next up previous
Next: About this document ... Up: The Open Problems Project Previous: Problem 77: Zipper Unfoldings


Problem 78: Rectangling a Rectangle

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≥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.


next up previous
Next: About this document ... Up: The Open Problems Project Previous: Problem 77: Zipper Unfoldings
The Open Problems Project - December 04, 2015