  Given the subjective preferences of n roommates in an n-bedroom apartment,
one can use Sperner's lemma to find a division of the rent such that each
roommate is content with a distinct room. At the given price distribution, no
roommate has a strictly stronger preference for a different room. We give a new
elementary proof that the subjective preferences of only n-1 of the roommates
actually suffice to achieve this envy-free rent division. Our proof, in
particular, yields an algorithm to find such a fair division of rent. The
techniques also give generalizations of Sperner's lemma including a new proof
of a conjecture of the third author.
