by Krešo Josić

Today, we talk about the mathematics of moving a couch. The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

Imagine you are helping your friend move into a new apartment. Your friend has a heavy sofa that you can't lift, but have to push along the floor. Unfortunately, the corridor leading to his apartment makes a right turn. Will you be able to push the sofa around the bend?

If the sofa is square and spans the corridor you will be fine: First push the sofa as far as it will go. Once it is all the way in the corner, push it past the bend. You've made it! A half-circular sofa, with radius equal to the width of the corridor will also fit. But change these shapes just a little, and you could be stuck.

Joseph Gerver's sofa shape is composed of 18 curve sections. Computational evidence suggests that this is the largest sofa that can clear the corner, but no-one has been able to prove this yet.

*Photo Credit:Wikipedia*.

Over 50 years ago the Canadian mathematician Leo Moser asked what is the shape of the largest possible sofa that can clear the bend. Mathematicians have found increasingly bigger sofas that do the job. But we still don't know the shape of the biggest sofa that does the trick.

The current record sofa is about 2.2 times larger than the square sofa. It is made of 18 curves glued together in just the right way. Many think that this is the largest sofa that will make it. But someone could still discover a larger sofa.

An animated gif of Hamersley's sofa clearing the corner

*Photo Credit:Wikipedia*.

There are other similar unresolved puzzles: For instance, suppose you need to patch some holes in a drywall. None of the holes is larger than 1 inch across: The holes are not all round, but when you measure any of them with a ruler in any direction, they are all less than an inch wide. What is the shape of the smallest patch that will cover any such hole?

This question is over 100 years old, and still no-one knows the answer. We do know that the smallest universal patch is has an area of at least 0.832 square inches. Mathematicians have created cleverly shaped patches that are just a little larger than this. But someone could still discover a smaller patch that works.

Why are these questions so hard? For one there are so many possible shapes of sofas and patches. It is hard to get a handle on all the possibilities. We are like the ancient Greeks who also tried to find the smallest or largest shape of one kind or another. Many questions these Greeks asked were not answered until over a milenium later when we developed the powerful tool we now call calculus. Indeed, calculus now lets our high school students answer questions that would have stumped the smartest ancients.

But calculus allows us to do so much more: We use it to engineer phones and cars, and understand the motion of stars. Calculus is a keystone of our civilization.

The sofa problem may seem frivolous. But such idle questions may not be idle at all. In answering them we may uncover new mathematical tools that hold the key to far more important questions. Like calculus before them, these tools can help us understand the universe, and improve our lives within it.

This is Krešo Josić at the University of Houston where we are interested in the way inventive minds work.

You can read more about the moving sofa problem, and see a movie of the different shapes here.

Universal cover problem here.

There are other simple problems that are either unanswered, or have answers that are really complicated. Some examples are the traveling salesman problem, Kepler's conjecture, and the four color theorem. Episode No. 3076 (all discussed in previous Engines episodes). All of these problems have lead to interesting and useful mathematical ideas.

This episode was first aired on May 5, 2020