The last IMO problem AI could not solve

The last IMO problem AI could not solve

by 3Blue1Brown · View the channel on PSI

Formal & Physical Sciences Science — General & History PDScience: general issuesPDZPopular science
🎙 3Blue1Brown 👥 8.6M 📅 September 18, 2026 ⏱ 51 min 👁 1K 📄 science communication 🧭 2026-09-18
Available in: English (current) Français

Keywords

IMO 2025tiling problemminimal tilesproofAI limitations

Summary

The video begins by recounting the 2025 International Mathematical Olympiad, where problem 6 was the only one that no AI system could solve, despite progress in 2024 and 2025. The presenter, Grant Sanderson, states that the video will focus on the problem itself rather than AI, but will later reflect on why AI struggled. He introduces a tiling problem: given an n×n grid, place rectangular tiles aligned to grid lines so that each row and column has exactly one uncovered unit square (an X). The goal is to find the minimum number of tiles and prove optimality. Sanderson draws an analogy with a cube-cutting puzzle to develop intuition, focusing on edges of X-squares that must be covered by tiles. He then constructs a candidate optimal tiling using k×k square tiles in a pinwheel pattern, yielding (k-1)^2 interior tiles plus 4(k-1) border tiles, totaling k^2 + 2k - 3. For n=2025, with k=45, this gives a conjectured minimum. He then proves a lower bound by noting that each tile touches at most four highlighted edges from X-squares, giving at least n^2-1 tiles—but this is weaker than needed. After many attempts, he demonstrates that a stronger lower bound can be obtained by considering only, say, right-hand edges, and then develops a more refined counting argument to prove the construction is optimal. He also mentions a final conjecture and connects to the Erdős–Szekeres theorem. The video concludes with reflections on AI in mathematics, quoting Thang Luong of DeepMind that models lack patience and a sense of beauty.

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Critical Evaluation

Value of the Information & Strength of the Argument

The video provides a rigorous and insightful exploration of a challenging mathematical problem. The value lies in the step-by-step development of intuition, the clever analogy with the cube-cutting puzzle, and the systematic approach to both constructing a candidate solution and proving its optimality. The argumentation is solid: the presenter carefully builds from simple observations to a full proof, addressing potential pitfalls and explaining the rationale behind each step. The use of visual animations enhances understanding, and the logical progression is clear and well-structured.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is high: the video presents a complete proof of the minimal tiling number, consistent with the official IMO solution. Sources are provided in the description, including Evan Chen’s solution notes and a related video by Dedekind Cuts, which are credible references. The title accurately reflects the content, as the video indeed centers on the last IMO problem that AI could not solve, and the body discusses both the problem and the AI context. There is no indication of publication bias or sensationalism. The quality of sources is appropriate, with direct links to relevant materials.

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Title / Content Match

The title accurately reflects the content: the video discusses the last IMO problem that AI models could not solve, and then provides a detailed analysis and solution.

Quality & Reliability

9/10

The video is from a highly reputable educational channel known for rigorous mathematical explanations. The content is a deep dive into an IMO problem, with careful step-by-step reasoning. Sources are provided for further reading. The only minor caveat is that the transcript provided is an AI translation, which might slightly affect nuances, but the mathematical content is reliable.

Chapters

Cited Sources

External References

Contribution & Novelties

The video offers an original pedagogical approach to a difficult IMO problem, emphasizing the role of intuition and experience in mathematical discovery. It introduces a novel analogy (the cube-cutting puzzle) and a visual framework that makes the solution accessible. The reflection on AI and the quote from a DeepMind researcher add a contemporary dimension. The video does not present new mathematical results but contributes to the exposition of an existing solution with clarity and depth.

Pour aller plus loin :

  • Erdős–Szekeres theorem — The theorem referenced in the video, relevant to the combinatorial structure.
  • AlphaProof — DeepMind’s AI system that solved four IMO 2024 problems, discussed as context.
  • Lean (proof assistant) — The formal proof language used in some AI attempts, mentioned in the video.

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Radar Profile

The scores are uniformly high, indicating excellent quality across all dimensions. The video is dense but well-structured, and the high technical level is matched by rigorous sourcing. The profile suggests a top-tier educational resource.

Reliability 9/10