'Moving sofa' problem: Mathematician solves a century-old puzzle

For almost 60 years, mathematicians have struggled to answer the age-old question: will your sofa fit around the corner of the new place you are moving to?

However, Jineon Baek of Yonsei University in Korea has made a breakthrough. His 100-page proof addresses the infamous “moving sofa problem,” a mathematical conundrum that’s perplexed scholars since the mid-20th century.

Finally, there’s clarity on what kind of furniture fits through tight corners without causing chaos.

Birth of the moving sofa problem

The story begins in 1966 when Austrian-Canadian mathematician Leo Moser formalized the “moving sofa problem.”

While it might seem trivial, the puzzle taps into deep mathematical principles of combinatorics and geometry.

It asks a deceptively simple question: What is the largest two-dimensional shape that can navigate an L-shaped corner in a corridor of unit width?

Imagine trying to move a perfectly rectangular piece of furniture through a narrow hallway. A square unit fits easily, but as the object’s dimensions grow, the maneuver becomes impossible. Moser’s challenge captured the frustration of movers everywhere.

Semi-circle sofa

British mathematician John Hammersley took an early crack at the problem in 1968. He designed a sofa that combined a semicircle with a square featuring a semicircular bite.

His creation was mathematically efficient and could maneuver around an L-shaped corner with an area of 2.2074 units.

Hammersley also established an upper limit of 2.8284 units, suggesting no sofa larger than that could fit.

Refining the solution

Nearly 25 years later, Joseph Gerver of Rutgers University refined Hammersley’s design. By rounding edges and incorporating additional arcs, Gerver proposed a new sofa shape with an area just over 2.2195 units.

His solution was “locally optimal,” meaning it was the best possible shape under the constraints he defined.

Yet, without a universal formula for all potential sofa shapes, mathematicians couldn’t rule out the possibility of an even larger design.

Solving the moving sofa problem

In 2018, Yoav Kallus of the Santa Fe Institute and Dan Romik from the University of California, Davis, added a modern twist.

Using computer simulations, the team theorized that a sofa shape with an area as large as 2.37 units might be feasible. Their work reignited interest in the problem and pushed the mathematical boundaries further.

Jineon Baek’s recent work builds on this foundation. By employing an advanced mathematical technique called an injective function, Baek mapped the key properties of Gerver’s sofa design.

Acceptable area of the sofa

Baek analyzed how these properties could be extended to larger dimensions, ensuring that the sofa’s shape would remain viable for navigating tight spaces.

Through this rigorous approach, Baek conclusively demonstrated that 2.2195 units is the absolute maximum area for a sofa that can successfully pass through an L-shaped corner in a 1-unit-wide corridor.

While his findings are still awaiting peer review, they currently represent the most thorough and convincing resolution to this longstanding mathematical problem.

The power of mathematical reasoning

The proof from Yonsei University might settle the question for a single L-shaped corner, but real-world challenges often involve more complexity.

For instance, if a second corner in the opposite direction appears, mathematicians recommend Romik’s “ambidextrous sofa,” a shape designed for dual-corner navigation.

While the moving sofa problem might seem niche, it highlights the power of mathematical reasoning in solving practical problems.

Baek’s work not only contributes to geometry and combinatorics but also showcases how abstract mathematics can have unexpected real-world applications.

Similar mathematical puzzles

The moving sofa problem is part of a broader category of mathematical puzzles exploring geometry, optimization, and navigation. Several other problems remain open, including:

1. The Piano Mover’s problem: This generalization asks how to maneuver any shape through a constrained environment, like a hallway with obstacles, using minimal movements.
2. Packing problems: How can irregularly shaped objects fit into a confined space most efficiently? This problem has practical applications in manufacturing and shipping industries.
3. The “Kissing Number” problem in higher dimensions: While solved in certain dimensions, this puzzle explores how many spheres can touch another sphere in higher-dimensional spaces.
4. Optimal pathfinding in dynamic environments: How do you determine the shortest path through a space when obstacles are constantly moving or changing?
5. The moving table problem: Similar to the moving sofa, this problem investigates how to rotate and translate a table-like object through tight spaces without tipping it.

These challenges highlight the interplay of creativity, computation, and mathematical theory. Solving them could unlock further insights with practical applications in robotics, logistics, and even game design.

The study is published in arXiv.

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