In 1992, Joseph Gerver from Rutgers University introduced an innovative curved shape with an area of approximately 2.2195, leading mathematicians to believe it might resolve Moser’s longstanding question. However, they lacked the necessary proof.

Now, a promising postdoctoral researcher, Jineon Baek of Yonsei University in Seoul, has achieved this breakthrough. In a comprehensive 119-page study, he demonstrated that Gerver’s sofa is, indeed, the largest shape that can navigate through a hallway.

This publication is significant not only for solving a problem that has persisted for six decades but also because researchers had anticipated that any conclusive proof would rely on computer assistance. Remarkably, Baek’s proof was accomplished without computational aid, inspiring mathematicians to consider how his methodologies might advance other complex optimization challenges.

Even more fascinating is the nature of Gerver’s sofa. Unlike more familiar geometric shapes, its area cannot be expressed with standard constants, like π or square roots. Yet, for the moving sofa problem—a seemingly straightforward query—it serves as the optimal solution. This finding highlights that even simple optimization tasks can produce surprisingly intricate results.

Sofas and Telephones

The initial significant advancement on the moving sofa question occurred in 1968, merely two years after Moser posed it. John Hammersley linked two quarter circles with a rectangle, subsequently removing a semicircle, creating a form reminiscent of an old telephone. This configuration had an area of π/2 + 2/π, roughly 2.2074.

Additionally, Hammersley established that any solution to the problem could not exceed an area of $latex 2\sqrt{2}$, which approximates to 2.8284.

A few years later, Gerver, then a graduate student at the University of California, Berkeley, was introduced to the question by a fellow graduate student. “He never mentioned it was unsolved. After pondering it for a few days, I returned and confessed, ‘I give up. What’s the solution?’ To which he said, ‘Just keep thinking about it. You’ll figure it out.’”

For the next two decades, Gerver intermittently considered the problem. It was only in 1990, when discussing it with the eminent mathematician John Conway, that he discovered it remained unresolved. This newfound motivation led him to dedicate more effort to the problem, resulting in his eventual discovery of a potential solution.

Gerver’s sofa bore similarity to Hammersley’s telephone but was significantly more complex, comprising 18 distinct components. Interestingly, Ben Logan, an engineer at Bell Labs, independently stumbled upon the same shape but never got around to publishing his findings. Among the segments, some were simple lines and arcs, while others proved to be more intricate and challenging to articulate.

Despite its complexity, Gerver was confident in this shape’s optimality, as it displayed numerous characteristics expected of the ideal sofa design. He successfully demonstrated that altering its contours slightly wouldn’t increase the area, reinforcing his belief in its optimal status.