s web site. The question remains open for tiling rectangles with rational edge lengths.


If all edge lengths of the pieces are required to be rational, no such partition is possible (R. Nandakumar and N. Ramana Rao).

Further questions

The question may be extended to higher dimensions d in the obvious way. The posers believe there is no solution in \mathbb{R}^d for d \ge 3.


packing; partitioning.

Entry Revision History

R. Nandakumar and N. Ramana Rao, Mar. 14, 2012; J. O’Rourke, 15 Mar. 2012; 25 Mar. 2012.