# NYT: Sperner's lemma defeats the rental harmony problem | Summary and Q&A

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February 10, 2017
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Mathologer
NYT: Sperner's lemma defeats the rental harmony problem

## TL;DR

Sperner's Lemma, a key result in abstract math, can be applied to solve fair division problems, such as splitting rent among friends with different preferences for rooms.

## Key Insights

• 🧚 Sperner's lemma can be applied to solve fair division problems, such as rent allocation, by using a triangulated triangle.
• 🔺 The lemma guarantees the existence of at least one little triangle that features all three colors.
• 🤵 By labeling the triangle with the friends' names and applying the coloring scheme of Sperner's lemma, a fair allocation of rooms and rent can be achieved.

## Transcript

Welcome to another Mathologer video. Today is about a math story that made it into The New York Times something that's really very rare. It's about a familiar problem. A couple of friends, Ashwin, Bret and Chad want to rent an apartment. Now the rooms are quite different and the friends have different preferences, different ideas about what's worth... Read More

### Q: What is Sperner's lemma and how does it relate to fair division problems?

Sperner's lemma is a mathematical result that guarantees the existence of a small triangle in a larger triangle that features all three colors in a certain coloring scheme. This lemma can be applied to fair division problems, such as splitting rent, by using the triangle to represent different divisions and finding a fair allocation of rooms and rent.

### Q: How does the proof of Sperner's lemma work?

The proof of Sperner's lemma involves considering the doors of the rooms in a triangulated triangle. It shows that each room can have zero, one, or two doors, with the rooms containing exactly one door being the special 3-colored ones. By entering and leaving the house through the doors, it is proven that at least one of the special rooms will be reached.

### Q: How is Sperner's lemma applied to the rent allocation problem?

The triangle representing rent divisions is labeled with the friends' names. By applying the coloring scheme of Sperner's lemma, where each little triangle has all three names at its vertices, the rooms can be assigned to the friends based on their preferences. The different rent divisions corresponding to the vertices of the little triangles allow for a fair allocation that satisfies all friends.

### Q: Can Sperner's lemma be applied to fair division problems with more or fewer rooms?

Yes, Sperner's lemma can be applied to fair division problems with any number of rooms. The dimensions of the lemma will vary accordingly. For example, if there are four rooms, a higher-dimensional version of Sperner's lemma would be used.

## Summary & Key Takeaways

• Sperner's lemma is a mathematical result that can be used to solve fair division problems, such as splitting rent among friends with different room preferences.

• The lemma guarantees that in a triangle with a certain coloring scheme, at least one of the little triangles will feature all three colors.

• By applying Sperner's lemma to a triangle representing different rent divisions, it is possible to find a fair allocation of rooms and rent that satisfies all friends.