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For those who have wrestled a bulky couch around a tight corner and lamented, “Will this even fit?” mathematicians have heard your pleas. Geometry’s “moving sofa problem” asks for the largest shape that can turn a right angle in a narrow corridor without getting stuck. The problem sat unsolved for nearly 60 years until November, when Jineon Baek, a postdoc at Yonsei University in Seoul, posted a paper online claiming to resolve it. Baek’s proof has yet to undergo thorough peer review, but initial passes from mathematicians who know Baek and the moving sofa problem seem optimistic. Only time will tell why it took Baek 119 pages to write what Ross Geller of the sitcom Friends said in one word.

The solution is unlikely to help you on moving day, but as frontier math grows more abstruse, mathematicians hold a special fondness for unsolved problems that anybody can understand. In fact, the popular math forum MathOverflow maintains a list of “Not especially famous, long-open problems which anyone can understand,” and the moving sofa problem currently ranks second on the list. Still, every proof expands our understanding, and the techniques used to resolve the moving sofa problem will likely lend themselves to other geometric puzzles down the road.

The rules of the problem, which Canadian mathematician Leo Moser first formally posed in 1966, involve a rigid shape—so the cushions don’t yield when pressed—turning a right angle in a hallway. The sofa can be any geometric shape; it doesn’t have to resemble a real couch. Both the shape and the hallway are two-dimensional. Imagine the sofa weighs too much to lift, and you can only slide it.

A quick tour through the problem’s history reveals the extensive effort that mathematicians have poured into it—they were no couch potatoes. Faced with an empty hallway, what is the largest shape you could squeeze through it? If each leg of the corridor measures one unit across (the specific unit doesn’t matter), then we can easily scoot a one-by-one square through the passage. Elongating the square to form a rectangle fails instantly, because once it hits the kink in the hallway, it has no room to turn.

Yet mathematicians realized they can go bigger by introducing curved shapes. Consider a semicircle with a diameter (the straight base) of 2. When it hits the turn, much of it still overhangs in the first leg of the hallway, but the rounded edge leaves just enough room to clear the corner.

Remember the goal is to find the largest “couch” that slides around the corner. Dusting off our high school geometry formulas, we can calculate the area of the semicircle as π/2, or approximately 1.571. The semicircle gives a significant improvement over the square, which had an area of only 1. Unfortunately, both would look strange in a living room.

Solving the moving sofa problem requires that you not only optimize the size of a shape but also the path that shape traverses. The setup permits two types of motion: sliding and rotating. The square couch only slid, whereas the semicircle slid, then rotated around the bend, and then slid again on the other side. But objects can slide and rotate at the same time. Mathematician Dan Romik of the University of California, Davis, has noted that a solution to the problem should optimize both types of motion simultaneously.

British mathematician John Hammersley discovered in 1968 that stretching the semicircle can buy you a larger sofa if you carve out a chunk to deal with that pesky corner. Furthermore, Hammersley’s sofa takes advantage of a hybrid sliding plus rotating motion. The resulting sofa looks like a landline telephone:

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