# Moving sofa problem

Unsolved problem in 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.[1] The area thus obtained is referred to as the sofa constant. The exact value of the sofa constant is an open problem. The currently 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.

## History

The first formal publication was by the Austrian-Canadian mathematician Leo Moser in 1966,[2] although there had been many informal mentions before that date.[1]

## Bounds

Work has been done to prove that the sofa constant (A) cannot be below or above specific values (lower bounds and upper bounds).

### Lower

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. ${\displaystyle A\geq \pi /2\approx 1.57}$ 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 ${\displaystyle A\geq \pi /2+2/\pi \approx 2.2074}$.[3] This can be achieved using a shape resembling a telephone handset, consisting of two quarter-disks of radius 1 on either side of a 1 by ${\displaystyle 4/\pi }$ rectangle from which a half-disk of radius ${\displaystyle 2/\pi }$ has been removed.[4][5]

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 (sequence A128463 in the OEIS).[6][7]

### Upper

Hammersley stated an upper bound on the sofa constant of at most ${\displaystyle 2{\sqrt {2}}\approx 2.8284}$.[3][1][8] Yoav Kallus and Dan Romik published a new upper bound in 2018, capping the sofa constant at ${\displaystyle 2.37}$. 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.[9]

## 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.[10][11]

## References

1. ^ 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. Archived from the original (PDF) on 2015-04-20. Retrieved 2009-07-25.
2. ^ Moser, Leo (July 1966). "Problem 66-11, Moving furniture through a hallway". SIAM Review. 8 (3): 381. JSTOR 2028218.
3. ^ a b 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.
4. ^ 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. Vol. II. Springer-Verlag. ISBN 978-0-387-97506-1. Retrieved 24 April 2013.
5. ^ Finch, Steven, Moving Sofa Constant, Mathcad Library (includes a diagram of Gerver's sofa).
6. ^ 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.
7. ^
8. ^ Stewart, Ian (January 2004). Another Fine Math You've Got Me Into... Mineola, N.Y.: Dover Publications. ISBN 0486431819. Retrieved 24 April 2013.
9. ^ 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.
10. ^ 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.
11. ^ Romik, Dan. "The moving sofa problem - Dan Romik's home page". UCDavis. Retrieved 26 March 2017.