It’s an exciting day: You’ve just signed the lease on a new apartment.There’s only one problem. The hallway leading to your apartment has a sharp corner and your couch won’t seem to make it, no matter how much you and your friends yell “pivot!”.
When confronted with this frustration, most people keep working at it until they make the corner (or they just can’t take it anymore and leave the couch on the curb). But mathematicians see something else. They look at this problem and wonder: What’s the largest possible couch you could get around this corner?
This is known as the Moving Sofa Problem, and it’s far from a real estate curiosity. The problem has been perplexing mathematicians for half a century. It was part of our list of unsolved problems in mathematics, and has been covered in more depth in this video by Numberphile:
With a little knowledge of geometry, it’s easy to come up with different shapes that will fit around the corner. With a lot of knowledge of geometry, it’s possible to come up with some very large shapes that still fit. But the problem compels us to find the largest possible solution, and it’s extremely difficult to show that any one possible shape is the largest possible. … (Popular Mechanics)