Grant Sanderson explores how machines learned to solve impossible math problems — and what it took to get there.
The most surprising thing about Alpha Geometry isn't that artificial intelligence can solve International Mathematical Olympiad geometry problems. It's that computers were already solving them without any AI at all.
In January 2024, Google DeepMind released a model that solved 25 out of 30 geometry problems from the IMO — the hardest high school math competition in the world. The buzz was immediate and enormous. But the storynobody's telling is what came before Alpha Geometry arrived: a brute-force approach using pure logic and equation solving already solved 18 out of 25 problems. That's nearly a bronze medal.
The Non-AI Breakthrough
The technique that achieved this was surprisingly simple. Two modules worked together. First, a "deductive database" — a hard-coded list of geometric rules derived from fundamental facts like "when two lines cross, opposite angles are equal" or "if two horizontal lines are parallel, the angles inside the Z are equal." Second, "algebraic reasoning" — the ability to solve systems of linear equations using basic linear algebra.
The researchers ran these modules alternately. Deductive database would deduce everything it could from known facts until it stopped. Then algebraic reasoning would pick up and solve any equations that emerged. Then deductive database again. They called this procedure "DD plus AR."
The results were impressive: DD alone solved 7 out of 30 IMO problems. DD plus human-coded heuristics pushed that to 14. Adding heuristics brought it to 18 — nearly a bronze medal at the IMO, achieved with no AI whatsoever.
But then the model hit a wall.
Where Logic Stops and Humans Begin
The fundamental limitation: DD plus AR couldn't make what mathematicians call "auxiliary constructions." This is the key insight that separates machines from humans.
Many hard geometry problems require drawing extra lines or shapes not present in the original diagram. To prove that angles in a triangle sum to 180°, for instance, you need to draw two parallel lines above and below the triangle — an auxiliary construction that creates new angles to work with. These constructions are where geometry becomes creative.
The problem is that there are infinitely many possible constructions at every step. A human mathematician might intuitively draw just the right two lines. A machine searching through all possibilities drowns in what mathematicians call "the infinite search space." This is exactly where artificial intelligence was needed.
How Alpha Geometry Works
DeepMind built a language model whose only job was to produce auxiliary constructions. The input was the problem statement and proof steps produced so far. The output was an extra point or figure on the diagram — essentially, a clever idea for how to proceed.
The system works by alternating between two minds: the creative brain (the language model) thinks of clever auxiliary constructions, while the logical brain (DD plus AR) deduces new facts from those ideas. The creative brain proposes another construction, and the logical brain deduces more consequences. This continues until either the problem is solved or time runs out.
But where did they get training data? There aren't many IMO geometry problems with solutions available online. So Alpha Geometry generated its own synthetic data — randomly plotting points and lines on a plane, using DD plus AR to deduce everything possible (various angles equal, various lines parallel), then erasing portions of the diagram to create new problems that would require auxiliary constructions to solve.
The result: 9 million synthetic proof examples. The most complex had 247 steps with two auxiliary constructions.
What This Means for Machines and Humans
Alpha Geometry solved 25 out of 30 IMO geometry problems — better than a silver medalist. But the deeper significance isn't about triangles or competition scores.
This represents machines that can think creatively while also reasoning logically. The combination of creativity and logic is by no means specific to geometry. It applies to problem-solving in science, medicine, engineering, and virtually any domain where humans need to think.
Critics might note that IMO problems represent a narrow slice of mathematical ability — solving competition problems doesn't necessarily translate to broader mathematical insight or real-world problem-solving. The auxiliary constructions Alpha Geometry generates are also fundamentally different from the intuitive insights that human mathematicians bring to bear.
"The true gems in geometry always feel like adding in the right idea seemingly came out of left field."
Bottom Line
Sanderson's deep dive into Alpha Geometry reveals something remarkable: non-AI approaches already solve nearly three-quarters of the hardest geometry problems in the world. The AI component isn't about solving what's impossible — it's about solving what was always possible but required creative insight humans took for granted. This is machines not just calculating, but constructing. And that's a glimpse into a future where AI doesn't just assist human reasoning but emulates it.