The moving sofa problem
What's the largest area of a flat shape that can be moved round a right-angle corner in a hallway of width 1? Leo Moser posed the question in 1966, and it's still an active topic.
A short history
- 1968: John Hammersley found a "telephone handset" shape with area π/2 + 2/π ≈ 2.2074, and showed that no sofa can exceed 2√2 ≈ 2.83.
- 1992: Joseph Gerver found a shape made of 18 curve sections with area about 2.2195. It's still the best known.
- 2018: Yoav Kallus and Dan Romik lowered the upper bound with a computer-assisted method.
- 2024: Jineon Baek posted a preprint (arXiv:2411.19826) claiming to prove that Gerver's sofa is optimal. As of 2026 it's a long, not yet peer-reviewed argument, and a 2026 note questions whether one step is complete. Treat it as a claimed proof.
Why Carry Clearance doesn't use it
The sofa problem looks for the best shape. Your sofa already has a shape. Carry Clearance answers the narrower question of whether a given rigid box gets round a given corner. It uses only motions it can verify, and it assumes the worst wherever it can't.
Sources
- Wikipedia, "Moving sofa problem" (history, Hammersley and Gerver constants, Kallus–Romik bound).
- J. Baek, "Optimality of Gerver's Sofa", arXiv:2411.19826 (2024 preprint).
- Z. Rafik, note on an unverified step in Baek's preprint, Zenodo (2026).