A shape with Heesch number 7

FrontierMath: Open Problems lists this problem with notability “Solid result”, field “Topology/Geometry” and type “Construction - Finite; Record Breaking”. Each problem has a bespoke verifier program, so a proposed answer can be checked automatically.

Verifiable task — Epoch prompt, verbatim

# Record-Setting Heesch Polyform

## Goal
Submit one connected, hole-free unmarked polyform with at most **200 cells**, compact enough for the verifier (`span_x + span_y <= 29`), and a witness patch showing **Heesch number at least 7**. The submitted polyform should not admit an isohedral tiling.

## Submission File
Return a single plain-text file containing one tile record in the compact `heesch-sat` text format.

The record must have this structure:

```text
GRID x1 y1 x2 y2 ... xn yn
~ hc hh P
N
level1 <a,b,c,d,e,f>
level2 <a,b,c,d,e,f>
...
levelN <a,b,c,d,e,f>

P is the number of witness patches; it is 1 in the common case (hh == hc) and 2 when you also certify hh == hc + 1. The block above shows a single patch. When P = 2, append a second count line M followed by M placement lines for the hole-allowed patch.

Line-by-Line Format

GRID x1 y1 x2 y2 ... xn yn

GRID is the grid abbreviation. Allowed abbreviations include O (polyomino/square grid), H (polyhex/hexagonal grid), and I (polyiamond/triangular grid). Use O unless you intentionally submit a different supported polyform family. The remaining integers are occupied cell coordinates, not boundary vertices and not an outline walk. For example:

O 0 0 1 0

describes a domino: one cell at (0,0) and one adjacent cell at (1,0).

Every coordinate pair must name a cell of the grid you chose. O and H have a cell at every pair, but the other families do not: in I, for instance, a cell needs x and y to be congruent mod 3 and congruent to 0 (upward triangle) or 1 (downward triangle), so I 1 0 names no cell. The verifier rejects a shape that lists a position which is not a cell, and likewise a placement whose transform is not a symmetry of the grid and so moves the tile off the grid's cells.

~ hc hh P

This status line declares a non-tiler witness record.

• hc is the claimed hole-free Heesch number.
• hh is the claimed Heesch number when holes are allowed in the outer corona. A non-tiler always has hh == hc or hh == hc + 1. If you are not demonstrating the hole variant, set hh = hc.
• P is the number of witness patches that follow. Set P = 1 when hh == hc (just the hole-free patch). Set P = 2 when hh == hc + 1: give the hole-free patch reaching corona hc first, then a second patch reaching corona hh whose outer corona may contain holes.
• Use 7 <= hc <= 12. Set hh = hc (with P = 1) unless you also want to certify the hole variant hh = hc + 1 (with P = 2). Acceptance depends only on the recomputed hole-free hc reaching the threshold.

N

The number of placed copies of the tile in the following patch, including the central copy. Each patch begins with its own count line and is followed by that many placement lines.

level <a,b,c,d,e,f>

Each placement line gives one copy of the tile.

• level is the corona index of that copy: 0 for the central seed, 1 for the first surrounding corona, 2 for the second, and so on.
• <a,b,c,d,e,f> is the affine lattice transform applied to every occupied cell (x,y) of the base tile:

x' = a*x + b*y + c
y' = d*x + e*y + f

The matrix part [[a,b],[d,e]] should be a grid symmetry: a rotation and/or reflection valid for the selected grid. The translation part is (c,f).

The central tile should normally be:

0 <1,0,0,0,1,0>

Constraints

• Use one tile record per submission file.
• Every listed coordinate pair must be a cell of the chosen grid (see the shape line above).
• The shape must be connected and hole-free.
• The shape must have at most 200 occupied cells.
• The shape must be compact enough for the current verifier: its coordinate bounding-box spans must satisfy span_x + span_y <= 29. Absolute translation does not matter; the solver normalizes position before classification. More elongated shapes may be mathematically interesting, but this verifier rejects them to avoid a known fixed-bitmap SAT failure mode.
• The witness patch must include all copies needed to demonstrate coronas through level hc.
• The patch must have no overlapping placed cells.
• Keep coordinates reasonably small so automated checking can finish.
• The verifier independently recomputes the classification with a maximum search level of 12. A shape that reaches that search bound is classified as inconclusive and is not accepted, even if its submitted witness is valid.

Toy Example

This example is not a solution to the challenge; it only illustrates syntax.

O 0 0 1 0
~ 1 1 1
3
0 <1,0,0,0,1,0>
1 <1,0,0,0,1,1>
1 <1,0,0,0,1,-1>

Submit your own record-setting high-Heesch construction instead of the toy example.

Epoch lists this problem as unsolved in its data archive updated 7 October 2026. Earlier AI attempts are not displayed by Epoch.