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triangles to solve rent problems

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The connection between Sperner’s lemma and rent division was first published by Dr. Su in a 1999 papertitled “Rental Harmony: Sperner’s Lemma in Fair Division.” He came to the problem while completing his doctorate at Harvard. A friend of his was facing the same predicament I was — moving into a shared apartment with rooms of different sizes and features — and had asked for his advice.

Dr. Su realized that it might be related to another problem he had heard about, in which a group has to divide a theoretical cake when some want frosted flowers or an edge with more frosting.

“The trick is to design a procedure to have everyone act in their own self-interest and have an outcome that’s fair,” he said in an interview.