Moving Sofa Constant
Description of constant
The moving sofa constant $C_{41}=A$ 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. 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 |
|---|---|---|
| $2 \sqrt{2}$ | [Hammersley1968] | |
| 2.37 | [KR2018] | Best published bound, using a computer-assisted proof scheme |
| 2.2195 | [Baek2024] | Announced bound, matching the Gerver construction |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $\pi/2 + 2/\pi$ | [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
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[Baek2024] Baek, J. (2024). Optimality of Gerver’s Sofa. arXiv preprint arXiv:2411.19826.
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[GGSWT2025] Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Mathematical exploration and discovery at scale. arXiv:2511.02864
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[Gerver1992] Gerver, Joseph L. (1992). On Moving a Sofa Around a Corner. Geometriae Dedicata. 42 (3): 267–283.
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[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.
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[KR2018] Kallus, Y., & Romik, D. (2018). Improved upper bounds in the moving sofa problem. Advances in Mathematics, 340, 960-982.
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[Moser1966] Moser, L. (1966). Problem 66-11, Moving furniture through a hallway. SIAM Review, 8(3), 381.
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[Romik2017] Romik, D. (2017). Differential equations and exact solutions in the moving sofa problem. Experimental Mathematics, 26(2), 316-330.
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[Wagner1976] Wagner, N. R. (1976). The Sofa Problem. The American Mathematical Monthly, 83(3), 188–189.
Contribution notes
Prepared with assistance from Gemini 3 Pro.