Moving Sofa Problem
Find the largest-area shape that can move around a right-angled hallway corner.
The moving sofa problem asks for the largest possible shape that can be maneuvered around a 90-degree turn in a unit-width hallway. Gerver's shape has long been the best-known candidate; recent claimed solutions make this an active-watch entry.
Formula
The supremum of areas among all rigid shapes that can be moved continuously around a 90° corner in a hallway of width 1.
Gerver's 1992 construction achieves this area, long conjectured optimal. Baek (2024) claims to have proved μ = |G|.
The best known upper bound, established by Hammersley (1968) via a simple geometric argument.
Summary
The moving sofa problem asks for the largest possible shape that can be maneuvered around a 90-degree turn in a unit-width hallway. Gerver's shape has long been the best-known candidate; recent claimed solutions make this an active-watch entry.
