Moving sofa problem
What is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?
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 is readily obtained.
John Hammersley derived a considerably higher lower bound 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 has been removed.
- 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.
- 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.
- Weisstein, Eric W., "Moving sofa problem", MathWorld.
- Wagner, Neal R. (1976). "The Sofa Problem" (PDF). The American Mathematical Monthly 83 (3): 188–189. doi:10.2307/2977022. JSTOR 2977022.
- Stewart, Ian (January 2004). Another Fine Math You've Got Me Into... Mineola, N.Y.: Dover Publications. ISBN 0486431819. Retrieved 24 April 2013.