Common-net existence for equal-area cuboids

Determine whether every pair of cuboids with the same surface area has a common polyomino net, in particular whether the 5×5×5 cuboid shares a net with the 37×1×1 cuboid.

Background

The authors report that every tested pair consisting of an N×1×1 cuboid and another cuboid of equal surface area had at least one common net. They do not know whether this phenomenon holds for arbitrary equal-area cuboid pairs.

They single out the 5×5×5 and 37×1×1 cuboids as a candidate pair with no common net and note that a human-readable nonexistence proof would be desirable. Prior work cited in the paper establishes only that the smallest area of a pair without a common net must exceed 86.

References

We are not sure whether this remains the case for any two cuboids sharing the same area. For this reason, we think it would be interesting if the $5\times 5 \times 5$ cuboid does not share a net with the $37\times 1\times 1$ cuboid.

— 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 “Are there two cuboid shape with the same surface area, but no common nets?” (Section 3.1.1)