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The moving sofa problem

#### Written by Lydia Nawas

Mathematics is undeniably beautiful in a certain way, providing us answers to many of life’s complex issues. It finds itself in all aspects of society, and has played a major role for thousands of years. However, despite our deep understanding, there are still many unsolved problems. For most of us, we can’t even begin to imagine the majority of these problems, but not all of them are so complicated. One of the most famous problems in mathematics is actually very simple to picture. Imagine a corridor, with width one metre, with a right angle in it, like this:

Now imagine moving a sofa through this corridor. The question is simple: what is the largest surface area of sofa that you can fit through this corridor? This problem was proposed in 1966 by Canadian mathematician Leo Moser, and has, officially, never been solved. You might think that this is very easy to solve, but what makes this problem so difficult to solve is trying to figure out what kind of shaped sofas we can make. We can start simply with a cube with edge length 1, like this:

This fits, and we have a surface area of 1, but we can surely go bigger. If we try a semicircle with radius 1, it also fits with a surface area approximately 1.57. This is also interesting because now we have curved edges. This means that our shape does not need to fully shift, or translate, but can rotate around the corner. What if our shape could do both? British mathematician John Hammersley tested this idea in 1968, giving birth to the Hammersley sofa. This sofa has a semi-circle pulled apart, with a rectangular block in between, which fits around the corner of the corridor. This has a record-breaking surface area of , which is approximately 2.2074, and looks like:

Hammersley also showed that the maximum area that can be fit into the corner is , giving an upper bound for our problem. To quickly summarise the proof, we look at trying to fill up the corner as much as possible. The easiest way to do this is by looking at the right-angle and imagine a triangle spurring from this. So, imagine our triangle was made out of something squishy, like play-dough. The proof for the lower bound begins with a neatly stuffed right-angle triangle in the corner. If we try pushing this triangle further out of the corner, into each of the corridors, at some point we will split into two quadrilaterals, meaning we no longer have one single sofa, but two separate pieces, which isn’t allowed. So, what is the maximum area we can fit before splitting our play-dough into 2? With some calculation that I won’t bore you with, we get an area of 2.828. This means that any solution we find cannot be bigger than this, but we do not know if this is the actual maximum. So, looking at our current solution (2.2074) the Hammersley sofa seems to come relatively close. For a long time, people thought that this was the best solution, until 1992, when Joseph Gerver adapted the Hammersley sofa. Simply put, Gerver rounded off some of the edges in Hammersley’s design. Now we have a shape made up of 3 straight edges and 15 curved segments. This yields an area of approximately 2.2195, which is even closer to our upper bound:

To this day, this is the best solution, however, there is no formal proof that we cannot do any better than this. This is the same for any problem similar to this. For example, mathematicians have further adapted this troublesome problem for a corridor that has 2 corners in it, like this:

In 2017, Dan Romik came up with the best solution yet for this version of the moving sofa. He suggested a symmetrical version of the “optimal” solution for the original problem. In practice, this looks like a pair of a glasses, or a bikini top, with a surface area of approximately 1.645:

As with the original problem, this has also not been formally proven. So hopefully now you understand why the moving sofa problem is one of the world’s most interesting problems. It is painstakingly simple, yet formally unsolvable. Maybe you can try your hand at finding the solution, or proving the current solution!