# Moving sofa problem

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{{Short description|Unsolved geometry question on moving a sofa through a 90° angle}}
thumb|upright=1.2|Diagram of the moving sofa problem
{{unsolved|mathematics|What is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?}}
In [mathematics](/source/mathematics), the '''moving sofa problem''' or '''sofa problem''' is a two-dimensional idealization of real-life [furniture-moving problems](/source/Motion_planning) and asks for the rigid two-dimensional shape of the largest [area](/source/area) that can be maneuvered through an L-shaped planar region with legs of unit width.<ref name="Neal Wagner">{{cite journal |last=Wagner |first=Neal R. |title=The Sofa Problem |journal=[The American Mathematical Monthly](/source/The_American_Mathematical_Monthly) |volume=83 |issue=3 |year=1976 |pages=188–189 |doi=10.2307/2977022 |url=http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |jstor=2977022 |access-date=2009-07-25 |archive-date=2015-04-20 |archive-url=https://web.archive.org/web/20150420160001/http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |url-status=dead }}</ref> The area thus obtained is referred to as the '''sofa constant'''. The exact value of the sofa constant is an [open problem](/source/Unsolved_problems_in_mathematics).

The leading solution, by Joseph L. Gerver, has a value of approximately 2.2195. In November 2024, Jineon Baek posted a 119-page [arXiv](/source/arXiv) preprint claiming that Gerver's value is optimal, which if true would solve the moving sofa problem.<ref name="Jineon Baek">{{cite arXiv|last=Baek|first=Jineon|date=2024-11-29|title=Optimality of Gerver's Sofa|eprint=2411.19826|class=math.MG}}</ref><ref>{{Cite web |last=Green |first=Richard |date=2025-02-14 |title=The Largest Sofa You Can Move Around a Corner |url=https://www.quantamagazine.org/the-largest-sofa-you-can-move-around-a-corner-20250214/ |access-date=2025-03-12 |work=[Quanta Magazine](/source/Quanta_Magazine) |language=en}}</ref>

==History==
The first formal publication was by the Austrian-Canadian mathematician [Leo Moser](/source/Leo_Moser) in 1966,<ref>{{cite journal
 | last = Moser | first = Leo | author-link = Leo Moser
 | date = July 1966
 | issue = 3
 | journal = [SIAM Review](/source/SIAM_Review)
 | jstor = 2028218
 | page = 381
 | title = Problem 66-11, Moving furniture through a hallway
 | volume = 8| doi = 10.1137/1008074 }}</ref> although there had been many informal mentions before that date.<ref name="Neal Wagner"/>

==Bounds==
Work has been done to prove that the sofa constant <math>A</math> cannot be below or above specific values ([lower bound](/source/lower_bound)s and [upper bound](/source/upper_bound)s).

===Lower===
thumb|The Hammersley sofa has an area of 2.2074, but is not the largest solution
{{multiple image
 | image1 = Gerver.svg
 | caption1 = Gerver's sofa of area 2.2195 with 18 curve sections
 | image2 = Telefono automatico a batteria centrale (BCA) - Museo scienza tecnologia Milano D0955 10.jpg
 | caption2 = A telephone [handset](/source/handset), a closer match than a sofa to Gerver's shape
 | total_width = 400
}}
A lower bound on the sofa constant can be proven by finding a specific shape with a high area and a path for moving it through the corner. <math>A \geq \pi/2 \approx 1.57</math> is an obvious lower bound. This comes from a sofa that is a half-[disk](/source/disk_(mathematics)) of unit radius, which can slide up one passage into the corner, rotate within the corner around the center of the disk, and then slide out the other passage.

In 1968, [John Hammersley](/source/John_Hammersley) stated a lower bound of <math>A \geq \pi/2 + 2/\pi \approx 2.2074</math>.<ref name=hammersley>{{cite journal|author=J. M. Hammersley|author-link=John Hammersley|title=On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities|pages=66–85|url=https://archive.org/details/hammersley1968|journal=[Bulletin of the Institute of Mathematics and Its Applications](/source/Bulletin_of_the_Institute_of_Mathematics_and_Its_Applications)|volume=4|year=1968}} See Appendix IV, Problems, Problem 8, p. 84.</ref> This can be achieved using a shape resembling an old-fashioned [telephone handset](/source/handset), consisting of two quarter-disks of radius 1 on either side of a 1 by <math>4/\pi</math> rectangle from which a half-disk of radius <math>2/\pi</math> has been removed.<ref>{{cite book |last1=Croft |first1=Hallard T. |authorlink1=Hallard Croft |last2=Falconer |first2=Kenneth J. |last3=Guy |first3=Richard K. |authorlink3=Richard K. Guy |title=Unsolved Problems in Geometry |series=Problem Books in Mathematics; Unsolved Problems in Intuitive Mathematics |volume=II |editor-last=Halmos |editor-first=Paul R. |publisher=[Springer-Verlag](/source/Springer-Verlag)|year=1994 |isbn=978-0-387-97506-1 |url=https://archive.org/details/unsolvedproblems0000crof |accessdate=24 April 2013 |url-access=registration }}</ref><ref>[Finch, Steven](/source/Steven_Finch), [https://web.archive.org/web/20080107101427/http://mathcad.com/library/constants/sofa.htm Moving Sofa Constant], ''[Mathcad Library](/source/Mathcad)'' (includes a diagram of Gerver's sofa).</ref>

