Department of Mathematics

The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width, and is conjectured to have as its solution a complicated shape derived by Gerver in 1992. We extend Gerver's techniques by deriving a family of six differential equations arising from the area-maximization property. We then use this result to derive a new shape that we propose as a possible solution to the "ambidextrous moving sofa problem," a variant of the problem previously studied by Conway and others in which the shape is required to be able to negotiate a right-angle turn both to the left and to the right. Unlike Gerver's construction, our new shape can be expressed in closed form, and its boundary is a piecewise algebraic curve. Its area is equal to $X+\arctan Y$, where $X$ and $Y$ are solutions to the cubic equations $x^2(x+3)=8$ and $x(4x^2+3)=1$, respectively.