# New twist on sofa problem that stumped mathematicians and furniture movers

Most of us have struggled with the mathematical puzzle known as the "moving sofa problem." It poses a deceptively simple question: What is the largest sofa that can pivot around an L-shaped hallway corner?

A mover will tell you to just stand the sofa on end. But imagine the sofa is impossible to lift, squish or tilt. Although it still seems easy to solve, the moving sofa problem has stymied math sleuths for more than 50 years. That's because the challenge for mathematicians is both finding the largest sofa and proving it to be the largest. Without a proof, it's always possible someone will come along with a better solution.

"It's a surprisingly tough problem," said math professor Dan Romik, chair of the Department of Mathematics at UC Davis. "It's so simple you can explain it to a child in five minutes, but no one has found a proof yet.

The largest area that will fit around a corner is called the "sofa constant" (yes, really). It is measured in units where one unit corresponds to the width of the hallway.

Inspired by his passion for 3-D printing, Romik recently tackled a twist on the sofa problem called the ambidextrous moving sofa. In this scenario, the sofa must maneuver around both left and right 90-degree turns. His findings are published online and will appear in the journal Experimental Mathematics.

Eureka Moment

Romik, who specializes in combinatorics, enjoys pondering tough questions about shapes and structures. But it was a hobby that sparked Romik's interest in the moving sofa problem—he wanted to 3-D print a sofa and hallway. "I'm excited by how 3-D technology can be used in math," said Romik, who has a 3-D printer at home. "Having something you can move around with your hands can really help your intuition."

The Gerver sofa—which resembles an old telephone handset—is the biggest sofa found to date for a one-turn hallway. As Romik tinkered with translating Gerver's equations into something a 3-D printer can understand, he became engrossed in the mathematics underlying Gerver's solution. Romik ended up devoting several months to developing new equations and writing computer code that refined and extended Gerver's ideas. "All this time I did not think I was doing research. I was just playing around," he said. "Then, in January 2016, I had to put this aside for a few months. When I went back to the program in April, I had a lightbulb flash. Maybe the methods I used for the Gerver sofa could be used for something else."

Romik decided to tackle the problem of a hallway with two turns. When tasked with fitting a sofa through the hallway corners, Romik's software spit out a shape resembling a bikini top, with symmetrical curves joined by a narrow center. "I remember sitting in a café when I saw this new shape for the first time," Romik said. "It was such a beautiful moment."

Finding Symmetry

Like the Gerver sofa, Romik's ambidextrous sofa is still only a best guess. But Romik's findings show the question can still lead to new mathematical insights. "Although the moving sofa problem may appear abstract, the solution involves new mathematical techniques that can pave the way to more complex ideas," Romik said. "There's still lots to discover in math."

Explore further

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

Feedback to editors

Mar 20, 2017
On the Gerver sofa, once you round a first corner, you can flip it over and go around the other. Flipping the sofa around to get it to fit is pretty common.

Mar 20, 2017
On the Gerver sofa, once you round a first corner, you can flip it over and go around the other. Flipping the sofa around to get it to fit is pretty common.

What if it's taller than the width of the corridor?

Mar 20, 2017
You can't flip it over -- that would be a trivial way to solve an otherwise interesting problem.

Mar 20, 2017
What if it's taller than the width of the corridor?

It's a 2-D problem. Doesn't anyone read the article?

Mar 20, 2017
I've had to install a floor to ceiling cabinet against a wall after bringing it into the room horizontally. The top of the cabinet gets hung up on the ceiling when righting it, but the solution is to cut a radius from the bottom corners at the back of the cabinet. These hypothetical couches are too ugly to make a plausible puzzle.

Mar 20, 2017
not even an attempt to describe the method?

Mar 20, 2017
The method is simple enough. Just make a rectangle to fill the hallway as long as the hall, three times the width will probably do the job though.

Once the object is hard up against the opposite wall allow it to move into the next path by deleting any parts that would stop it from moving. effectively making an arc inside the curve and on the outside against the wall.

Do the same with the trailing edge and you have the required shape.

By placing another bend in the corridor you repeat the process around the next bend.

Mar 20, 2017
What is the sofa constant of the Romik 'bikini top' sofa?

Mar 21, 2017
Seems like an ideal problem to tackle with a genetic algorithm.

Mar 21, 2017
The method is simple enough. Just make a rectangle to fill the hallway as long as the hall, three times the width will probably do the job though.

Once the object is hard up against the opposite wall allow it to move into the next path by deleting any parts that would stop it from moving. effectively making an arc inside the curve and on the outside against the wall.

Do the same with the trailing edge and you have the required shape.

By placing another bend in the corridor you repeat the process around the next bend.

Yeah, they did that in the first version of the solution, but then someone found an even better shape.

Mar 22, 2017
Furniture movers have a corridor and a sofa to move. It either fits or not, there are about 4 ways to turn or twist it.
Machomaticians on the other hand can have all the fun they want, but I don't want a bikini-shaped or a phone-shaped sofa; that what was 'stumping' the honest guys - they had to redefine a 'sofa' to have a ridiculous shape, to get it fit when longer. This is violating the original premise of a sofa - making it a largest fitting 'object'. No reward points for this.

Mar 22, 2017
When it becomes a 2D problem, it loses all significance. Relevance matters. Standing the sofa on end happens, and when it happens there is another shape to consider. Flipping the sofa in the hall when approaching another turn also happens, and is relevant. Perhaps 2D only problems can have relevance somewhere somehow, but that relevant somewhere/somehow has got to be very small.

Mar 23, 2017
What is the sofa constant of the Romik 'bikini top' sofa?

From the article: ≈ 1.644955218425440.

They even have a closed form expression for it. If you're interested, here's the article: https://arxiv.org...8111.pdf

Mar 23, 2017
Oh good grief NoStrings & Zzzzzzzz. It is a mathematical way of solving a geometrical problem!
It might have wider use and application, it really has nothing to do with a physical couch or a corridor.

Mar 23, 2017
Yeah, they did that in the first version of the solution

The first version of the solution is a tiiiiiiny bit more complex (it consist of 14 areas with different curvature).
Takeaway message: seemingly simply problems are often a bit more complex than a first "knee-jerk" analysis would suggest