Square-Peg Problem for Polyomino Boundaries

Definitions and mathematical statement

Use the conventional finite square-grid definition of a polyomino: let $S\subset{\mathbb{Z}}^2$ be finite and nonempty, with any two elements joined by a sequence of elements of $S$ whose successive differences belong to $\{(1,0),(-1,0),(0,1),(0,-1)\}$. Set \[ P=\bigcup_{(m,n)\in S}[m,m+1]\times[n,n+1],\qquad J=\partial P. \] Require that $J$ be a Jordan curve, meaning that it is the image of a continuous injective map from the unit circle into ${\mathbb{R}}^2$.

The proposed assertion is that for every such $P$, there exist $x,u,v\in{\mathbb{R}}^2$ satisfying \[ \|u\|=\|v\|>0,\qquad u\cdot v=0,\qquad \{x,x+u,x+u+v,x+v\}\subseteq J. \] Thus the four distinct points are the vertices of a nondegenerate Euclidean square. They may lie anywhere on $J$; no lattice-point, boundary-corner, axis-alignment, or square-interior containment condition is imposed by the supplied statement.

The record does not explicitly define "polyomino." The conventional definition above is an interpretive convention requiring confirmation against the original report, particularly if a more restrictive grid variant was intended.

Short English statement

Does every polyomino whose entire boundary is a Jordan curve have four boundary points forming a nondegenerate square? The draft uses the conventional finite, edge-connected square-grid meaning of polyomino; the precise intended grid conventions require source review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30001695$, source problem code catalogue entry, title Square-Peg Problem for Polyomino Boundaries; collection: Oberwolfach Reports Open Problems. The supplied proposed year is 2011; no individual proposer is identified.
• Supplied source citation and literature reference labelled "original OWR report": Topological and Geometric Combinatorics (2011), Oberwolfach Reports, pp. 349--423, DOI 10.4171/owr/2011/08. https://doi.org/10.4171/owr/2011/08. The supplied background describes these reports as documenting workshops and their open-problem discussions.
• Supplied literature reference labelled "current square-peg progress": https://arxiv.org/abs/2301.01340. No author, title, or specific supporting passage is supplied for this reference.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open, states that the underlying Toeplitz square-peg problem remains open for arbitrary Jordan curves, and reports that the grid/polyomino variant was not resolved by the located literature. This is supplied dated curation derived from the OWR record, not an independently verified assessment; time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified, with a supplied claim of manual semantic, typographic, and LaTeX review confirming a meaning-preserving, self-contained cleanup. This editorial label is recorded as provenance and does not constitute independent mathematical verification of this draft.