Rectangling a Rectangle

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A rectangular partition has finitely many rectangular pieces with disjoint interiors and union equal to the outer rectangle. Each piece has positive edge lengths; its area is the product of its edge lengths and its perimeter is twice their sum.

Statement recorded in the supplied source.

Do there exist rectangular partitions into $n\ge2$ pieces of equal area with pairwise different perimeters? The source reports a seven-piece construction with real edge lengths and asks whether such a partition is possible with all edge lengths rational, conjecturing that it is not. The source also asks for the analogous equal-volume, distinct-boundary-measure partition into boxes in $\mathbb R^d$ for $d\ge3$, with a conjectured negative answer. [S2]

Short English statement

Do there exist rectangular partitions into $n\ge2$ pieces of equal area with pairwise different perimeters? The source reports a seven-piece construction with real edge lengths and asks whether such a partition is possible with all edge lengths rational, conjecturing that it is not. The source also asks for the analogous equal-volume, distinct-boundary-measure partition into boxes in $\mathbb R^d$ for $d\ge3$, with a conjectured negative answer.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5500078 (AMR-054-0078). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke (editors), The Open Problems Project, maintained original source, Problem 78, Statement and complete Partial/Related Results. https://github.com/edemaine/topp/blob/main/Problems/P.000078.