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? (more unsolved problems in mathematics)

The Hammersley sofa has area 2.2074 but is not the largest solution

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). One lower bound is ${\displaystyle \scriptstyle A\,\geq \,\pi /2\,\approx \,1.57079}$. 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 ${\displaystyle \scriptstyle A\,\geq \,\pi /2+2/\pi \,\approx \,2.2074}$ based on a handset-type shape 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 $\scriptstyle 2/\pi \,$ has been removed.^[2][3]

Joseph Gerver found a sofa that further increased the lower bound for the sofa constant to approximately ${\displaystyle \scriptstyle 2.2195}$.^[4][5]

Hammersley also found an upper bound on the sofa constant, showing that it is at most $\scriptstyle 2{\sqrt {2}}\,\approx \,2.8284$.^[1][6]

See also

• Mountain climbing problem
• Moser's worm problem

References

1. Wagner, Neal R. (1976). "The Sofa Problem" (PDF). The American Mathematical Monthly. 83 (3): 188–189. doi:10.2307/2977022. JSTOR 2977022.
2. 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.
3. Moving Sofa Constant by Steven Finch at MathSoft, includes a diagram of Gerver's sofa
4. Gerver, Joseph L. (1992). "On Moving a Sofa Around a Corner". Geometriae Dedicata. 42 (3): 267–283. doi:10.1007/BF02414066. ISSN 0046-5755.
5. Weisstein, Eric W. "Moving sofa problem". MathWorld.
6. Stewart, Ian (January 2004). Another Fine Math You've Got Me Into... Mineola, N.Y.: Dover Publications. ISBN 0486431819. Retrieved 24 April 2013.