Ambidextrous Moving Sofa Constant

The ambidextrous moving sofa constant C_41b asks for the maximum area of a sofa, as defined in C_41a that can navigate both left and right corners inside a Z-shaped corridor of width 1, where the corners are sufficiently far apart.

Current state (tracked automatically)

Best known bounds for constant 41b: 1.64495521 ≤ C ≤ 2.2195. Upper bound ([Baek2024], Naive bound from C\_41a); lower bound ([Romik2016]).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 41b)	1.64495521 [Romik2016]	Tao's optimization constants
Best upper bound with an explicit value (constant 41b)	2.2195 [Baek2024] — Naive bound from C\_41a	Tao's optimization constants

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Cite

@misc{cairn-constant-41b-ambidextrous-moving-sofa-constant,
  title        = {Ambidextrous Moving Sofa Constant},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-41b-ambidextrous-moving-sofa-constant}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-13}
}

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

Description of constant

The ambidextrous moving sofa constant $C_{41 b}$ asks for the maximum area of a sofa, as defined in $C_{41 a}$ 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, $C_{41 a} \geq C_{41 b}$

Known upper bounds

Bound	Reference	Comments
$22$	[Hammersley1968]	Intended for $C_{41 a}$ , but can be modified for $C_{41 b}$
2.2195	[Baek2024]	Naive bound from $C_{41 a}$

Known lower bounds

Bound	Reference	Comments
1.64495	[Gibbs2014]	Numerical result
1.64495521	[Romik2016]

Additional comments

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

References

[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.

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 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.