# Moving sofa problem

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 What is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?
The Hammersley sofa has area 2.2074... but is not the largest solution

The moving sofa problem was formulated by the Austrian-Canadian mathematician Leo Moser in 1966. The 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. The area A thus obtained is referred to as the sofa constant. The exact value of the sofa constant is an open problem.

## Lower and upper bounds

As a semicircular disk of unit radius can pass through the corner, a lower bound for the sofa constant $\scriptstyle A\,\geq\,\pi/2\,\approx\, 1.570796327$ is readily obtained.

John Hammersley derived a considerably higher lower bound $\scriptstyle A\,\geq\,\pi/2 + 2/\pi\,\approx\,2.207416099$ based on a handset-type shape consisting of two quarter-circles of radius 1 on either side of a 1 by 4/π rectangle from which a semicircle of radius $\scriptstyle 2/\pi\,$ has been removed.[1][2]

Joseph Gerver found a sofa that further increased the lower bound for the sofa constant to 2.219531669.[3][4]

In a different direction, an easy argument by Hammersley shows that the sofa constant is at most $\scriptstyle 2\sqrt{2}\,\approx\, 2.8284$.[5][6]

## References

1. ^ Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Hamos, 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.
2. ^ Moving Sofa Constant by Steven Finch at MathSoft, includes a diagram of Gerver's sofa
3. ^ Gerver, Joseph L. (1992). "On Moving a Sofa Around a Corner". Geometriae Dedicata 42 (3): 267–283. doi:10.1007/BF02414066. ISSN 0046-5755.
4. ^
5. ^ Wagner, Neal R. (1976). "The Sofa Problem" (PDF). The American Mathematical Monthly 83 (3): 188–189. doi:10.2307/2977022. JSTOR 2977022.
6. ^ Stewart, Ian (January 2004). Another Fine Math You've Got Me Into... Mineola, N.Y.: Dover Publications. ISBN 0486431819. Retrieved 24 April 2013.