Differential equations and exact solutions in the moving sofa problem

TL;DR

This paper derives differential equations related to the moving sofa problem and proposes a new, explicitly defined shape that can navigate a right-angled turn in a hallway, advancing the understanding of optimal shapes in this classic puzzle.

Contribution

The authors extend Gerver's techniques by deriving differential equations and introduce a closed-form shape solution for the ambidextrous moving sofa problem.

Findings

Derived six differential equations from area-maximization.

Proposed a new explicit shape with algebraic boundary.

Calculated the shape's area using solutions to specific cubic equations.

Abstract

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.