A shape with Heesch number 7

Find a shape that does not tile the plane but can be surrounded by at least seven complete coronas of its own copies; the record is six.

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State the problem, from FrontierMath: Open Problems

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	1	+# A shape with Heesch number 7
	2	+
	3	+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.
	4	+
	5	+## Problem
	6	+
	7	+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.
	8	+
	9	+Task. Find a shape whose Heesch number is finite and at least 7.
	10	+
	11	+## Background
	12	+
	13	+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.
	14	+
	15	+## What is known
	16	+
	17	+- Record: Heesch number 6 (Bašić, 2021) [1]; previously 5 (Mann and Thomas) [2].
	18	+- Exhaustive SAT-based classification of small unmarked polyforms found nothing above the record [2].
	19	+- No shape in the plane with finite Heesch number 7 or more is known.
	20	+- In other spaces the question is settled: finite Heesch numbers are unbounded for convex monotiles in the hyperbolic plane (Maiti, 2026, [arXiv:2603.27827](https://arxiv.org/abs/2603.27827)) and, by an explicit construction, in Euclidean 3-space (Bašić, October 2026, [arXiv:2610.24281](https://arxiv.org/abs/2610.24281)). Both are unrefereed preprints; the Euclidean plane remains open.
	21	+
	22	+## What would count as progress
	23	+
	24	+- 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.
	25	+- 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.
	26	+- A family of shapes with unbounded finite Heesch numbers would answer Heesch's problem.
	27	+
	28	+## References
	29	+
	30	+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](https://doi.org/10.1007/s00283-020-10034-w)
	31	+2. C. S. Kaplan, Heesch numbers of unmarked polyforms, Contributions to Discrete Mathematics 17(2) (2022). [arXiv:2105.09438](https://arxiv.org/abs/2105.09438)
	32	+3. C. Mann, Heesch's tiling problem, Amer. Math. Monthly 111 (2004), 509–517. [doi:10.1080/00029890.2004.11920105](https://doi.org/10.1080/00029890.2004.11920105)
	33	+
	34	+## Source
	35	+
	36	+Imported from [FrontierMath: Open Problems, heesch-number](https://epoch.ai/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.