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OP-12053OpenGeometry and topology

Ambidextrous moving sofa constant (C41bC_{41b})

Ambidextrous moving sofa constant: determine C41bC_{41b} or improve its bounds; the repository records 1.64495521 (lower) and 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 ambidextrous moving sofa constant C41bC_{41b} asks for the maximum area of a sofa, as defined in C41aC_{41a} that can navigate both left and right corners inside a Z-shaped corridor of width 1, where the corners are sufficiently far apart. That is, 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). Trivially, C41a≥C41bC_{41a} \ge C_{41b}

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Background

  • Wikipedia section
  • Another question posed by Conway considers a T-shaped intersection. It is unknown whether the two problems are equivalent.

References (from the repository page)

  • [Baek2024] Baek, J. (2024). Optimality of Gerver's Sofa. arXiv:2411.19826.

  • [Gibbs2014] Philip Gibbs (2014). A Computational Study of Sofas and Cars. vixra:1411.0038v2.

  • [Hammersley1968] 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.

  • [Romik2016] Dan Romik (2016). Differential equations and exact solutions in the moving sofa problem. arxiv:1606.08111.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1.64495521; best known upper bound 2.2195. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
222 \sqrt{2} [Hammersley1968] Intended for C41aC_{41a}, but can be modified for C41bC_{41b}
2.2195 [Baek2024] Naive bound from C41aC_{41a}

Known lower bounds

Bound Reference Comments
1.64495 [Gibbs2014] Numerical result
1.64495521 [Romik2016]
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Ambidextrous moving sofa constant ($C_{41b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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