# Moving sofa problem

 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]