**Previous:** Problem 77: Zipper Unfoldings of Convex Polyhedra

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

- Origin
Posed by R. Nandakumar, Feb. 2012: http://nandacumar.blogspot.in/2012/02/packing-rectangles.html. The phrase “rectangling a rectangle” was introduced by Michael Brand at http://brand.site.co.il/riddles/201203q.html.

- Status
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.

- Conjecture
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.

- Categories
packing; partitioning.

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