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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?

Abstract

Determine whether a rectangle with rational side lengths can be tiled by finitely many rectangles of equal area whose perimeters are all distinct.

Background and progress

A spiral layout of seven rectangles gives a partial solution when real edge lengths are allowed. The rational-edge-length case remains open, with the conjecture that no such partition exists; higher-dimensional analogues are also open. The original problem appears in the Open Problems Project at https://topp.openproblem.net/p78.

Reference papers

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Organizer

Boyuan Wang

Minghan Wang

Bochao Li

Hongwei Hu