# P2892: Plane tilings by every five-cell lattice animal

- ID: `P2892`
- Reference: `five-cell-lattice-animal-plane-tiling`
- Page: https://theoremdb.org/statements/P2892
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(A\subset\mathbb Z^2\) have exactly five elements, with no connectivity assumption. Must \(\mathbb Z^2\) admit a partition into sets of the form \(g(A)+t\), where \(t\in\mathbb Z^2\) and \(g\) is a rotation or reflection preserving the square lattice?

### Definitions

- **Definition.** The set A represents the union of the five closed unit cells whose lower-left corners are its elements; such a possibly disconnected set is called a five-cell lattice animal.
- **Definition.** A tiling is an exact partition of lattice cells: every element of Z^2 belongs to exactly one transformed copy, so overlaps and uncovered cells are forbidden.

### What counts as a solution

- Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists.
- A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow and Mathematics Stack Exchange pages have no solution. The source author reported in 2021 that the author of the four-cell theorem knew of no later progress, and no subsequent exact classification was found. Exact unresolved remainder: Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists. A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient. [1](#reference-1) [2](#reference-2) [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow and Mathematics Stack Exchange pages have no solution. The source author reported in 2021 that the author of the four-cell theorem knew of no later progress, and no subsequent exact classification was found. Exact unresolved remainder: Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists. A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: The MathOverflow and Mathematics Stack Exchange pages have no solution. The source author reported in 2021 that the author of the four-cell theorem knew of no later progress, and no subsequent exact classification was found.

Exact unresolved remainder: Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists. A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient.

### Background and intake notes

Disconnected animals have unbounded relative offsets, so the central issue is finding a finite reduction or an invariant. Periodic tiling certificates and failed offset families are directly reusable.

- Original intake status: UNKNOWN as of 2026-07-27. The MathOverflow and Mathematics Stack Exchange pages have no solution. The source author reported in 2021 that the author of the four-cell theorem knew of no later progress, and no subsequent exact classification was found.
- On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for MathOverflow question 380490; comments settle only special shapes such as a 2 by 2 block plus one cell.
- Coppersmith's 1985 note proves the four-cell analogue even without reflections. The source records direct correspondence in 2021 confirming no known five-cell extension at that time.
- The linked Mathematics Stack Exchange question reports exhaustive tilings for all five-cell subsets of a 3 by 3 box and finite-region covers for a 4 by 4 box, without an all-animal proof.
- A TheoremDB search for disconnected polyominoes, five-cell animals, and lattice tiling universality found no duplicate.

- Recorded example: Every five-cell subset of a 3 by 3 lattice box was reported to tile using translations and rotations, but that finite family does not cover animals with widely separated cells.

### Open directions

- **Route 1** (reported): Prove that every five-element subset A of Z^2 tiles by lattice translations and square-lattice symmetries, or exhibit a specific five-element A and prove that no such tiling exists. A positive computational classification must state a finite reduction for arbitrarily separated cells and provide periodic fundamental-domain certificates for every reduced class; bounded-box enumeration alone is insufficient. [1](#reference-1)

### Computational notes

- The source reports that every five-cell subset of a 4 by 4 box covers a 10 by 10 square; a finite cover is weaker than a plane tiling and should be stored separately.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `five-cell-lattice-animal-plane-tiling`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>MathOverflow question 380490, “Plane tilings by every five-cell lattice animal,” checked 2026-08-01. Question 380490 and all visible comments, checked through the Stack Exchange API on 2026-07-27. https://mathoverflow.net/questions/380490/does-every-5-celled-animal-tile-the-plane
   - Also cited at Full question, answers, and visible comments concerning Plane tilings by every five-cell lattice animal; checked 2026-08-01.
   - Also cited at Editorial research route recorded 2026-08-01.
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For Plane tilings by every five-cell lattice animal: This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.
   - Source named by the research packet.
2. <a id="reference-2"></a>Mathematics Stack Exchange question 3955595, related discussion for “Plane tilings by every five-cell lattice animal,” checked 2026-08-01. question statement and update requesting progress on the five-cell case after Coppersmith's four-cell theorem https://math.stackexchange.com/questions/3955595/does-every-5-celled-animal-tile-the-plane
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Plane tilings by every five-cell lattice animal, this source is an exact duplicate and discovery discussion for the five-cell target, with no accepted resolution.
3. <a id="reference-3"></a>Don Coppersmith, “Each Four-Celled Animal Tiles the Plane,” Journal of Combinatorial Theory, Series A 40(2) (1985), 444-449. DOI 10.1016/0097-3165(85)90105-0. main theorem that every four-celled animal tiles the plane https://www.science.smith.edu/~jorourke/MathOverflow/Coppersmith4Animal.pdf
   - website; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Plane tilings by every five-cell lattice animal, this source establishes the sharp known lower-size boundary and does not handle five-celled animals.