Congruent Partitions of Polygons

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.

Let $P$ be a polygon and $n$ a positive integer. Choose $n$ polygonal pieces contained in $P$, with disjoint interiors and mutually congruent under translations, rotations or reflections. Minimize the uncovered area $\operatorname{area}(P\setminus\bigcup_i Q_i)$. A minimizing choice is an optimal congruent partition; a choice with no uncovered area is perfect.

Statement recorded in the supplied source.

Find an optimal congruent partition of a given polygon $P$ into $n$ pieces. Decide whether a perfect congruent partition exists (especially for $n>2$); determine how the problem changes when reflections are prohibited; and bound the least uncovered area in terms of $P$ and $n$. [S2]

Short English statement

Find an optimal congruent partition of a given polygon $P$ into $n$ pieces. Decide whether a perfect congruent partition exists (especially for $n>2$); determine how the problem changes when reflections are prohibited; and bound the least uncovered area in terms of $P$ and $n$.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5500073 (AMR-054-0073). 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 73, Statement and complete Partial/Related Results. https://github.com/edemaine/topp/blob/main/Problems/P.000073.