New variant of the sofa problem that baffles mathematicians and movers

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The moving sofa problem asks what is the largest shape that can move around a right-angled curve? UC Davis’ mathematician Dan Romik has extended this problem to a two-turn hallway and shows that a bikini-shaped sofa is the largest sofa ever found that can move in such a hallway. Photo credit: Dan Romik, UC Davis

Most of us have struggled with the math puzzle known as the “moving sofa problem”. A deceptively simple question arises: which is the largest sofa that can rotate around an L-shaped hallway corner?

A moving helper tells you to just set up the sofa. But imagine if the sofa cannot be lifted, squeezed or tilted. While it still seems easy to solve, the moving sofa problem has been holding back math detectives for more than 50 years. Because the challenge for mathematicians is both to find the largest sofa and to prove that it is the largest. Without proof, it is always possible that someone will come up with a better solution.

“It’s a surprisingly difficult problem,” said math professor Dan Romik, chair of the math department at UC Davis. “It’s so easy you can explain it to a child in five minutes, but no one has found any evidence yet.

The largest area that fits around a corner is called the “sofa constant” (yes, really). It is measured in units, with one unit being the width of the hallway.

Inspired by his passion for 3D printing, Romik recently tackled a twist on the sofa problem known as the ambidextrous sofa. In this scenario, the sofa has to maneuver through 90 degree turns to the left as well as to the right. His results are published online and appear in the journal Experimental Mathematics.

The Gerver sofa is the largest sofa that fits a single turn. It has a “sofa constant” of 2.22 units, with one unit representing the width of the hallway. Photo credit: Dan Romik / UC Davis

Eureka moment

Romik, who specializes in combinatorics, likes to deal with difficult questions about shapes and structures. But it was a hobby that sparked Romik’s interest in the moving sofa problem – he wanted to 3D print a sofa and a hallway. “I’m excited about how 3-D technology can be used in math,” says Romik, who has a 3-D printer at home. “If you have something that you can move with your hands, it can really help your intuition.”

The Gerver sofa – which resembles an old telephone receiver – is the largest sofa ever found for a hallway with one turn. When Romik was tinkering with translating Gerver’s equations into something a 3D printer could understand, he immersed himself in the math that underlies Gerver’s solution. In the end, Romik devoted several months to developing new equations and writing computer code that refined and expanded Gerver’s ideas. “The whole time I didn’t think I was doing research. I was just messing around,” he said. “Then in January 2016 I had to put that aside for a few months. When I returned to the program in April, I had a lightbulb flash. “

Romik decided to tackle the problem of a two-turn hallway. When tasked with assembling a sofa through the hallway corners, Romik’s software spat out a shape resembling a bikini top, with symmetrical curves connected by a narrow center. “I remember sitting in a cafe when I first saw this new shape,” said Romik. “It was such a beautiful moment.”

Find symmetry

Like the Gerver sofa, Romik’s ambidextrous sofa is still just a guess. However, Romik’s results show that the question can still lead to new mathematical knowledge. “While the moveable sofa problem may seem abstract, the solution involves new mathematical techniques that can pave the way for more complex ideas,” said Romik. “There’s still a lot to discover in mathematics.”

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More information:
Dan Romik, Differential Equations and Exact Solutions in the Moving Sofa Problem, Experimental Mathematics (2017). DOI: 10.1080 / 10586458.2016.1270858

Quote: New variant of the sofa problem that took mathematicians and furniture transporters by surprise (2017, March 20), accessed on July 6, 2021 from https://phys.org/news/2017-03-sofa-problem-stumped-mathematicians-furniture. html

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