Any fan of ‘Friends’ knows there’s only one way to get a sofa around a tight corner: Pivot!
However, while this might not have worked out so well in that iconic scene, mathematicians have now revealed how Ross Geller could have moved his sofa up the stairs.
Dr Jineon Baek, a mathematician from Yonsei University, Korea, has found the biggest possible sofa that can be pivoted around a 90-degree corner.
Unlike Ross’ sofa, that optimal design looks a lot like an old telephone receiver with a flat back, rounded corners, and a semicircular cutout at the front.
This design was first proposed by Joseph Gerber of Rutgers University in 1992 but, until now, no one has been able to prove that a bigger sofa wasn’t possible.
In a 100-page proof, Dr Baek has now confirmed that for a corridor one ‘unit’ wide the biggest sofa you can possibly get around the corner has an area of 2.2195 units exactly.
That means that, if the Friend’s stairwell was 2m-wide, the biggest sofa Ross could have ever moved in would have an area of 4.439 metres squared.
Beyond helping people move into tiny apartments, this solution also lays to bed a 60-year-old mathematical puzzle.
While Ross Gellar might have failed to get his sofa up the stairs in this iconic Friend’s scene, mathematicians have now proven how he could have pivoted his sofa around the corner
Dr Jineon Baek, a mathematician from Yonsei University, Korea, has identified the ideal shape for a sofa you need to move around a corner. This shape (pictured) allows for the biggest area of sofa based on the size of the corridor
The moving sofa problem was first proposed in 1966 by Austrian-Canadian mathematician Leo Moser.
This essentially set out in mathematical terms a puzzle that almost everyone has tried to solve at least once in their life.
The question asks: For a corridor with a 90-degree turn, ignoring its height, what is the biggest sofa you can possibly get around the corner and how big is that sofa?
While it might seem intuitive, it turns out that this is actually a fiendishly difficult mathematical puzzle.
In the most basic case, you might imagine pushing a square along the corridor; that square could have a width and length equal to the size of the corridor.
That means, for a corridor one arbitrary ‘unit’ across, your square sofa has an area of one unit.
That’s a good start but anyone who has ever moved a sofa can quickly see that you could still move a much larger sofa.
For example, if you have a semi-circular sofa with a radius the same width as the corridor you can easily bump the area up to 1.57 units.
This problem, known as the moving sofa problem, was first proposed in 1966 and has stumped mathematicians since. The question is: For a 2D corridor one unit wide, what is the biggest sofa you could move around a 90-degree turn and how big is that sofa? (stock image)
Soon mathematicians noticed that the sofas shaped like bananas or old telephone receivers could be even bigger.
In 1992, Professor Joseph Gerver proposed a design known as Gerver’s sofa, a complex shape made up of 18 curved sections.
For over 30 years, nobody had been able to find a shape which allows for a bigger area but, on the other hand, no one could prove that a bigger shape wasn’t possible.
Now, after working on his proof for seven years, Dr Baek has finally managed to prove that Gerver’s sofa is indeed the optimal shape.
Dr Baek’s breakthrough came from looking at a small set of sofa shapes and asking what properties they all have in common.
These properties include a fairly smooth outer edge, a mathematical property called balance which is similar to symmetry and the ability to rotate the full 90 degrees around the corner.
By combining all these properties, Dr Baek invented a new mathematical quantity called Q which was closely related to area.
This transformed the open-ended question of how big a sofa could be into a problem with one definite answer.
In 1992, a mathematician called Joseph Gerver proposed ‘Gerver’s sofa’, a telephone-like shape which he believed was the biggest possible sofa you could still move around a 90-degree corner. Pictured: An illustration of how Gerver’s sofa might look
More than 30 years later, Dr Baek has finally proven that Gerver’s sofa (pictured) is actually the biggest possible sofa shape. While his proof will need to be checked, Dr Baek is confident that he will be found correct
By finding the highest possible value of Q, Dr Baek could then show what shape would fit that value.
And, when he had finished crunching the numbers, the optimal sofa shape turned out to be exactly the same as the one proposed by Professor Gerver three decades ago.
Dr Baek told New Scientist: ‘I dedicated a lot of time to this, without any publication so far.
‘The fact that now I can say to the world that I committed something valuable to this problem is validating.’
If Dr Baek’s working is right, this will mathematically prove that Professor Gerver was right to say that his sofa was the biggest possible shape.
Professor Gerver says: ‘I am of course very happy about all of this. I am 75 years old, and Baek can’t be more than 30.
‘He has a lot more energy, stamina and surviving brain cells than I do, and I am glad that he picked up the baton. I am also very happy that I lived long enough to see him finish what I started.’
Dr Baek’s results will need to be fully checked over by other mathematicians before we can know for certain, but he remains confident that his results will be found to be correct.
