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A moving-day question becomes geometry

A couch stuck at a hallway corner is an ordinary frustration. Strip away the furniture, gravity and extra dimensions, however, and it becomes a problem that resisted mathematicians for almost six decades: what is the largest rigid two-dimensional shape that can pass around a right-angle bend in a corridor one unit wide?

Canadian mathematician Leo Moser formally posed the moving sofa problem in 1966. The “sofa” may have any shape, whether or not anyone would want it in a living room. It cannot bend or be lifted; it can only slide and rotate through the L-shaped passage. The unit width merely sets the scale, so the unknown quantity is the greatest possible area of a shape that can complete the turn.

A one-by-one square establishes an easy starting point with area 1. Making that square into a longer rectangle does not help, because the extra length leaves it unable to pivot at the corner. Curves do better. A semicircle with diameter 2 can swing around the bend, using the space in the first leg of the corridor while its rounded edge clears the inside corner. Its area is π/2, about 1.571.

That improvement reveals the problem’s central difficulty. It is not enough to optimize the sofa’s outline; the path of the sofa must be optimized at the same time. A shape may translate, rotate or do both simultaneously, and its best trajectory depends on its geometry. The object and its motion form one coupled puzzle.

Better sofas, smaller gains

In 1968 British mathematician John Hammersley found a much larger candidate. He stretched the semicircle and removed material where the inside corner would otherwise block it. The resulting shape resembles an old telephone handset and turns by sliding and rotating together. Its area is π/2 + 2/π, approximately 2.2074.

Progress then stopped for 24 years. In 1992 Rutgers University mathematician Joseph L. Gerver introduced a subtler construction made from 18 distinct curves. It looks broadly like Hammersley’s sofa, but details such as the beveled edges around its central cutout let it use the corridor more efficiently. Gerver’s shape has area about 2.2195, just 0.0121 larger than Hammersley’s comparatively simple design.

Gerver suspected that his sofa was not merely the largest known example but the largest possible one. That distinction is crucial. Constructing a shape with area 2.2195 proves that the optimum is at least that large; establishing the exact answer also requires ruling out every larger imaginable shape and every path it might follow. For another 32 years, nobody could do that.

Researchers used intensive computer simulations to tighten the bounds on what might fit. Jineon Baek, then a postdoctoral researcher at Yonsei University in Seoul, was prominent in that work and developed several incremental results while writing his doctoral thesis on the problem. Yet the proof he posted online in November 2024 did not ultimately rely on computer calculations. Its 119 pages assembled those ideas into an argument that no sofa larger than Gerver’s can negotiate the turn.

A claimed solution, with scrutiny still ahead

Baek’s result would settle the moving sofa problem after 58 years and confirm that Gerver had found the optimal shape in 1992. The article is careful about the status of that conclusion, though. At the time of publication, the proof had not completed thorough peer review. Early reactions from mathematicians familiar with both Baek and the problem were optimistic, but a long and intricate proof must survive detailed checking before the result can be treated as final.

The problem’s appeal lies partly in the contrast between its accessibility and its difficulty. Anyone who can picture a couch rounding a corner can understand the question, yet the answer depends on delicately coordinating a complex boundary with a continuous sequence of motions. The eventual gain over Hammersley’s sofa is tiny, but proving that no hidden improvement remains demanded decades of work.

Its value also extends beyond recreational geometry. Each successful attack develops ways to reason about shapes constrained by motion, and those techniques may transfer to other optimization problems. The moving sofa is therefore less a practical guide for furniture delivery than a vivid example of what mathematical proof contributes: not just a design that seems best, but an argument meant to exclude every better possibility.