About the problem
Given a two-dimensional shape, Heesch’s problem asks: how many times can the shape surround itself?
If a shape tiles the plane, then its Heesch number is infinite; otherwise, it has finite Heesch number. There are examples of shapes with Heesch numbers up to 6; these can all be viewed on the Wikipedia page. The record, a shape with Heesch number exactly 6, is due to Bašić from 2021. It is unknown whether there is a shape with Heesch number \(n\) for every positive integer \(n\). Read more about Heesch’s problem in Kaplan’s survey here.
This problem asks for a shape with finite Heesch number at least seven. The submission requires a drawing certifying that the shape has Heesch number at least seven. The verifier checks that the certificate correctly shows that the Heesch number is at least seven, and also checks whether the shape tiles the plane (in which case it has infinite Heesch number and is rejected). Testing whether the shape tiles the plane involves some risk, as it can be difficult to decide in general if a shape tiles the plane. Otherwise, it may also be the case that there are no shapes with Heesch number 7.
Prompt
# 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: ```text O 0 0 1 0 ``` describes a domino: one cell at `(0,0)` and one adjacent cell at `(1,0)`. `~ 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: ```text 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: ```text 0 <1,0,0,0,1,0> ``` ## Constraints - Use one tile record per submission file. - 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. ```text 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.