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 idealisation of real-life furniture-moving problems and asks for the rigid two-dimensional shape of largest area *A* that can be maneuvered through an L-shaped planar region with legs of unit width.^{[1]} The area *A* thus obtained is referred to as the *sofa constant*. The exact value of the sofa constant is an open problem.

## History

The first formal publication was by the Austrian-Canadian mathematician Leo Moser in 1966, although there had been many informal mentions before that date.^{[1]}

## Lower and upper bounds

Work has been done on proving that the sofa constant cannot be below or above certain values (lower bounds and upper bounds).

### Lower bounds

An obvious lower bound is . This comes from a sofa that is a half-disk of unit radius, which can rotate in the corner.

John Hammersley derived a lower bound of based on a shape resembling a telephone handset, consisting of two quarter-disks of radius 1 on either side of a 1 by 4/π rectangle from which a half-disk of radius has been removed.^{[2]}^{[3]}

Joseph Gerver found a sofa described by 18 curve sections each taking a smooth analytic form. This further increased the lower bound for the sofa constant to approximately 2.2195.^{[4]}^{[5]}

A computation by Philip Gibbs produced a shape indistinguishable from that of Gerver's sofa giving a value for the area equal to eight significant figures.^{[6]} This is evidence that Gerver's sofa is indeed the best possible but it remains unproven.

### Upper bounds

Hammersley also found an upper bound on the sofa constant, showing that it is at most .^{[1]}^{[7]}

Yoav Kallus and Dan Romik proved a new upper bound in June 2017, capping the sofa constant at .^{[8]}

## Ambidextrous sofa

A variant of the sofa problem asks the shape of largest area that can go round both left and right 90 degree corners in a corridor of unit width. A lower bound of area approximately 1.64495521 has been described by Dan Romik. His sofa is also described by 18 curve sections.^{[9]}^{[10]}

## See also

*Dirk Gently's Holistic Detective Agency*– novel by Douglas Adams, a subplot of which revolves around such a problem.- Mountain climbing problem
- Moser's worm problem
- "The One with the Cop" - an episode of the American TV series
*Friends*a subplot revolving around such a problem.

## References

- ^
^{a}^{b}^{c}Wagner, Neal R. (1976). "The Sofa Problem" (PDF).*The American Mathematical Monthly*.**83**(3): 188–189. doi:10.2307/2977022. JSTOR 2977022.^{[dead link]} **^**Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Halmos, Paul R. (ed.).*Unsolved Problems in Geometry*. Problem Books in Mathematics; Unsolved Problems in Intuitive Mathematics.**II**. Springer-Verlag. ISBN 978-0-387-97506-1. Retrieved 24 April 2013.**^**Moving Sofa Constant by Steven Finch at MathSoft, includes a diagram of Gerver's sofa.**^**Gerver, Joseph L. (1992). "On Moving a Sofa Around a Corner".*Geometriae Dedicata*.**42**(3): 267–283. doi:10.1007/BF02414066. ISSN 0046-5755. S2CID 119520847.**^**Weisstein, Eric W. "Moving sofa problem".*MathWorld*.**^**Gibbs, Philip, A Computational Study of Sofas and Cars**^**Stewart, Ian (January 2004).*Another Fine Math You've Got Me Into...*Mineola, N.Y.: Dover Publications. ISBN 0486431819. Retrieved 24 April 2013.**^**Kallus, Yoav; Romik, Dan (December 2018). "Improved upper bounds in the moving sofa problem".*Advances in Mathematics*.**340**: 960–982. arXiv:1706.06630. doi:10.1016/j.aim.2018.10.022. ISSN 0001-8708. S2CID 5844665.**^**Romik, Dan (2017). "Differential equations and exact solutions in the moving sofa problem".*Experimental Mathematics*.**26**(2): 316–330. arXiv:1606.08111. doi:10.1080/10586458.2016.1270858. S2CID 15169264.**^**Romik, Dan. "The moving sofa problem - Dan Romik's home page".*UCDavis*. Retrieved 26 March 2017.

## External links

- Romik, Dan (March 23, 2017). "The Moving Sofa Problem" (video).
*YouTube*. Brady Haran. Retrieved 24 March 2017. - SofaBounds - Program to calculate bounds on the sofa moving problem.
- A 3D model of Romik's ambidextrous sofa