Moving sofa problem

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'.

As a semicircular disk of unit radius can pass through the corner, a lower bound for the sofa constant $\scriptstyle A\, =\, \pi/2$ or 1.570796327 is readily obtained. Hammersley derived a considerably higher lower bound $\scriptstyle A\, =\, \pi/2 + 2/\pi$ or 2.207416099 based on a handset-type shape consisting of two quarter-circles on either side of a 1 by 4/π rectangle from which a semicircle of radius $\scriptstyle 2/\pi\,$ has been removed.^[1][2]

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 $2\sqrt{2}$ or 2.8284.^[5][6] The exact value of the sofa constant is still an open problem.

See also

• Mountain climbing problem
• Moser's worm problem

References

1. ^ H.T. Croft, K.J. Falconer, and R.K. Guy, Unsolved Problems in Geometry, Springer-Verlag, 1994
2. ^ Moving sofa problem on Mathsoft includes a diagram of Gerver's sofa
3. ^ Joseph L. Gerver (1992). "On Moving a Sofa Around a Corner". Geometriae Dedicata 42 (3): 267–283. doi:10.1007/BF02414066. 
4. ^ Weisstein, Eric W., "Moving sofa problem" from MathWorld.
5. ^ Neal R. Wagner (1976). "The Sofa Problem". The American Mathematical Monthly 83 (3): 188–189. doi:10.2307/2977022. JSTOR 2977022. http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf. 
6. ^ I. Stewart, Another Fine Math You've Got Me Into, Courier Dover Publications, 2004.