What is the largest sofa that you can squeeze around the corner of a hallway? A horseshoe-shaped piece of furniture known as Gerver’s sofa has officially taken the crown, solving a mathematical question first put forward almost 60 years ago.
The difficulties of getting large furniture into your home will be familiar to many people, but mathematicians have taken a particular interest in the moving sofa problem ever since it was first posed by Leo Moser in 1966. It supposes that you are trying to navigate a two-dimensional sofa (so ignoring its height) through a hallway with a 90-degree turn, and asks if it is possible to calculate the largest area shape for which this is possible.
Mathematicians soon determined that a shape resembling an old telephone handset seemed to allow the largest areas, and in 1992 Joseph Gerver at Rutgers University in New Jersey discovered a specific one of these shapes, made of 18 curved sections, that took the top spot. In the years after, researchers struggled to find any larger sofas and began to suspect that Gerver’s might be the largest possible solution, but no one could prove that there wasn’t an even bigger sofa lurking around the corner.
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Now, in a proof spanning more than 100 pages, at Yonsei University in South Korea has shown that Gerver’s solution is indeed the largest possible, officially solving the 58-year old problem. Baek, who started working on the puzzle seven years ago, says he came up with the outlines of the proof after around two years of working on the problem, but that it took a further five years to iron out all the details. “I dedicated a lot of time to this, without any publication so far,” says Baek. “The fact that now I can say to the world that I committed something valuable to this problem is validating.”
Baek’s proof first focuses on a small selection of possible sofas and uses these to prove certain properties that the largest sofa must have. These include a relatively smooth exterior, a quantity called balance that is related to symmetry and of course the crucial ability to rotate a full 90 degrees around the corner. Using these properties, Baek then came up with a new mathematical quantity closely related to area, called Q, that made analysing the problem easier. This meant it could be turned from an open-ended scenario, called a non-convex problem, to one with definite, or convex, solutions, like a ball rolling into the bottom of a bowl, rather than down either side of a hill. Baek then found that the largest value for Q was the exact solution that Gerver had proposed, proving that no sofa could be larger.
“I am of course very happy about all of this,” says Gerver. “I am 75 years old, and Baek can’t be more than 30. He has a lot more energy, stamina and surviving brain cells than I do, and I am glad that he picked up the baton. I am also very happy that I lived long enough to see him finish what I started.”
at the University of California, Davis, who has worked on the moving sofa problem and published his own solutions, says it is a “wonderful development”. “I know I could never have done this,” says Romik. “I don’t have a feeling of regret, or like, how could I miss this, because it’s clear it’s just not the sort of thinking that I think I would have been able to. [Baek] was just coming at it from a completely different direction.”
The proof has yet to be fully checked over by other mathematicians, so there is the possibility that it contains a mistake, but Baek is hopeful he is correct. “I can’t say that I’m confident 100 per cent, because we are humans, we make errors, but still, I did my best to be as confident as I can,” he says.
arXiv
