The Last IMO Problem AI Could Not Solve

Study Guide

About This Video

A 51:55 video from 3Blue1Brown (published 2026-09-18) on the last IMO problem AI could not solve: Problem 6 from the 2025 International Mathematical Olympiad. The host says the video is "mostly not about AI" and is "a deep dive into a very beautiful problem," followed by an argument about what AI changes for mathematics. Timestamps in the table of contents are estimated from the transcript, so expect a few seconds of drift.

Summary

In 2025, over 600 teenagers competed in the International Math Olympiad on Australia's Sunshine Coast. Problem 6 was the hardest by far, and less than 1% of students earned full marks on it. The host walks through the problem step by step: he finds a construction, then proves nothing beats it. He closes by arguing that what mathematics really needs is a motivated explanation, an account that makes a proof feel almost obvious, and that it deserves the academic credit proofs have always received.

AI and the IMO, 2021 to 2026

  • 2021: many people saw the IMO as "a different ballgame entirely." It mixes rigor and creativity, and the search space of logical moves in a proof is effectively infinite.
  • 2023: the host recalls being asked on a podcast whether an AI getting gold at the IMO would "just be AGI."
  • 2024: Google DeepMind's AlphaProof answered four of six questions. The asterisk: humans had to translate the problems from English into Lean, a language where proofs are programs.
  • 2025: several organizations tried the test, some in Lean and some purely in natural language. Models answered every question except Problem 6.
  • 2026: simply prompting publicly available reasoning models was enough to get correct answers on all six problems. The host says the IMO now "feels like ancient history as a benchmark" as AI has moved into research math.

The Problem

  • Take a 2025x2025 grid of unit squares. Matilda places rectangular tiles along grid lines, with no overlaps.
  • Each row and each column must have exactly one unit square not covered by any tile. Mark these gaps with X's, so there are 2025 gaps in total.
  • Find the minimum number of tiles. The IMO needs more than the number: it needs a construction and a rigorous proof that nothing smaller works.
  • On a 10x10 grid the host shows tilings with 21, 17, and 16 tiles. The simple diagonal construction uses 2n - 1 tiles, which is "unlikely to be what the IMO is asking us for."

The Solution Intuition

  • The cube detour: cutting a 3x3x3 cube into 27 unit cubes with rearrangement needs six slices. The inner cube has six faces and one slice can free only one of them. The lesson is that the insight is about proving the number optimal, not about finding a clever cutting strategy.
  • Edges to tiles: every edge of an X square must touch a tile, and since no two X's share a row or column, each side of a tile touches at most one X.
  • Conjecture: an efficient tiling has as many tiles as possible touching four X's, placed at the tiles' corners in a "windmill" pattern.
  • Construction: square tiles of side k fit together into a slightly tilted pattern on a grid of side k squared. Since 2025 = 45 squared, k = 45. The host notes this is also the pattern on the floor of the Sunshine Coast airport.
  • Count: (k - 1)^2 interior tiles plus 4(k - 1) boundary tiles gives k^2 + 2k - 3. With k = 45 this works out to 2112 (my arithmetic from the host's formula; the transcript does not state the number).

The Proof, Step by Step

  1. Highlighting all edges of the X's gives only a weak bound: at least k^2 - 1 tiles. Highlighting just one direction, such as right edges, also gives k^2 - 1 and misses tiles clustered on one side.
  2. Use symmetry instead. Draw one path through the X's along a longest increasing subsequence and another along a longest decreasing subsequence. They divide the grid into four regions. Highlight right edges in the right region, top edges in the top region, and so on, with two edges for X's on the boundaries.
  3. No tile can touch two highlighted edges, so the number of tiles is at least the number of highlighted edges. That count is k^2 plus the two path lengths, minus 4, with a correction when the paths cross.
  4. It remains to show that the two path lengths sum to at least 2k. By the arithmetic mean versus geometric mean inequality, this follows if the product of the longest increasing and longest decreasing subsequence lengths is at least n.
  5. That product claim is the Erdos-Szekeres theorem. The host proves it by labeling each term with a pair (longest increasing ending here, longest decreasing ending here). The pairs are all distinct, so the bounding box holds at least n lattice points.

Patience and Beauty

  • Patience: the host asked Thang Luong, a director of research at Google DeepMind, why the model failed. Luong's answer: "We didn't really have a way to teach the model to be patient." The model did not take time to get a feel for the problem before trying to solve it.
  • Beauty: the host adds his own ingredient. Models, "at least as I've interacted with them," do not seem to have a great sense of what makes a problem or strategy beautiful, and finding this solution benefits from appreciating beauty.
  • Experience: a recurring line is that "what can look like genius is usually the residue of experience," which is why the cube puzzle comes first.

The Closing Argument: Motivated Explanation

  • The proof already existed, including on YouTube, so the host and his collaborator asked what new thing a video could add. Knowing a proof is "only one very small part" of understanding.
  • He proposes the name motivated explanation for a narrative by which a viewer feels the proof is almost obvious, even though only six students solved the problem.
  • Mathematics has always been about furthering human understanding. Proofs and open problems act as a guiding light, so proofs generated without human understanding "defeat the point."
  • Because proof is so ingrained as the proxy for academic credit, he sees a "stark need" for a new way to measure whether published work advances understanding. His proposal: formalize motivated explanation and give it the same academic status and credit proof has had. He points to a guest post on Terry Tao's blog, also recorded as a video on the channel.
  • He ends with a quote from a supporter's friend who solved the problem: the question's value is that it warmed their heart, and "like a good book or a touching song, the value here is human."

Reflect & Discuss

  • The host says the patience of understanding before solving was hard to teach a model. Where do you rush to an answer before you understand the question?
  • What is the difference between knowing that a proof works and knowing how you would have come up with it? Where have you felt that gap in your own field?
  • If AI can produce correct results without human understanding, what should schools and journals credit instead? Is "motivated explanation" a workable answer?
YouTube