Existence of common striped nets for depth-three cuboids

Establish whether three cuboids with dimensions N×1×1, (4q+3)×3×h₁, and (4r+3)×3×h₂, with q>r, always have a common striped net whenever h₁ and h₂ exceed 2, h₁ satisfies the required q- and r-dependent minimum, the perimeter ratio p(4q+3,3)/p(4r+3,3) is odd, p(4q+3,3)p(h₁,3)=p(4r+3,3)p(h₂,3), and the surface areas agree.

Background

The paper extends its striped-net framework from depth-one cuboids to cuboids of depth three. The proposed conditions combine lower bounds on the heights, an odd perimeter ratio, a perimeter-product equality, and matching surface areas.

The authors explicitly identify this result as a conjecture and state immediately afterward that it is not fully proven.

References

Although not fully proven, these conjectures help reveal locations in the $w$, $h$, and $d$ space where solutions are more densely populated.

— 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 2 (labelled conj:depth3)

We were unable to verify whether the area-102 cuboids ($25\times 1\times 1$) have a simple-phase net that covers three cuboids.

— Polyomino Nets Covering Three Different Boxes of Area 106 and Related Results  (2608.19910 - Demaine et al., 20 Aug 2026) in Section “Simple-Phase nets and the Matrix Equation,” subsection “Basic Observations of the Results,” item 2

We especially look forward to getting answers to the two main questions that remain open: What is the smallest possible net by unit area in which a polyomino could be used to cover 3 cuboid shapes with orthogonal folds?

— Polyomino Nets Covering Three Different Boxes of Area 106 and Related Results  (2608.19910 - Demaine et al., 20 Aug 2026) in Section “Conclusion”