After a twenty-five-year delay in helping Ross get a new sofa in his apartment on “Friends,” a mathematician has finally solved the vexing “sofa problem.”
The math problem refers to the largest size sofa that can fit into a corner of a given width – exactly the problem the characters faced in an episode of “Friends” that aired in 1999. Ross’ pleas to “Pivot!” could have been avoided, it seems, if they had only considered Gerver’s sofa with 18 curve segments and a maximum area of 2.2195 units. (Okay, so maybe that wouldn’t have been so helpful.)
The solution to the sofa problem is a math first. The problem was posed by Austrian-Canadian mathematician Leo Moser in 1966. Moser asked for the largest possible area of a single figure in a plane that could pivot around the right-angle corner of a hallway with a width of one unit. While this may sound simple, the math is quite complex, as the problem involves both area maximization and the movement of a shape.
Now, Jinheon Baek, a postdoctoral researcher in mathematics at Yonsei University in South Korea, has discovered the answer. Baek posted his solution on the preprint website ArXiv on December 2.
In more than 100 pages of mathematical proofs, Baek found that for a hallway with a width of 1 unit, the maximum area the hypothetical couch could have is 2.2195 units — narrowing the answer with precision from the previously known range of between 2.2195 and 2.37 units.
This proof has not yet been published in a peer-reviewed journal and will need to be worked on by other mathematicians to determine whether it is truly optimal.
The “Gerver” of Gerver’s sofa is mathematician Joseph Gerver, an emeritus professor at Rutgers University, who set a lower limit of 2.2195 in 1992. But there was debate over whether the sofa could be larger, with one team using a computer-aided proof in 2018 to suggest that 2.37 was actually the upper limit.
Gerver’s sofa is a wide U-shaped sofa with a curved “seat” that can shrink around the corner without sagging. The question was whether this painstakingly designed sofa – made by joining 18 separate curves together – was actually the largest, most optimal shape that could bend. Beck worked through the geometry of the shape and its motion and found that Gerver’s solution was indeed correct.