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OP-12052OpenGeometry and topology

Moving sofa constant (C41aC_{41a})

Moving sofa constant: determine C41aC_{41a} or improve its bounds; the repository records 2.2195 (lower) and 2.37 (2.2195*) (upper).

MSC 49Q10 · Optimization of shapes other than minimal surfaces

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

The moving sofa constant C41a=AC_{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. 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).

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Background

  • 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 (from the repository page)

  • [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.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 2.2195; best known upper bound 2.37 (2.2195*). Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
222 \sqrt{2} [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
π/2+2/π\pi/2 + 2/\pi [Hammersley1968]
2.2195 [Gerver1992] The Gerver sofa
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Moving sofa constant ($C_{41a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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