Edge-unfolding polycubes

Glue unit cubes face to face into a solid with no tunnels. Can its surface always be cut along cube edges and unfolded into one flat, non-overlapping piece? This is one of the most concrete open cases among the unfolding questions that grew out of Dürer's problem, and small cases can be explored by computer.

Problem

A polycube is a polyhedron formed by gluing finitely many unit cubes of the integer lattice face to face, whose surface is homeomorphic to a sphere (genus zero). Every edge of a unit cube that lies on the surface is available for cutting, including edges that lie inside a flat face of the polycube. An edge unfolding cuts the surface along such edges so that the surface, developed into the plane, forms a single simple polygon whose unit squares have pairwise disjoint interiors.

Question. Does every polycube have an edge unfolding? Equivalently: is there a polycube that cannot be edge-unfolded?

Background

The question was posed by George Hart and Joseph O'Rourke in 2004. For general orthogonal polyhedra, whose faces meet at right angles but are not subdivided into unit squares, the answer is no: a simple box-on-box example cannot be edge-unfolded, which is why the unit-square refinement is allowed here. The Open Problems Project describes this as among the most specific questions whose answer would make progress on unfolding orthogonal polyhedra; O'Rourke's 2019 survey lists it as one of six unsettled cases in its table of unfolding questions [1].

What is known

• Layered polycubes. Damian and Meijer showed that every polycube whose horizontal layers are orthogonally convex has an edge unfolding [2].
• Zipper unfoldings fail. Demaine, Demaine, Eppstein and O'Rourke constructed polycubes with no edge zipper unfolding, one in which the cut edges form a single path [3]. So any general solution needs branching cut trees.
• With extra cuts. If faces may be cut along more lines than the unit grid, every genus-zero orthogonal polyhedron can be unfolded; polynomially many extra cutting planes suffice (Damian, Demaine and Flatland, 2014) [4].
• Related variant. If the faces shared by adjacent cubes are kept as part of the object, every such "solid polycube" has an edge unfolding (Harismendy, da Silva, Largeteau-Skapin and Andrès, 2026) [5]. This does not settle the surface version above.

What would count as progress

• A polycube with a proof that it has no edge unfolding. For a small candidate, the claim can be checked by an exhaustive or SAT-based search over cut trees.
• An exhaustive computation showing that every polycube with at most $k$ cubes has an edge unfolding, for $k$ as large as possible, with the nets published so that each one can be checked.
• An unfolding algorithm, with a proof, for a broader class, such as polycubes whose layers are arbitrary (not orthogonally convex) or all polycubes of height two.

References

1. J. O'Rourke, Unfolding polyhedra, Proc. 31st Canadian Conf. Comput. Geom. (2019), 85–86. arXiv:1908.07152
2. M. Damian and H. Meijer, Edge-unfolding polycubes with orthogonally convex layers, in Combinatorial Optimization and Applications, LNCS, Springer (2025), 63–74. doi:10.1007/978-981-96-4448-3_6, arXiv:2407.01326
3. E. D. Demaine, M. L. Demaine, D. Eppstein and J. O'Rourke, Some polycubes have no edge zipper unfolding, Geombinatorics 31(3) (2022), 101–109. arXiv:1907.08433
4. M. Damian, E. D. Demaine and R. Flatland, Unfolding orthogonal polyhedra with quadratic refinement: the delta-unfolding algorithm, Graphs Combin. 30 (2014), 125–140. doi:10.1007/s00373-012-1257-9
5. B. Harismendy, T. da Silva, G. Largeteau-Skapin and É. Andrès, Unfolding solid polycubes, Discrete Math. Theor. Comput. Sci. 28(2) (2026). doi:10.46298/dmtcs.16116

Source

Imported from The Open Problems Project, Problem 64, curated by Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.