This pattern breaks, but for a good reason | Moser's circle problem

Item: This pattern breaks, but for a good reason | Moser's circle problem
Rating: 5
Author: PostSingularity Institute

by 3Blue1Brown

Formal & Physical Sciences Mathematics PBMathematics PBKCalculus and mathematical analysis

🎙 3Blue1Brown 👥 8.6M 📅 July 2, 2023 ⏱ 16 min 👁 2.8M 📄 science communication 🧭 2026-08-28

Available in: English (current) Français

Keywords

Moser's circle problemcombinatoricsEuler's characteristicPascal's trianglepower of two

Summary

The video explores Moser’s circle problem: placing n points on a circle and connecting all pairs with chords, then counting the regions created. The sequence of regions for n=1 to 5 gives 1, 2, 4, 8, 16, suggesting powers of two, but for n=6 it is 31, breaking the pattern. The video then derives the general formula for the number of regions: 1 + C(n,2) + C(n,4). It explains how to count chords (C(n,2)) and intersection points (C(n,4)) by associating each intersection with a unique set of four points. Using Euler’s characteristic formula for planar graphs, it shows how to count regions by considering intersections as vertices and counting edges. Finally, it connects the formula to Pascal’s triangle, explaining why the pattern holds for n≤5 (summing the first five entries of the previous row gives a power of two) and why it fails for n=6 (missing one term). It also notes that for n=10, the sum again gives a power of two due to symmetry.

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Critical Evaluation

Value of the Information & Strength of the Argument

The video provides a clear and rigorous derivation of the formula, building from simple counting arguments to the use of Euler’s characteristic. The argumentation is solid, with each step logically justified. The explanation of why the pattern breaks is particularly insightful, linking the combinatorial formula to Pascal’s triangle and showing the exact reason for the deviation. The video also encourages viewers to think about the problem independently and offers a challenge regarding further powers of two.

Scientific Rigor, Source Quality, Title Accuracy

The video is scientifically rigorous, presenting a well-known mathematical problem and its solution. The sources cited are primarily the creator’s own resources (Patreon, GitHub, website) and music credits, but the mathematical content is self-contained and based on established principles. The title accurately reflects the content, and the video includes a correction note for a minor error. The explanation of Euler’s formula is intuitive and correct.

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Title / Content Match

The title accurately reflects the content: it presents a pattern that appears to hold but breaks, and explains the underlying reason.

Quality & Reliability

9/10

High-quality mathematical exposition with rigorous reasoning, clear visualizations, and a well-known result (Euler's formula, Pascal's triangle). The video is produced by a reputable mathematics educator and includes a correction note.

Chapters

Cited Sources

3Blue1Brown Patreon — Funding support for the channel.
3Blue1Brown FAQ — Information about the manim library used for animations.
Manim GitHub repository — Source code for the animation library.
Manim Community GitHub repository — Community version of the animation library.
3Blue1Brown videos GitHub repository — Code for specific videos.
Music by Vincent Rubinetti — Composer of the background music.
The Music of 3Blue1Brown on Bandcamp — Album download.
The Music of 3Blue1Brown on Spotify — Streaming the music.
3Blue1Brown website — General channel information.
3Blue1Brown Reddit — Community discussions.

Concurring Sources

Moser's circle problem - Wikipedia — Confirms the problem statement and the formula for the number of regions.

External References

3b1b.co

Contribution & Novelties

The video provides a clear and intuitive explanation of Moser’s circle problem, emphasizing the underlying combinatorial structure and the connection to Pascal’s triangle. It offers a fresh perspective on why the pattern of powers of two breaks, making the mathematical reasoning accessible. The use of Euler’s characteristic formula is elegantly presented.

Pour aller plus loin :

Moser’s circle problem — Wikipedia article with historical context and further details.
Euler characteristic — Generalization of Euler’s formula for planar graphs.
Pascal’s triangle — Properties and applications.
Binomial coefficient — Definition and combinatorial interpretation.

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Radar Profile

The radar profile shows high scores in quality, reliability, and technical level, with a slightly lower score in quantity of information, reflecting the focused nature of the video. The overall balance indicates a well-crafted educational content.

Reliability 9/10

💬 Très positif. Sur les 30 commentaires analysés, les spectateurs expriment une admiration unanime pour la clarté de l'explication, la qualité des animations et la profondeur pédagogique, avec des témoignages personnels sur l'accessibilité pour les enfants et les débutants.