Mathematician Solves Long-Standing Moving Sofa Puzzle

When it comes to moving furniture, mathematicians aren’t usually the first people we think to consult. And why would they be? For nearly six decades, they couldn’t definitively answer whether your stylish three-seater could navigate that tricky turn into your apartment hallway.

However, Jineon Baek, a mathematics enthusiast from Yonsei University in South Korea, might just change that perception. Baek has unveiled a groundbreaking 100-page proof addressing this exact conundrum, potentially saving us from countless moving-day frustrations. His work offers a solution to a puzzle that has fascinated the math community for years: how to choose furniture that won’t get stuck halfway up a narrow stairwell or at a sharp corner.

The Origins of the Moving Sofa Problem

The problem, formally introduced by Austrian-Canadian mathematician Leo Moser in 1966, tackles a seemingly simple question: what’s the largest two-dimensional object that can maneuver through an L-shaped turn in a one-unit-wide corridor?

A one-square-unit chair can glide through effortlessly, but a rectangle measuring two square units is guaranteed to get stuck. Anything longer? Forget it; that’s where it stays. Things get trickier when you consider irregularly shaped furniture—think IKEA designs named after fantasy characters, with curves resembling old-school phone receivers.

In 1968, British mathematician John Hammersley proposed a shape made of a semicircle joined to a square with a semicircular notch. This design, he found, could pass through the corner if it had an area of up to 2.2074 units. He also established an upper limit, noting that no shape larger than 2.8284 units could fit.

Decades later, in 1992, Joseph Gerver of Rutgers University refined Hammersley’s concept. By smoothing some edges and introducing additional curves, Gerver discovered a shape with an area of slightly over 2.2195 units. His solution was deemed “locally optimal,” meaning it was the best outcome within the constraints of that particular shape.

But the quest for a definitive answer persisted. Without a universal formula to account for all possible sofa designs, there was no guarantee a slightly larger or differently curved sofa wouldn’t work. In 2018, researchers Yoav Kallus and Dan Romik pushed the boundary further using computer-assisted techniques, suggesting that a sofa could theoretically be as large as 2.37 units.

Baek’s recent breakthrough relies on an advanced mathematical concept known as an injective function. This approach allowed him to map and analyze the properties of Gerver’s sofa design, systematically expanding the dimensions to confirm the maximum possible size. His conclusion? The optimal sofa for a one-unit-wide corridor and an L-shaped turn is indeed 2.2195 units, matching Gerver’s 1992 proposal.

Although Baek’s findings are yet to undergo peer review, they could represent the final chapter of this longstanding mathematical challenge—at least for single-corner scenarios. If your hallway features a second turn in the opposite direction, you might need to consider Romik’s “ambidextrous sofa.”

Read the original article on: Science Alert

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