Moving Sofa Problem Finally Solved?
A mathematician from South Korea’s Seoul National University claims to have solved the 58-year-old moving sofa problem by proving that Gerver’s sofa shape from 1992 is indeed optimal, with a detailed 119-page proof.
The moving sofa problem, first posed by mathematician Leo Moser in 1966, asks a deceptively simple question: what is the largest possible area of a rigid shape that can navigate a right-angled corridor of width 1? Despite its straightforward formulation, this geometric optimization problem has challenged mathematicians for nearly six decades.
The mathematical community was intrigued when Jineon Baek from Seoul National University uploaded a 119-page paper titled “Optimality of Gerver’s Sofa” in November 2024. In this comprehensive work, Baek presents a proof that the shape discovered by Joseph Gerver in 1992, with an area of approximately 2.2195 units, is indeed the optimal solution.
The journey to this solution has been marked by several milestone achievements. The simplest possible solution - a unit square with area 1 - was quickly surpassed. In 1968, John Hammersley constructed a shape with a larger area. However, the most significant breakthrough came in 1992 when Gerver discovered his sofa shape, bounded by 18 curved segments.
Baek’s proof employs sophisticated mathematical techniques to establish the optimality of Gerver’s sofa. The paper introduces novel concepts such as monotone sofas and develops a framework using convex geometry and variational methods. The proof combines elements of geometric measure theory with careful analysis of the sofa’s boundary curves.
This breakthrough has broader implications for geometric optimization problems. The techniques developed in proving the moving sofa problem’s solution may provide new tools for solving other long-standing geometric puzzles. The resolution of this problem demonstrates how seemingly simple questions in mathematics can require deep and complex analysis to solve completely.
For the wider mathematical community, this solution represents more than just answering a geometric puzzle. It exemplifies how modern mathematics can definitively resolve questions that bridge pure geometry and practical optimization. The moving sofa problem has also spawned variations, including Dan Romik’s ambidextrous sofa problem, which considers shapes that can turn both left and right.
The length and complexity of Baek’s proof underscores why this problem remained unsolved for so long. It required developing new mathematical machinery while building upon decades of previous work by other mathematicians. As the mathematical community reviews this proof, it may open new avenues for solving similar geometric optimization problems.