Moving Sofa Problem

So you're moving into your new apartment, and you're trying to bring your sofa. The problem is, the hallway turns and you have to fit your sofa around a corner. If it's a small sofa, that might not be a problem, but a really big sofa is sure to get stuck. If you're a mathematician, you ask yourself: What's the largest sofa you could possibly fit around the corner? It doesn't have to be a rectangular sofa either, it can be any shape.





This is the essence of the moving sofa problem. Here are the specifics: the whole problem is in two dimensions, the corner is a 90-degree angle, and the width of the corridor is 1. What is the largest two-dimensional area that can fit around the corner?





The largest area that can fit around a corner is called—I kid you not—the sofa constant. Nobody knows for sure how big it is, but we have some pretty big sofas that do work, so we know it has to be at least as big as them. We also have some sofas that don't work, so it has to be smaller than those. All together, we know the sofa constant has to be between 2.2195 and 2.8284.