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.

Gerver found a sofa that further increased the lower bound for the 'sofa constant' to 2.219531669. In a different direction, an easy argument by Hammersley shows that the 'sofa constant' is at most $2\sqrt{2}$ or 2.8284. The exact value of the sofa constant is still an open problem.

See also

• Mountain climbing problem
• Moser's worm problem