A computer study has reported 40 simple-phase polyomino nets with surface area 106 that can fold into three different cuboids: 1 × 1 × 26, 1 × 2 × 17 and 1 × 5 × 8. Four of the 40 are rotationally symmetric, while 25 are new non-stacked nets.
Those are not the only shared patterns in the result. The same three cuboids have 15 common stacked nets, according to the preprint. In this setting, a common net is a pattern that folds into each of the specified shapes under the study’s rules.
Polyomino nets are flat, grid-based shapes considered as possible unfoldings of a cuboid. The model restricts folds to the polyomino grid and requires a flat, non-overlapping net. The findings therefore apply to a defined geometric setting rather than to every possible way of cutting and folding a box.
The study is an enumeration exercise rather than a sample-based experiment: its material consists of computer-generated net configurations, with comparisons made between search procedures, net classes and cuboid configurations.
The geometry behind the search
The authors approached the problem with several computer-enumeration strategies. Their algorithms exploit hidden structures in N × 1 × 1 nets, striped folding patterns and a variant of the Redelmeier algorithm.
For simple-phase nets, the researchers represented layer states in a matrix. Each entry records the number of ways to add layer state i onto layer state j. The matrix’s largest eigenvalue—a summary number extracted from that system—was approximately 9.4956. The paper describes it as the real solution to λ³ − 10λ² + 5λ − 2 = 0.
Another tool was a breadth-first-search iterator, a procedure that works through possible next additions. It recursively tries every possible next-tile addition and does not need to save previous configurations. The search applies the same grid and no-overlap conditions used in the broader model.
The thorough-search procedure came with a practical trade-off. It was reported to be 3 to 10 times slower than the comparison method, and it did not update the lowest possible area for a net covering three cuboids.
A tally that grows quickly
A separate part of the work examined the N × 1 × 1 family, the notation used for cuboids whose dimensions are N, 1 and 1. The search was designed to count the corresponding nets across a series of N values.
For N = 1 through 7, the thorough search counted 11, 723, 14,978, 228,547, 3,014,430, 36,175,381 and 407,023,305 nets, respectively. The largest reported count was for N = 7.
The results for N = 1 through 4 were marked not new, while those for N = 5 through 7 were marked new in the comparison table.
For another pair of cuboids—11 × 1 × 1 and 5 × 3 × 1—the search found 4,233,426 nets without invisible cuts, after around 3 weeks of computation.
The paper also reports infinite families of nets that fold into three cuboid shapes, along with counts of common nets between smaller cuboids.
What the numbers leave open
Every count depends on the model’s boundaries. Folds are restricted to the polyomino grid, and the nets must be flat and non-overlapping. The paper also reports a non-standard-cut analysis with 36 new solutions, making that result part of a changed cutting setup rather than the main count.
The area-106 examples do not settle the smallest-area question. The conclusion asks for the smallest-area net that can cover three cuboid shapes with orthogonal, or right-angle, folds.
It also asks whether a polyomino net can cover four cuboid shapes with orthogonal folds. That question remains separate from the reported three-cuboid configurations.
Taken together, the work provides a catalogue of reported configurations, counts for increasingly large searches and several methods for exploring structured nets. Its stated questions concern how small a three-shape net can be and whether four shapes can share one under the same folding framework.
A preprint with supporting files
The document is an arXiv version 1 preprint dated 20 August 2026.
The paper says that reference images, raw output files and code were uploaded to Zenodo, providing supporting materials for the reported computational work.
Paper data and sources
Original title: Polyomino Nets Covering Three Different Boxes of Area 106 and Related Results
Authors: Erik D. Demaine, Jenny Diomidova, Nicole Jacobus et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text