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Do you know Joe, Jay and Jim? They are renting a 3-bedroom apartment in Cambridge and have to split the $3,000 monthly rent.
$1,000 each?
But the trouble is that all 3 bedrooms are of a different size.
Divide the rent based on a room size?
Good idea! However one room has two windows, second has only one and the third one has a fire escape in front of its window. Joe really wants the last one for his cigarette breaks but he is wiling to pay no more than $900 for it.
Apparently math could help them!

Recent New York times article describes an algorithm that could be used in such situations. Take an equal-sided triangle where every vertex represents a totally unfair situation when one person pays for one room only the whole $3,000 rent. Subdivide this triangle into smaller ones and interpolate rent linearly into every new vertex. See below:


Now go over all the vertices and for each vertex ask either one of the roommates which room he prefers at such rent split. Apparently there always will be a triangle where every roommate has picked a different room. The fair rent subdivision lies somewhere in this triangle. Read more about the algorithm in the New York Times article: To divide the rent, start with a triangle.


And try the New York Times interactive rent division calculator: Rent Division Calculator.

Top image from Rentcafe.