# Science Solves Rent Division Dilemma

A few years ago, I moved into the perfect apartment with the perfect roommate. The only problem? One of us was getting stuck in a shoebox while the other lived it up in the master suite. It was also an older building where only the smaller room was equipped with an in-unit air conditioner and ceiling fan. Keeping the harmony was going to be a pain. What to do? Some folks simply decide to divide the costs evenly, or perhaps based on the bedrooms’ square footage or even each renters income. Science, however, may have solves this pressing issue using mathematics.

The theorem, Sperner’s Lemma, could hold the key to dividing rents fairly, according to The New York Times. It calculates price points at which every roommate finds their happy place, helping to eliminate room envy. Using a triangle with three points: Point A, B, and C. The interior of the triangle is divided to represent the variety of ways to split rent.  The rent is more accurately and “fairly” divided the more the triangle is divided. Sounds complicated? A bit, but the NYT breaks it down for us using this handy  rent division calculator.

Part of the beauty of the approach is that you don’t have to come up with numbers yourself. All it requires is that at each step you pick which room you like best based on the prices assigned to each room at that moment. As the method proceeds, the prices get closer and the decisions become harder, but it contains no surprises. You’re never stuck with a price that you haven’t chosen.

So how do you apply it? Roommates take turns answering a series of questions, picking a room they like best, based on a specific price. With each step, the hypothetical prices get closer, and the decisions are trickier. Each roommate continues answering until there’s a fair division. Try it out the next time you’re leasing.