Using topology for discrete problems | The Borsuk-Ulam theorem and stolen necklaces

Using topology for discrete problems | The Borsuk-Ulam theorem and stolen necklaces

by 3Blue1Brown

Formal & Physical Sciences Mathematics PBMGeometryPBMPDifferential and Riemannian geometry
🎙 3Blue1Brown 👥 8.6M 📅 November 18, 2018 ⏱ 19 min 👁 958K 📄 science communication 🧭 2026-08-28
Available in: English (current) Français

Keywords

Borsuk-Ulamnecklace splittingtopologycontinuous mappingantipodal points

Summary

The video introduces the necklace splitting problem, a discrete fair division puzzle, and demonstrates its solution using the Borsuk-Ulam theorem from topology. The presenter first defines the problem: given a necklace with an even number of each of n types of gems, it is always possible to split it fairly between two thieves with at most n cuts. The video then explains the Borsuk-Ulam theorem, which states that any continuous function from a sphere to a plane maps some pair of antipodal points to the same point. A proof of the theorem is sketched using a loop around the equator and its deformation to a point. The key insight is to translate the discrete necklace problem into a continuous one by representing the necklace as a line segment and considering continuous cuts. This allows a correspondence between allocations of the necklace and points on a sphere, where antipodal points correspond to swapping the pieces between the two thieves. Applying Borsuk-Ulam to a function that measures the amount of each gem type given to thief 1 guarantees a fair division with at most n cuts. The video concludes by discussing the generalization to higher dimensions and the broader significance of topology in solving discrete problems.

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Critical Evaluation

Value of the Information & Strength of the Argument

The video provides a clear and compelling argument for the necklace splitting theorem. The value lies in its ability to make a sophisticated topological proof accessible through intuitive visualizations and step-by-step reasoning. The argumentation is solid: it builds from the discrete problem to the continuous analogue, establishes a bijection between necklace allocations and points on a sphere, and then applies the Borsuk-Ulam theorem. The proof of Borsuk-Ulam itself is presented with a convincing topological argument involving loops and their deformations. The logical flow is excellent, and the connection between the two seemingly unrelated areas is made explicit and satisfying.

Scientific Rigor, Source Quality, Title Accuracy

The video demonstrates high scientific rigor. It cites the original 1986 paper by Alon and West, which contains the proof, and also references related work in electrical engineering. The presentation is mathematically accurate and the proof is complete. The title accurately reflects the content, which is a demonstration of how topology can be used to solve a discrete problem. The video is well-structured with clear timestamps, and the creator’s reputation for accuracy is well-established. The sources provided in the description are relevant and credible.

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Title / Content Match

The title accurately reflects the content, which demonstrates the application of the Borsuk-Ulam theorem to solve the discrete necklace splitting problem.

Quality & Reliability

9/10

The video is produced by a well-known mathematics educator with a strong reputation for accuracy and clarity. The proof is rigorous and well-explained, and the video cites original sources, including the 1986 paper by Alon and West.

Chapters

Cited Sources

  • The Borsuk-Ulam Theorem and bisection of necklaces (Alon & West, 1986) — Original paper presenting the proof of the necklace splitting theorem using Borsuk-Ulam.
  • ACM paper on related ideas — Electrical engineering paper applying ideas related to the necklace splitting problem.
  • Mathologer video on fair division — Recommended video for more fair division math fun.
  • VSauce video on fixed points — Related video on fixed points, which is a related topological concept.
  • Quora post by Alon Amit — Quora post that inspired the video.

Concurring Sources

  • The Borsuk-Ulam Theorem and bisection of necklaces (Alon & West, 1986) — The original paper proves the same theorem presented in the video.

External References

Contribution & Novelties

The video’s original contribution lies in its pedagogical approach: it makes a sophisticated topological proof accessible to a broad audience through clear visualizations and a step-by-step narrative. It effectively demonstrates the power of topology in solving discrete problems, a connection that is often not emphasized in standard treatments. The video also provides a clear explanation of the Borsuk-Ulam theorem and its proof, which is valuable for learners.

Pour aller plus loin :

  • Borsuk-Ulam theorem — Wikipedia article providing a comprehensive overview of the theorem and its variants.
  • Necklace splitting problem — Wikipedia article detailing the problem and its solutions.
  • Fair division — Wikipedia article on the broader field of fair division, which encompasses the necklace problem.
  • Topology — Wikipedia article on topology, the branch of mathematics central to the video.

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Radar Profile

The radar profile shows high scores across all dimensions, indicating a video that is both informative and reliable. The slightly lower score for technical level reflects its accessibility, but it still provides a rigorous proof. Overall, this is an excellent educational resource.

Reliability 9/10

💬 Très positif. Sur les 30 commentaires analysés, les spectateurs expriment une admiration et un enthousiasme marqués pour la clarté de l'explication et la beauté de la démonstration, certains mentionnant avoir passé beaucoup de temps à comprendre chaque étape.