It seems that the old mathematical problem of the sofa has finally been solved

You already know the problem: You move house and have to fit the sofa through a fairly narrow space. You try to tilt it, turn it, flip it, but nothing. So you equip yourself with paper, pen and a plan of the house and try to evaluate if there is any possibility, at least theoretical, that the sofa will fit. If you can’t find a solution, it’s not your fault: the “couch problem” was formulated in the 1960s and until a few days ago no mathematician had managed to find a solution, to the point that it was included in the list of “great unsolved problems of mathematics.” But things have just changed: Jineon Baekscientist at Yonsei University in Seoul (South Korea), published in ArXiv a review of more than 100 pages in which he claims to have found a solution to the old problem. Or more precisely, Baek has shown that the solution proposed by one of his colleagues, Joseph Gerver, is indeed the optimal one.

Let’s move that sofa

The formalization of the problem, which dates back to 1966 and is due to the Austro-Canadian mathematician Leo Moseris the following: “What is the largest rigid surface that can move through a hallway at right angles, that is, L-shaped, with both arms of unit width?” Let’s try a simplified version: suppose we have a hallway one unit wide, which forms a right angle. If you want to pass a chair one unit wide, that is, as wide as the hallway, the solution is trivial: you push it to the corner and then pull it in a perpendicular direction. If, on the other hand, you have a two-by-one rectangular piece of furniture, there is little you can do: unless you disassemble it, it will be impossible to move it around the corner. An intermediate and more complex case, the one that interests Moser, is that of a C-shaped sofa: well, in this case, with a little maneuver, it can be done. In 1968, two years after the problem was formulated, the mathematician John Hammersley He was able to demonstrate, in fact, that an object of this type could pass through the right angle, as long as it had an area no greater than 2.2074 units.

Inconvenient, but functional

but the story it doesn’t end here. Twenty-five years later, another mathematician, Joseph Gerversuggested a small “tweak” to Hammersley’s sofa, softening some of its edges and slightly modifying its shape: in this way, Gerver explains, it is even possible to fit a sofa with an area of ​​2.2195 units. Gerver’s It is a so-called locally optimal solution.that is, the best within the conditions defined by that specific shape of the sofa. But at this point something is still missing: a “general” answer to the problem, which also takes into account any other variations in the shape of the sofa. That is precisely what Baek has worked on, addressed the problem by defining a strict “topology” of the sofastudying all the key properties of the shape imagined by Gerver and trying to apply small modifications and extensions to it: in this way, he has shown that, indeed, the solution found by his colleague is the optimal one, and any change in shape and size makes it impossible move the sofa The work still has to go through the peer review process: if Baek’s proof is validated, the problem can probably be considered definitively solved. But the move won’t be any less stressful.

Article originally published in WIRED Italy. Adapted by Mauricio Serfatty Godoy.