February 4, 2025
4 Min Read
Mathematicians solve the infamous “movable sofa”
What is the biggest sofa that may turn the corner? After 58 years we finally know
For those that fought with a bulky sofa around a decent corner and lamented: “Does it fit at all?” Mathematicians have heard your requests. The “moving problem with the sofa” of geometry asks for the largest shape, which may turn the right angle in a narrow corridor without getting stuck. The problem was unsolved for nearly 60 years to November, when Jineon Baek, Postdoc at the University of Yonsei in Seoul, published Online paper claiming to solve it. The proof of Baka has not yet been subjected to an intensive review, but the initial transitions from mathematicians who know Baka and the moving problem with the sofa seem optimistic. Only time will tell why Baek 119 pages wrote what Ross Geller from sitcom Friends He said in a single word.
The solution is unlikely to make it easier to on the day of moving, but because the Math frontier becomes more skilled, mathematicians have a special passion for unsolved problems that everybody can understand. In fact, the popular mathematical forum Mathoverflow keeps the list “Seniorly speaking, especially known, long -open problems that everybody can understand“, And the problem with the sofa now ranks second on the list. Despite this, every proof extends our understanding, and the techniques used to solve the moving sofa problem are probably suitable for other geometric puzzles down the road.
Principles of the problem that Canadian mathematician Leo Moser first he formally posed In 1966 it features a stiff shape – so the pillows will not be after pressing – turning the right angle in the corridor. The sofa could have any geometric shape; It doesn’t should resemble an actual couch. Both shape and corridor are two -dimensional. Imagine that the sofa weighs an excessive amount of to lift, and you possibly can only move it.
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A fast trip through the history of the problem reveals the extensive effort that mathematicians poured into it – weren’t potatoes on the couch. In the face of an empty corridor, what’s the largest shape that you may squeeze through it? If each leg of the corridor measures one unit across (a particular unit doesn’t matter), we are able to easily move the square one after the other through the passage. Extending the square with a view to create a rectangle fails immediately, because when it hits a breakdown in the corridor, there isn’t a place to rotate.


However, mathematicians realized that they may grow by introducing curved shapes. Consider a semi -circle with a diameter (easy base) 2. When it gets right into a corner, a big part that also hangs in the first stage of the corridor, but the rounded edge leaves enough space to wash the corner.

Remember that the goal is to seek out the largest “sofa” that slides around the corner. By buying our geometry patterns at highschool, we are able to calculate the semi -circle area as π/2 or about 1.571. The hemisphere gives a big improvement in relation to the square, which was just one 1. Unfortunately, each would look strange in the lounge.
Solving an issue with a movable sofa requires not only optimization of shape size, but additionally path This shape is traveled. Configuration allows two kinds of movement: sliding and rotation. The square sofa only moved, while the semicircle moved after which turned around the bend, after which moved on the other side. But objects can move and rotate at the same time. Mathematician Dan Romik from the University of California, Davis, ma excellent that the solution to the problem should concurrently optimize each kinds of movement.
British mathematician John Hammersley discovered in 1968 that stretching a semicircle Power You will buy a bigger sofa when you sculpt a bit to cope with this unbearable corner. In addition, Hammersley’s sofa uses a hybrid shifted movement plus rotary movement. The created sofa looks like a landline phone:

Amanda Montañez; Source: “On Moving a Sofa Around a Horn” Joseph L. Gerver, w Dedicated geometry; Volume. 42, No. 3; June 1992 (reference)
Optimization of varied variables gives a settee with the π/2 + 2/π area or around 2.2074. This is a large semicircle improvement, just like the transition from a love seating to the section. But progress got stuck there for twenty-four years. The last significant improvement can be the last. In 1992 Joseph Gerver presented The place of mathematical joinery, which we all know that it’s the biggest possible sofa.

Amanda Montañez; Source: “On Moving a Sofa Around a Horn” Joseph L. Gerver, w Dedicated geometryVolume. 42, No. 3; June 1992 (reference)
You have been forgiven that you are feeling déjà vu now. The Gervera sofa looks an identical to Hammersley, but it surely is a far more complicated design. Gerver sewed 18 clear curves, creating his shape. During closer control, you possibly can detect some differences, especially the chapted edges at the base of the rounded cut -out.

Amanda Montañez; Source: “On Moving a Sofa Around a Horn” Joseph L. Gerver, w Dedicated geometryVolume. 42, No. 3; June 1992 (reference)
The area of Gerver’s triumph is measured at 2.2195 units. Surprisingly, the relatively easy Hammersley sofa fell only about 0.012 optimal. Although only SKOSH greater than his predecessor, Gerver suspected that his discovery reached the maximum possible size. He couldn’t prove it, nevertheless. And nobody else might have been for the next 32 years.
Baek finished his doctorate in 2024 and wrote his work about the moving problem with the sofa, contributing to several incremental insights. In the same yr he sewed all his fresh ideas into impressive opus This seems to be no sofa larger than Gerver can squeeze through the corridor. Covering a protracted -term open problem is a dream for each mathematics, not to say that so early of their profession. If the work of Baka withstands the control, it’ll probably be in the state of the professor. Unless it turns into furniture production.