General matrix characterization of all N×1×1 polyomino nets

Construct a matrix representation of the full search for all polyomino nets covering an N×1×1 cuboid, and determine whether the largest eigenvalue of that matrix equals the largest eigenvalue, approximately 9.496, of the 7×7 matrix governing simple-phase nets; subsequently extend such matrix representations to N×A×B cuboids with fixed A and B.

Background

The paper represents simple-phase nets for an N×1×1 cuboid by transitions among seven possible layer states, encoded in a 7×7 matrix. The authors suggest that unrestricted N×1×1 nets might admit an analogous, larger finite-state matrix description.

Such a representation could accelerate searches for nets covering multiple cuboids. The authors further conjecture that the dominant eigenvalue of the unrestricted matrix would coincide with that of the simple-phase matrix, and identify extension to fixed-cross-section N×A×B cuboids as a longer-term generalization.

References

We conjecture that the full search of all polyomino nets covering the $N\times 1 \times 1$ cuboids could also be mapped out by a bigger matrix. Once the matrix is found, we could potentially use it to speed up the search for nets that fold into the $N\times 1 \times 1$ cuboid and other cuboids. We also conjecture that the top eigenvalue of that matrix will match the top eigenvalue of the $7\times 7$ matrix, which is approximately 9.496...

— Polyomino Nets Covering Three Different Boxes of Area 106 and Related Results  (2608.19910 - Demaine et al., 20 Aug 2026) in Section “Open Questions Raised While Studying The Results,” subsection “Can the simple-phase search algorithm for N×1×1 cuboids algorithm be extended to include all N×1×1 nets?”

Three cuboids with dimensions $N\times 1 \times 1$, $(4q+1)\times 1\times h_1$, and $(4r+1)\times 1\times h_2$ (where $q > r$) have a common striped net if the following conditions are met simultaneously:

— Polyomino Nets Covering Three Different Boxes of Area 106 and Related Results  (2608.19910 - Demaine et al., 20 Aug 2026) in Section “Generalization of the Area 106 nets,” Conjecture 1 (labelled conj:depth1)