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Even mathematicians can’t figure out how to move a sofa around a corner

One of the biggest frustrations a person usually has after moving into a new flat is finding that the sofa is not placed where you want it to be.

You find yourself trying to lift the couch up and then carrying it around the corner screaming ‘PIVOT! You see your friends, exhausted, holding the other end.

Friends aside, mathematicians can still be stumped when it comes to moving the couch. This problem has been a puzzle for many decades. However, they are still searching for the right solution.

First posed by mathematician Leo Moser in 1996, it is commonly known as the ‘moving sofa problem’.

Even mathematicians are stumped
Even mathematicians are stumped

Moser asked: ‘What is the shape of largest area in the plane that can be moved around a right-angled corner in a two-dimensional hallway of width 1?’

The crux of it is, even with little knowledge of geometry, it can be easy to find different shapes that will fit around the corner, but even for those with a lot of knowledge, it’s hard to come up with large shapes that will still fit.

In a YouTube video by Numberphile, professor of mathematics at The University of California, Dan Romik, talks about the problem in more detail.

He explains how it’s all about the area of the sofa. It’s not the longest or the heaviest, but the biggest.

A simple semi-circular sofa shape would need the width of at least one unit, and as the semi circle would have a radius of one also, the formula would be:

π over two, which = 1.57.

John Hammersley, a mathematician, noticed that if the semicircle were cut into quarter-circles and then pulled apart with a rectangle block, it would create a bigger sofa shape that could be moved around a corner.

In his blog, Romik explained that “Hammersley’s idea would work for every value between 0 and 1 of the radius of the semi-circular hole at the bottom.

“The shape of maximal area in this family is obtained when the radius is chosen to be 2/ᴨ (approximately 0.637), which gives an area of 2/ᴨ+ᴨ/2, or approximately 2.2074.

“This is much better than the area of our ‘idiot’s sofa,’ the unit square. Hammersley thought his construction may be optimal, but this turned out to be false.”

So, it looks like we still haven’t found the perfect realistic shape solution to move a sofa around a corner effectively.

John Horton Conway and others even came up with an idea of a sofa that can be moved around at a 90-degree turn both to the right and to the left, known as an ‘ambidextrous sofa’ in Romik’s words.

The hallway is divided into two corners. One that goes left and one that goes right. Although it’s important to mention that the sofas in this problem look slightly different than the standard sofa problem.

It seems that currently, Dan Romik has a pretty solid solution explained in the video which he calls ‘The Romik Ambiturner’ but unfortunately, we’re still far away from solving the moving sofa problem for good.

Basically, you’re probably best either renting or buying somewhere with straight hallways or completely taking apart your sofa. Ah, the joys associated with moving house.

Even mathematicians can't figure out how to move a sofa around a corner
Anurag Reddy
I'm a 29-year-old travel enthusiast, travel and nature photographer, Computer Science graduate, and Mass Communication student. I have seen different shades of life through traveling and lived different lives through reading. With every word I write, I travel within, and I understand myself better. Writing helps me discover myself, and that paved roads for me to choose writing as a profession.


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