A computer search reports 40 polyomino nets—flat, grid-based layouts that can be folded without overlap—that fold into three different cuboids, or box-shaped solids, with a surface area of 106. The result comes from a preprint and is conditional on the stated search model; it does not establish that 106 is the smallest possible area for such a net.
Searching the folding patterns
The search enumerated grid-aligned, non-overlapping layouts. Its algorithms exploited structural features in N × 1 × 1 nets, striped folding patterns and a variant of Redelmeier’s algorithm.
The authors also describe a breadth-first-search iterator that recursively tries possible next-tile expansions and avoids storing previous configurations. They say it can check trillions of configurations without memory overhead.
The area-106 result
The paper reports 15 common stacked nets—layouts built in layers—for the 1 × 1 × 26, 1 × 2 × 17 and 1 × 5 × 8 cuboids, all with surface area 106.
A separate simple-phase search reported 40 area-106 nets covering 26 × 1 × 1, 17 × 2 × 1 and 8 × 5 × 1 cuboids. Four were rotationally symmetric, and 25 were new non-stacked nets.
The numbers grow fast
The paper also gives a conditional expression for how striped solution counts grow as cuboid height increases, using an N × 1 × 1 cuboid as the layering cuboid. It writes that relationship as f(h) ≈ (p/q)7h. The authors say the striped-net conjectures behind it are not fully proven.
The selected depth-1 cases show the scale of the enumeration: reported solutions rise from 15 for 17 × 2 × 1 to 613,862 for 17 × 7 × 1, passing through 161, 1,387, 10,884 and 82,794 for the intervening configurations.
A matrix model of the layer states—a way of tracking how one layer can follow another—has a largest eigenvalue, or growth factor, of approximately 9.4956. The paper says each newly added layer increases the number of nets by close to a multiple of that value.
A bigger search, and a bigger cost
In the later thorough search, counts of normal nets in the N × 1 × 1 family rose from 11 for N = 1 to 407,023,305 for N = 7. The reported counts for N = 2 through N = 6 were 723, 14,978, 228,547, 3,014,430 and 36,175,381; the paper marks the N = 5 through N = 7 counts as new.
The N = 6 and N = 7 searches required logical duplicate handling to avoid memory overflow, according to the paper. A separate paired search reported 4,233,426 nets covering the 11 × 1 × 1 and 5 × 3 × 1 cuboids after around three weeks of computation.
Definitions also affect what is counted. Using different cut rules for a hole-bearing net, the researchers reported 36 new solutions; another search found four nets with two distinct cut ways, none for 3 × 3 × 1 and 7 × 1 × 1, and three for 5 × 2 × 1 and 8 × 1 × 1.
What the search does not settle
The 40-net result is a computational finding within the stated grid-aligned, non-overlapping model. It does not establish that area 106 is minimal, and the paper says its striped-net generalizations remain conjectural.
Reference images, raw output files and code were uploaded to Zenodo. The minimum-area question remains open.
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