Moving Sofa Problem

Gerver's sofa is the unique sofa that attains the sofa constant, up to a rigid motion. The motion is needed: horizontalHallway is (-∞, 1] × [0, 1], so a leftward translate of any moving sofa is again one, obtained by sliding right and then following the original motion.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-moving-sofa,
  title        = {Moving Sofa Problem},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/moving-sofa}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Gerver's sofa is the unique sofa that attains the sofa constant, up to a rigid motion.

The motion is needed: horizontalHallway is $(- \infty, 1] \times [0, 1]$ , so a leftward translate of any moving sofa is again one, obtained by sliding right and then following the original motion. It has the same area, so uniqueness cannot hold on the nose.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.MovingSofa.

theorem volume_eq_sofaConstant_iff_congruent_gerversSofa (s : Set ℝ²)
    (hs : ∃ m, IsMovingSofa s m) :
    volume s = sofaConstant ↔ ∃ g : E(2), s = g '' gerversSofa

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
[Ge92] Gerver, J. L., _On moving a sofa around a corner_. Geometriae Dedicata 42.3 (1992): 267-283.
[Ro18] Romik, D. _Differential equations and exact solutions in the moving sofa problem_. Experimental mathematics 27.3 (2018): 316-330.
[Ba24] Baek, J. _Optimality of Gerver's Sofa_. arXiv preprint arXiv:2411.19826 (2024).

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.