{{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, the '''moving sofa problem''' or '''sofa problem''' is a two-dimensional idealization of real-life furniture-moving problems and asks for the rigid two-dimensional shape of the largest 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 |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.
The leading solution, by Joseph L. Gerver, has a value of approximately 2.2195. In November 2024, Jineon Baek posted a 119-page 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 |language=en}}</ref>
==History== The first formal publication was by the Austrian-Canadian mathematician Leo Moser in 1966,<ref>{{cite journal | last = Moser | first = Leo | author-link = Leo Moser | date = July 1966 | issue = 3 | journal = 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 bounds and upper bounds).
===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, 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 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 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|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, 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|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, [https://web.archive.org/web/20080107101427/http://mathcad.com/library/constants/sofa.htm Moving Sofa Constant], ''Mathcad Library'' (includes a diagram of Gerver's sofa).</ref>
In 1992, Joseph L. Gerver of 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 |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 |location=Mineola, N.Y. |isbn=0486431819 |url=http://store.doverpublications.com/0486431819.html |accessdate=24 April 2013}}</ref> Yoav Kallus and 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|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. 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'' – A novel by 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" – An episode of the American TV series ''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|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