362:. 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.
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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
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63:. The leading solution, by Joseph L. Gerver, has a value of approximately 2.2195 and is thought to be close to the optimal, based upon subsequent study and theoretical bounds.
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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 (sequence
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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.
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A lower bound on the sofa constant can be proven by finding a specific shape of a high area and a path for moving it through the corner.
551:"On the enfeeblement of mathematical skills by 'Modern Mathematics' and by similar soft intellectual trash in schools and universities"
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that can be maneuvered through an L-shaped planar region with legs of unit width. The area thus obtained is referred to as the
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585:. Problem Books in Mathematics; Unsolved Problems in Intuitive Mathematics. Vol. II. Springer-Verlag.
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What is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?
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Work has been done to prove that the sofa constant (A) cannot be below or above specific values (
739:
Kallus, Yoav; Romik, Dan (December 2018). "Improved upper bounds in the moving sofa problem".
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Romik, Dan (2017). "Differential equations and exact solutions in the moving sofa problem".
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in 1966, although there had been many informal mentions before that date.
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The first formal publication was by the
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The
Hammersley sofa has area 2.2074 but is not the largest solution
506:(July 1966). "Problem 66-11, Moving furniture through a hallway".
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published a new upper bound in 2018, capping the sofa constant at
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Hammersley stated an upper bound on the sofa constant of at most
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is an obvious lower bound. This comes from a sofa that is a half-
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Gerver, Joseph L. (1992). "On Moving a Sofa Around a Corner".
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Bulletin of the
Institute of Mathematics and Its Applications
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and asks for the rigid two-dimensional shape of the largest
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903:- Program to calculate bounds on the sofa moving problem.
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area approximately 1.64495521 has been described by
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Gerver's sofa of area 2.2195 with 18 curve sections
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220:. This can be achieved using a shape resembling a
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841:"The moving sofa problem - Dan Romik's home page"
213:{\displaystyle A\geq \pi /2+2/\pi \approx 2.2074}
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47:is a two-dimensional idealization of real-life
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59:. The exact value of the sofa constant is an
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383:. 18 curve sections also describe his sofa.
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252:rectangle from which a half-disk of radius
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331:{\displaystyle 2{\sqrt {2}}\approx 2.8284}
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573:Croft, Hallard T.; Falconer, Kenneth J.;
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418:" - an episode of the American TV series
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148:{\displaystyle A\geq \pi /2\approx 1.57}
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907:A 3D model of Romik's ambidextrous sofa
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708:Another Fine Math You've Got Me Into...
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394:Dirk Gently's Holistic Detective Agency
30:(more unsolved problems in mathematics)
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622:(includes a diagram of Gerver's sofa).
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16:Geometry question on motion planning
711:Mineola, N.Y.: Dover Publications.
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577:(1994). Halmos, Paul R. (ed.).
21:Unsolved problem in mathematics
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927:Unsolved problems in geometry
869:Romik, Dan (March 23, 2017).
818:10.1080/10586458.2016.1270858
581:Unsolved Problems in Geometry
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283:In 1992, Joseph L. Gerver of
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410:Square packing in a square
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871:"The Moving Sofa Problem"
765:10.1016/j.aim.2018.10.022
374:Romik's ambidextrous sofa
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49:furniture-moving problems
932:Recreational mathematics
796:Experimental Mathematics
447:Wagner, Neal R. (1976).
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166:stated a lower bound of
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742:Advances in Mathematics
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273:{\displaystyle 2/\pi }
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245:{\displaystyle 4/\pi }
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682:"Moving sofa problem"
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614:Moving Sofa Constant
416:The One with the Cop
405:Moser's worm problem
355:{\displaystyle 2.37}
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635:Geometriae Dedicata
41:moving sofa problem
937:1966 introductions
679:Weisstein, Eric W.
648:10.1007/BF02414066
449:"The Sofa Problem"
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338:. Yoav Kallus and
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285:Rutgers University
280:has been removed.
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592:978-0-387-97506-1
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890:. Retrieved
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848:. Retrieved
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480:the original
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89:upper bounds
85:lower bounds
82:
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61:open problem
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45:sofa problem
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882:Brady Haran
749:: 960–982.
509:SIAM Review
37:mathematics
916:Categories
901:SofaBounds
809:1606.08111
756:1706.06630
718:0486431819
516:(3): 381.
504:Moser, Leo
489:2009-07-25
429:References
773:0001-8708
687:MathWorld
664:119520847
656:0046-5755
381:Dan Romik
340:Dan Romik
323:≈
268:π
240:π
205:≈
202:π
180:π
177:≥
162:In 1968,
140:≈
129:π
126:≥
73:Leo Moser
892:24 March
886:Archived
850:26 March
826:15169264
724:24 April
598:24 April
561:: 66–85.
549:(1968).
387:See also
942:Couches
878:YouTube
874:(video)
845:UCDavis
781:5844665
530:2028218
476:2977022
421:Friends
293:in the
290:A128463
67:History
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326:2.8284
208:2.2074
79:Bounds
39:, the
822:S2CID
804:arXiv
777:S2CID
751:arXiv
660:S2CID
526:JSTOR
483:(PDF)
472:JSTOR
452:(PDF)
301:Upper
95:Lower
894:2017
852:2017
769:ISSN
726:2013
713:ISBN
652:ISSN
600:2013
587:ISBN
350:2.37
295:OEIS
157:disk
143:1.57
87:and
53:area
814:doi
761:doi
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518:doi
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43:or
35:In
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123:A
23::
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