In 1992, Joseph L. Gerver of [Rutgers University](/source/Rutgers_University) described a sofa with 18 curve sections, each taking a smooth analytic form. This further increased the lower bound for the sofa constant to approximately 2.2195 {{OEIS|A128463}}.<ref>{{cite journal |last=Gerver |first=Joseph L. |title=On Moving a Sofa Around a Corner |journal=[Geometriae Dedicata](/source/Geometriae_Dedicata) |issn=0046-5755 |volume=42 |issue=3 |pages=267–283 |year=1992 |doi=10.1007/BF02414066|s2cid=119520847 }}</ref><ref>{{MathWorld|urlname=MovingSofaProblem|title=Moving sofa problem}}</ref>
thumb|Overlap of Hammersley’s sofa (red) and Gerver’s sofa (blue).

===Upper===
Hammersley stated an upper bound on the sofa constant of at most <math>2\sqrt{2} \approx 2.8284</math>.<ref name=hammersley/><ref name="Neal Wagner"/><ref>{{cite book |last=Stewart |first=Ian |authorlink=Ian Stewart (mathematician) |title=Another Fine Math You've Got Me Into... |date=January 2004 |publisher=[Dover Publications](/source/Dover_Publications) |location=Mineola, N.Y. |isbn=0486431819 |url=http://store.doverpublications.com/0486431819.html |accessdate=24 April 2013}}</ref> Yoav Kallus and [Dan Romik](/source/Dan_Romik) published a new upper bound in 2018, capping the sofa constant at <math>2.37</math>. Their approach involves rotating the corridor (rather than the sofa) through a finite sequence of distinct angles (rather than continuously) and using a computer search to find translations for each rotated copy so that the intersection of all of the copies has a connected component with as large an area as possible. As they show, this provides a valid upper bound for the optimal sofa, which can be made more accurate using more rotation angles. Five carefully chosen rotation angles lead to the stated upper bound.<ref>{{Cite journal|last1=Kallus|first1=Yoav|last2=Romik|first2=Dan|date=December 2018|title=Improved upper bounds in the moving sofa problem|journal=[Advances in Mathematics](/source/Advances_in_Mathematics)|volume=340|pages=960–982|arxiv=1706.06630|doi=10.1016/j.aim.2018.10.022|s2cid=5844665|issn=0001-8708}}</ref>

== Ambidextrous sofa ==
right|280px|thumb|Romik's ambidextrous sofa
A variant of the sofa problem asks the shape of the largest area that can go around both left and right 90-degree corners in a corridor of unit width (where the left and right corners are spaced sufficiently far apart that one is fully negotiated before the other is encountered). A lower bound of area approximately 1.64495521 has been described by [Dan Romik](/source/Dan_Romik). 18 curve sections also describe his sofa.<ref name="Dan Romik">{{cite journal |last=Romik |first=Dan |title=Differential equations and exact solutions in the moving sofa problem |journal=Experimental Mathematics |volume=26 |issue=2 |year=2017 |pages=316–330 |doi=10.1080/10586458.2016.1270858 |arxiv=1606.08111 |s2cid=15169264 }}</ref><ref name="UCDavis">{{cite web|last1=Romik|first1=Dan|title=The moving sofa problem - Dan Romik's home page|url=https://www.math.ucdavis.edu/~romik/movingsofa/|website=UCDavis|accessdate=26 March 2017}}</ref>

==See also==
* ''[Dirk Gently's Holistic Detective Agency](/source/Dirk_Gently's_Holistic_Detective_Agency)'' – A novel by [Douglas Adams](/source/Douglas_Adams), with a subplot that revolves around such a problem.
* {{anl|Moser's worm problem}}
* {{anl|Square packing in a square}}
* "[The One with the Cop](/source/The_One_with_the_Cop)" – An episode of the American TV series ''[Friends](/source/Friends)'' with a subplot pivoting around such a problem.

==References==
{{reflist}}

==External links==
*{{cite web|last1=Romik|first1=Dan|title=The Moving Sofa Problem|url=https://www.youtube.com/watch?v=rXfKWIZQIo4 |archive-url=https://ghostarchive.org/varchive/youtube/20211221/rXfKWIZQIo4 |archive-date=2021-12-21 |url-status=live|website=YouTube|publisher=[Brady Haran](/source/Brady_Haran)|accessdate=24 March 2017|format=video|date=March 23, 2017}}{{cbignore}}
*[https://github.com/ykallus/SofaBounds SofaBounds] - Program to calculate bounds on the sofa moving problem.
*[https://www.thingiverse.com/thing:2191347 A 3D model of Romik's ambidextrous sofa]
*{{cite web |title=Mathematician solves the moving sofa problem |website=Phys.org |date=2024-12-11 |url=https://phys.org/news/2024-12-mathematician-sofa-problem.html |ref={{sfnref|Phys.org|2024}} |access-date=2024-12-12}}
*[https://www.quantamagazine.org/the-largest-sofa-you-can-move-around-a-corner-20250214/ The Largest Sofa You Can Move Around a Corner]

Category:Discrete geometry
Category:Unsolved problems in geometry
Category:Recreational mathematics
Category:1966 introductions
Category:Couches

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Adapted from the Wikipedia article [Moving sofa problem](https://en.wikipedia.org/wiki/Moving_sofa_problem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Moving_sofa_problem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.