Designing a sofa to navigate a narrow hallway's corner has been a challenging puzzle for mathematicians over the years.

The problem, known as the "sofa-moving problem," centers on finding the largest sofa that can smoothly traverse the corner of an L-shaped hallway. While it might sound straightforward, this problem has baffled mathematicians for many years.

Typically, when moving furniture, people are advised to lift, squeeze, or tilt the sofa. But what if the sofa cannot be maneuvered in such ways? Surprisingly, this seemingly simple issue, known as the "sofa-moving problem," has perplexed mathematicians for over 50 years.

Mathematicians not only need to locate the largest sofa that fits but also provide rigorous mathematical proof that it is indeed the largest possible. Without this proof, there's always the chance that someone might discover a better solution in the future.

Dan Romik, a professor in the Department of Mathematics at the University of California, Davis, remarked, "This is an exceedingly challenging problem. It's so simple that you can explain it to a child in five minutes, but finding the optimal solution and proving it has stumped everyone."

In mathematical terms, the largest sofa that can fit into a corner is referred to as the "sofa constant," where one unit corresponds to the hallway's width.

Recently, Romik introduced a solution to a variant of the sofa problem, the "two-sided dexterity sofa problem," using 3D printing technology. This problem requires the sofa to pass through successive 90-degree turns in both clockwise and counterclockwise directions.

Romik specializes in combinatorial mathematics and enjoys tackling problems related to shape and structure.

His interest in the sofa-moving challenge originated from his hobby – the desire to 3D print a sofa and a hallway. He finds it exciting to apply 3D printing technology to mathematical research as manual manipulation enhances one's intuition.

The "Gerver Sofa," resembling an old-fashioned telephone receiver, is currently the optimal solution for moving a corner sofa in a single hallway. Romik became interested in 3D printing the Gerver Sofa, sparking his fascination with the mathematics behind Gerver's solution. He spent the next seven months working on creating new solutions. Romik then decided to tackle the problem of two corners.

To design a sofa capable of passing through both corners, Romik employed software to create a shape that combined two symmetrical curves with a slender center.

While not definitively proven, Romik's discovery provides valuable mathematical insights into this intricate problem.

Although the sofa-moving problem might appear abstract, solving it involves the development of innovative mathematical techniques that can serve as the foundation for more complex ideas in the future.

The journey of mathematical exploration continues, as mathematicians seek further revelations in this intriguing field.