# 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?

In mathematics, 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. 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.

## 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).

### Lower bounds

An obvious lower bound is $A\geq \pi /2\approx 1.57$ . 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 $A\geq \pi /2+2/\pi \approx 2.2074$ based on a shape resembling a telephone handset, 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 $2/\pi$ has been removed.

Joseph Gerver found a sofa described by 18 curve sections each taking a smooth analytic form. This further increased the lower bound for the sofa constant to approximately 2.2195.

A computation by Philip Gibbs produced a shape indistinguishable from that of Gerver's sofa giving a value for the area equal to eight significant figures. This is evidence that Gerver's sofa is indeed the best possible but it remains unproven.

### Upper bounds

Hammersley also found an upper bound on the sofa constant, showing that it is at most $2{\sqrt {2}}\approx 2.8284$ .

Yoav Kallus and Dan Romik proved a new upper bound in June 2017, capping the sofa constant at $2.37$ .

## Ambidextrous sofa

A variant of the sofa problem asks the shape of largest area that can go round both left and right 90 degree corners in a corridor of unit width. A lower bound of area approximately 1.64495521 has been described by Dan Romik. His sofa is also described by 18 curve sections.