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^{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.