The smallest polyomino nets that fold into their box in several essentially different ways
Abstract
A polyomino that folds onto the surface of a box may do so in more than one essentially different way. We make the count precise – the rotation group Rot(B) acts freely on the set of foldings of a net P into B, so |Rot(B)| divides that set's cardinality and the number of orbits ways(B,P) is well defined – and then determine the smallest surface areas at which multiplicity occurs. That 10 is the smallest area at which multiple folding ways exist is asserted, without proof and without a definition of "different", by Uehara; we prove it, together with the statement that 10 is also the smallest area at which some net folds three ways, and the exclusion below area 10 uses no enumeration of polyominoes at all. We then exhibit a 16-cell net folding into 1 × 2 × 2 in four ways and three 18-cell nets folding into 1 × 1 × 4 in four ways – the smallest four-way net previously exhibited has area 28 – and prove that no polyomino of surface area below 14 folds into its box in four essentially different ways, so the smallest such area is 14 or 16. That bound exhausts the cube and 1 × 1 × 2 over enumerations of their developments whose completeness is itself proved rather than programmed. All statements are machine-checked in Lean 4.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
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Claim ledger
Stated results
BU1known data2026-08-22
Headline: a formal definition of polyomino-into-box folding, and a decision procedure for it proved sound and complete
BU2known2026-08-22
The definition is the source's: exactly 11 of the 35 free hexominoes fold into a cube, and the other 24 provably do not
BU3known data2026-08-22
Demaine et al.'s 40 nets of area 106 each fold into all three boxes of that area
BU4known data2026-08-22
All 2263 common unfoldings of the two boxes of area 22 – the smallest area with two boxes – verified
BU5routine2026-08-22
The frontier of the smallest three-box area, derived: exactly ten candidate areas carrying 37 boxes
BU6routine2026-08-22
Negative controls: three independent kinds, one of them needing no computation at all
BU7routine2026-08-30
The number of ways a polyomino folds into a box, made well defined: the rotation group of the box acts freely on the foldings, so its order divides the fold count
BU8known2026-08-30
10 is the smallest surface area at which a polyomino folds into its box in two essentially different ways – and also the smallest for three
BU9candidate2026-08-30
A polyomino of area 16 that folds into 1x2x2 in four essentially different ways, and three of area 18 that fold into 1x1x4 in four
BU10measurement2026-08-30
The complete fold-multiplicity census of every box of surface area at most 18, and of 1x1x5 at area 22: 3,353,080 nets, exactly one folding four ways at area 16, and none folding five ways
This ledger entry is reported in prose and is not bound to a Lean theorem.BU11known data2026-08-30
Tadaki-Amano's six published fold counts, including their area-52 five-way record, re-derived from the formal definition of folding
BU12routine2026-08-30
Negative controls for the multiplicity layer, and the definition re-validated against two independent published solution counts
BU13routine2026-08-30
The keystone: every polyomino admits a chained listing beginning at any of its cells, so folding is decidable without a connectivity certificate and the fold count is an invariant of the cell set
BU14routine2026-08-30
A complete enumeration of the developments of an arbitrary box, with completeness proved rather than programmed
BU15candidate2026-08-30
The cube and 1x1x2 exhausted inside the kernel, and the first bound on the smallest four-way area: no polyomino of surface area below 14 folds into its box in four essentially different ways
BU16routine2026-08-30
Negative controls for the enumeration layer, including a net whose listing defeats the old decision procedure
BU17known data2026-08-30
The rooted-development counts 384 and 12,124 reproduce the #SAT column of arXiv:2506.01079 Table 5 from the formal definition of folding
BU18routine2026-08-30
Counting foldings one rotation orbit at a time: an eightfold cheaper upper bound on the fold count, and the area-10 census re-derived from it
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- A polyomino is a finite, edge-connected set of unit cells of ℤ². A box a × b × c has integer sides; its surface is 2(ab + ac + bc) unit squares. The polyomino folds into the box when you can wrap it onto the surface, covering every square exactly once, folding only along the grid lines of the cells. A common unfolding of several boxes is one polyomino that folds into all of them.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7