The Largest Sofa You Can Move Around a Corner

In 1992, Joseph Gerver of Rutgers University proposed a particularly clever curved shape with an area of approximately 2.2195. Mathematicians suspected that it answered Moser’s question. But they couldn’t prove it.

Now a young postdoctoral researcher has. In a 119-page paper, Jineon Baek of Yonsei University in Seoul showed that Gerver’s sofa is the largest shape that can successfully pass through the hallway.

The paper isn’t just noteworthy for resolving a 60-year-old problem. It has also garnered attention because mathematicians had expected any eventual proof of the conjecture to require a computer. Baek’s proof did not. Mathematicians now hope that the techniques he used might help them make progress on other kinds of optimization problems.

Perhaps even more intriguing, Gerver’s sofa, unlike more familiar shapes, is defined in such a way that its area can’t be expressed in terms of known quantities (such as π or square roots). But for the moving sofa problem — a very simple question — it’s the optimal solution. The result illustrates that even the most straightforward optimization problems can have counterintuitively complicated answers.

Sofas and Telephones

The first major progress on the moving sofa problem came in 1968, just two years after Moser posed it. John Hammersley connected two quarter circles with a rectangle, then cut a semicircle out of it to form a shape that resembled an old telephone. Its area was π/2 + 2/π, or approximately 2.2074.

Hammersley also showed that any solution to the problem could have an area of at most $latex  2\sqrt{2}$, or about 2.8284.

A couple of years later, Gerver, then a graduate student at the University of California, Berkeley, learned about the question. “Another grad student told me this problem and challenged me to find the answer,” he said. “He never said anything about it being unsolved. So I thought about it for a few days. Finally, I came back to him and said ‘OK, I give up. What’s the solution?’ And he refused to tell me! He said, ‘Just keep thinking about it. You’ll get it.’”

Gerver thought about it sporadically over the next 20 years. But it wasn’t until 1990, when he mentioned it to the renowned mathematician John Conway, that he found out it had never been solved. The realization motivated him to spend more time with the problem, and he soon came up with a potential solution.

Gerver’s sofa looked a lot like Hammersley’s telephone, but it was far more complicated to describe, consisting of 18 different pieces. (As it turned out, Ben Logan, an engineer at Bell Labs, independently uncovered the same shape but never published his work.) Some of the pieces were simple line segments and arcs. Others were more exotic, and tougher to describe.

Still, Gerver suspected that this complicated shape was optimal: It possessed many features that mathematicians expected the optimal sofa to have, and he was able to prove that making small perturbations to its contours wouldn’t yield a suitable shape with a bigger area.