What's the biggest sofa you can carry around a hallway corner?
A hallway one unit wide turns a sharp right angle. What is the largest-area two-dimensional shape — the 'sofa' — that can be slid flat around the corner without ever lifting it? The shape can be any outline at all, not just a rectangle. Mathematicians have chased this simple-sounding question since 1966.
Reveal the answer
The exact maximal shape is still not fully settled. The best confirmed lower bound is 'Gerver's sofa,' an 18-curve shape of area ≈2.2195 found by Joseph Gerver in 1992; in 2024 Jineon Baek posted a proof, still under review, that Gerver's sofa is in fact optimal. Leo Moser first posed the problem formally in 1966 for the Canadian Mathematical Congress's list of unsolved problems, and the hunt for the exact 'sofa constant' has remained a running joke and a serious open question in geometry ever since.
— Leo Moser, Moving furniture through a hallway (Problem 66-11) — SIAM Review, 1966