Moving Sofa Constant
The moving sofa constant C_41a=A is the maximum area of a connected, rigid planar shape that can maneuver through an L-shaped corridor of unit width. The corridor is formed by two semi-infinite strips of width 1 meeting at a right angle.
From the catalogue. Imported from Terence Tao and contributors (optimizationproblems repository) (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.
Cite
@misc{cairn-constant-41a-moving-sofa-constant,
title = {Moving Sofa Constant},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/constant-41a-moving-sofa-constant}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
} Also: CITATION.cff · Atom feed of results
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The problem
Description of constant
The moving sofa constant is the maximum area of a connected, rigid planar shape that can maneuver through an L-shaped corridor of unit width. The corridor is formed by two semi-infinite strips of width 1 meeting at a right angle. The problem asks for the shape of the largest area (the "sofa") that can be moved from one end of the corridor to the other by a continuous rigid motion (translation and rotation).
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| [Hammersley1968] | ||
| 2.37 | [KR2018] | Best published bound, using a computer-assisted proof scheme |
| 2.2195* | [Baek2024] | Announced (unverified) bound, matching the Gerver construction |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| [Hammersley1968] | ||
| 2.2195 | [Gerver1992] | The Gerver sofa |
Additional comments and links
- First appears in print in [Moser1966].
- It was claimed in a recent preprint [Baek2024] that Gerver's sofa [Gerver1992] is the optimal solution, which if true would solve the moving sofa problem.
- AlphaEvolve was able to numerically locate Gerver's sofa as a proposed maximizer, though without a proof of optimality [GGSWT2025].
- Wikipedia entry on this problem
References
- [Baek2024] Baek, J. (2024).
Optimality of Gerver's Sofa. arXiv preprint arXiv:2411.19826.
- [GGSWT2025] Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Mathematical exploration and discovery at scale. arXiv:2511.02864
- [Gerver1992] Gerver, Joseph L. (1992).
On Moving a Sofa Around a Corner. Geometriae Dedicata. 42 (3): 267–283.
- [Hammersley1968] Dr. J. M. Hammersley (1968).
On the enfeeblement of mathematical skills by modern mathematics and by similar soft intellectual trash in schools and universities. Bulletin of the Institute of Mathematics and Its Applications. 4: 66–85. See Appendix IV, Problems, Problem 8, p. 84.
- [KR2018] Kallus, Y., & Romik, D. (2018).
Improved upper bounds in the moving sofa problem. Advances in Mathematics, 340, 960-982.
- [Moser1966] Moser, L. (1966).
Problem 66-11, Moving furniture through a hallway. SIAM Review, 8(3), 381.
- [Romik2017] Romik, D. (2017).
Differential equations and exact solutions in the moving sofa problem. Experimental Mathematics, 26(2), 316-330.
- [Wagner1976] Wagner, N. R. (1976).
The Sofa Problem. The American Mathematical Monthly, 83(3), 188–189.
What counts as progress
- A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
- A formal proof (Lean) of a known bound, or a precise error in a claimed one.
- New references for the tables above (literature claims).
Source and licence
Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.