How to move a sofa around a corridor? This is a problem that has puzzled mathematicians for more than 60 years.

It is probably too early to assert that the "movable sofa problem" has been completely solved.

Author | Denovo

Producer | China Science Expo

You may have had this experience: when moving house, you want to move furniture in a narrow space, but it gets stuck when turning a corner and cannot be turned around. Mathematicians call this problem the "moving sofa problem."

On December 2, 2024, Korean mathematician Jineon Baek announced on social media that he had solved the problem, which immediately sparked widespread discussion in domestic and foreign media and mathematics circles. You may be curious: How difficult is a seemingly small problem closely related to daily life?

The problem of moving the sofa involves the math of how the shape fits into the corner. Image credit: University of California, Davis

What is the "moving sofa problem"?

In fact, the "moving sofa problem" is a problem that has undergone many discussions and explorations. As early as the 1960s, some mathematicians had begun to explore geometric optimization problems related to this problem.

In 1966, Austrian-Canadian mathematician Leo Moser first proposed a clear mathematical definition and description of the "moving sofa problem" in a formal mathematical journal: In an L-shaped corridor with a width of 1, what is the maximum area of ​​a "sofa" that can pass a right-angle turn without colliding? This problem has since attracted widespread attention in the mathematical community and has become one of the classic geometric optimization problems.

Demonstration of the mobile sofa problem | Image source: Reference [1]

In 1968, British mathematician John Michael Hammersley proposed a solution based on the simplest case. He designed the "sofa" to be shaped like a telephone receiver, consisting of two quarter circles and a rectangular block in the middle, with a semicircle dug out of the rectangular block in the middle, and the maximum area of ​​the "sofa" was 2/π+π/2≈2.2074.

"Sofa" designed by Hammersley丨Image source: Wikipedia

In 1992, American mathematician Joseph Gerver improved on the "sofa" designed by Hamersley and proposed a "sofa" surrounded by 18 smooth curves. The calculated maximum area of ​​the sofa was approximately 2.2195, further improving the lower limit of the solution to this problem.

"Sofa" designed by Jeff | Image source: Wikipedia

In 2014, amateur mathematician Philip Gibbs calculated an optimal sofa shape through computer calculation, which is almost the same as the "Gerver Sofa" designed by Joseph Gerver, and the calculated area is the same under eight significant figures. This discovery shows that the "sofa" designed by Gerver is likely to be the optimal solution to the mobile sofa problem, but this has not yet been formally proved mathematically.

However, scientists have at least determined an upper limit to the area of ​​the "sofa", that is, the maximum area that this area cannot exceed. Hammersley pointed out that the upper limit of the sofa constant is at most

In 2018, Yoav Kallus and Dan Romik successfully narrowed the upper limit of "sofa" to 2.37 by rotating the corridor (not the sofa) at several different angles so that the intersection of the rotated corridors formed the largest possible connected area, and used computer search.

In other words, the optimal solution to the "moving sofa problem" is between 2.2195 and 2.37.

What is so difficult about the “moving sofa problem”?

Seeing this, you may ask: The "moving sofa problem" seems so intuitive and simple, why has it puzzled mathematicians for more than half a century?

Although Joseph Jeff has proposed an approximate optimal solution, it is still very difficult to prove that it is the true optimal solution, because this requires excluding all possible better shapes. In a plane, the shape of the "sofa" can vary greatly, and the optimal solution is likely to be an asymmetrical, complex and irregular polygon.

To explore all possible shapes and evaluate their area and mobility involves a very large amount of calculation, which makes it impossible to exhaust all possibilities. In addition, shapes that lack symmetry and regularity and can be flexibly rotated and moved are inherently very complex in geometry, so mathematicians have difficulty finding a universal formula to solve this problem.

After entering the new century, with the rapid development of computer technology, mathematicians began to widely use computer-aided design and motion path simulation to explore the possible shapes of "sofas". However, even with the use of computer-aided numerical methods and optimization algorithms, existing algorithms often face the problems of long calculation time and excessive consumption of computing resources when excluding all potential better shapes and exploring and verifying the feasibility and area of ​​various complex shapes, which largely limits the progress of further research.

Machine learning, which has been very popular in recent years, is also greatly limited in solving the "moving sofa problem". Machine learning models usually require a large amount of data for training, and the solution to the "moving sofa problem" mainly relies on limited data sets generated by theoretical derivation and optimization algorithms, which is difficult to meet the training needs of large-scale models.

In addition, mathematical optimization problems often require highly interpretable and precise solutions, and the "black box" nature of machine learning models may only give answers but not the solution process , which makes it difficult to directly apply them to solving such problems.

The "sofa" not only needs to make right-angle turns, but also must avoid colliding with the walls of the corridor. These multiple constraints make the optimization process extremely complicated. The "moving sofa problem" involves knowledge from multiple disciplines such as geometry, optimization theory, and computational geometry, so interdisciplinary research is needed to find a solution.

Has the “moving sofa problem” really been solved?

Let's turn our attention to the 119-page paper by Jineon Baek, which has attracted much attention recently. He claims that he has proved that the "sofa" designed by Joseph Jeff is the optimal solution.

Bai Zhenyun first proposed the shape constraints of the optimal "sofa": ① The shape of the sofa can be defined by the intersection of the rotating corridors; ② The side length of the sofa must meet specific balance conditions; ③ It must be able to rotate 90 degrees to complete the movement.

Then, he proved that the trajectories of the key points of the "sofa" during its movement do not intersect with themselves (that is, there is no repetition or overlap), forming a simple closed curve on the plane, thus ensuring the rigor of the area calculation.

Illustration of the definition of Q(S) | Image source: Reference [1]

He then constructed a quadratic function Q(S) as the upper bound of the area of ​​the "sofa", and used the Mamikon theorem and the Brunn-Minkowski theory to prove that Q(S) is a concave function, which means that its local maximum is also the global maximum .

Finally, he verified that the "sofa" designed by Jeff fully met these conditions, and the value of Q(S) reached its maximum here, confirming that its area of ​​2.2195 is the theoretical maximum value.

However, this paper has not yet been published in an authoritative journal or been widely peer-reviewed, and the academic community is still waiting to see whether its proof is correct and rigorous. It is probably too early to assert that the "moving sofa problem" has been completely solved.

Conclusion

So, what is the point of solving this problem completely?

In addition to providing new ideas for other geometric optimization problems, the tools and construction methods developed in the process of solving the problem can also be abstracted as an extreme optimization model of space utilization, which has important reference value for practical fields such as architectural design, furniture manufacturing, and logistics management. For example, the path planning of express robots when carrying objects in narrow spaces and the spatial path planning of robotic arms on production lines when carrying irregular objects can be inspired by the research on this problem.

Let us wait for mathematicians to carefully verify Bai Zhenyun's paper, and look forward to the successful resolution of this problem that has plagued the scientific community for more than 60 years.

References

[1] Baek J. Optimality of Gerver's Sofa[J]. arXiv preprint arXiv:2411.19826, 2024.

[2] Gibbs P. A computational study of sofas and cars[J]. Computer Science, 2014, 2: 1-5.

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