# 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 ${\displaystyle \scriptstyle 2/\pi \,}$ 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 ${\displaystyle \scriptstyle 2{\sqrt {2}}\,\approx \,2.8284}$.[1][6]