Moser's worm problem

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 worm is the image of a continuous rectifiable curve $\gamma:[0,1]\to\mathbb R^2$ of length one, where length is the supremum of sums $\sum_j|\gamma(t_j)-\gamma(t_{j-1})|$ over finite partitions. A planar region $K$ accommodates it if a translate and rotation of that image is contained in $K$. Area is Lebesgue area.

Statement recorded in the supplied source.

Find a planar region of minimum area that accommodates every worm, and determine the minimum area. [S2]

Short English statement

Find the least-area planar region into which every continuous curve of length one can be moved and rotated.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200075 (AMR-071-0075). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Zhipeng Deng, Universal Triangle Covering Curve and Polygonal Chain: Escaping Forest and Fitting Worm (2026), v2; arXiv:2608.01393. https://arxiv.org/pdf/2608.01393.