A shape with Heesch number 7

A shape that does not tile the plane can sometimes still be surrounded by copies of itself, layer after layer, before every further layer fails; its Heesch number counts the complete layers. Shapes with Heesch numbers up to 6 are known, and nothing higher has ever been found. The task is to find a shape with finite Heesch number at least 7.

Problem

Let $T$ be a shape in the plane (a closed topological disk). A corona around a patch of non-overlapping congruent copies of $T$ is a ring of further copies that completely surrounds the patch, leaving no gaps and no overlaps. The Heesch number of $T$ is the largest $k$ such that a single copy of $T$ can be surrounded by $k$ successive coronas. If $T$ tiles the plane, its Heesch number is infinite. Variants differ in whether the outermost corona may enclose holes; we use the hole-free version.

Task. Find a shape whose Heesch number is finite and at least 7.

Background

Heesch's problem asks which positive integers are Heesch numbers [2]. Writing in 1987, Grünbaum and Shephard knew no example with finite Heesch number above 1. A few isolated examples up to 4 followed [3]; Mann and Thomas reached 5 by a systematic computer search of polyforms with marked edges; and in 2021 Bašić exhibited a figure with Heesch number 6 [1]. Kaplan introduced a SAT-solver method and computed the Heesch numbers of all polyominoes with up to 19 cells, polyhexes with up to 17 and polyiamonds with up to 24, without breaking the record [2]. It is unknown whether every positive integer occurs as a Heesch number.

What is known

• Record: Heesch number 6 (Bašić, 2021) [1]; previously 5 (Mann and Thomas) [2].
• Exhaustive SAT-based classification of small unmarked polyforms found nothing above the record [2].
• No shape in the plane with finite Heesch number 7 or more is known.
• In other spaces the question is settled: finite Heesch numbers are unbounded for convex monotiles in the hyperbolic plane (Maiti, 2026, arXiv:2603.27827) and, by an explicit construction, in Euclidean 3-space (Bašić, October 2026, arXiv:2610.24281). Both are unrefereed preprints; the Euclidean plane remains open.

What would count as progress

• A shape with a witness patch: the placements of copies, corona by corona, through the seventh, plus evidence that the shape does not tile the plane. The patch is easy to check; certifying that the shape does not tile is the delicate part.
• Epoch scores submissions with its own verifier: a connected, hole-free polyform with at most 200 cells on the square, hexagonal or triangular grid, fitting a bounding box with $\mathrm{span}_x + \mathrm{span}_y \le 29$, with a witness patch in the heesch-sat text format. The verifier recomputes the Heesch number by SAT up to 12 coronas and rejects tilers and inconclusive shapes.
• A family of shapes with unbounded finite Heesch numbers would answer Heesch's problem.

References

1. B. Bašić, A figure with Heesch number 6: pushing a two-decade-old boundary, Math. Intelligencer 43(3) (2021), 50–53. doi:10.1007/s00283-020-10034-w
2. C. S. Kaplan, Heesch numbers of unmarked polyforms, Contributions to Discrete Mathematics 17(2) (2022). arXiv:2105.09438
3. C. Mann, Heesch's tiling problem, Amer. Math. Monthly 111 (2004), 509–517. doi:10.1080/00029890.2004.11920105

Source

Imported from FrontierMath: Open Problems, heesch-number, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.