AlphaGeometry
25/30
Olympiad geometry problems solved, out of 30
14: symbolic engine alone
The AlgoGens Library
Real systems where algorithms and AI are woven into one hybrid. 40 of them, each checked against its sources.
The ladder
Is it a script?
Flies the same route every time.
Is it a skill?
Improvises, follows instructions.
Is it a harness?
Model plus machinery: tools, memory, guardrails.
It's an AlgoGen.
Algorithms and AI woven into one hybrid. A harness can be an AlgoGen. An AlgoGen can go much further.
The people building AI keep saying the same thing
“I think we are all ... not dreaming big enough ... because we're all kind of used to last year's models.”
Sam Altman, OpenAIADIPEC, Abu Dhabi, Nov 2026 “They may be unrealistically utopian. But the important thing is to have an ambitious vision, to be willing to dream big and try things out.”
Dario Amodei, AnthropicMachines of Loving Grace (essay), Oct 2024 “So I think in my mind, there's this activity of just stepping back and thinking about what's a slightly more ambitious thing I can ask my AI to do than it's done for me to date. Let's see if it works.”
Greg Brockman, OpenAISilicon Valley Girl podcast, Sep 2026 “The fundamental change that we have to make, and I have to make, is whatever expectations I have for the company, you've got to increase it by about 100x.”
Jensen Huang, NVIDIAMilken Global Conference, May 2026 “One of the things I tell a lot of people is probably the place where the most interesting products are, are where you've made the phase change from impossible to hard.”
Kevin Scott, MicrosoftNo Priors podcast, May 2023 “I actually need to go 10 steps further because the technology enables me, and resetting my level of ambition I think is something that I've also spent a bunch of time thinking about.”
Logan Kilpatrick, Google DeepMindSequoia Training Data podcast, Jun 2026 “When execution is abundant, the only bottleneck is the scope of your own ambition”
May Habib, WriterTEDAI San Francisco, Oct 2025 “The biggest prize is in figuring out how you can keep ascending the layers of abstraction...”
Andrej Karpathypost on X, Feb 2026
Aim higher.
So who is already doing it, and what did they get?
Four results from Google DeepMind and Microsoft Research
Here is what came out. How they got there comes next.
FunSearch
512
points in a cap set in 8 dimensions, larger than any known before
- Lower bound on cap set capacity
- 2.2180previous best (Tyrrell, 2023) to 2.2202FunSearch
AlphaEvolve
49Strassen, 196948AlphaEvolve
Multiplications to multiply two 4x4 complex-valued matrices
For complex-valued matrices, nobody had beaten Strassen's 49 in 56 years, by the paper's account.
0.7%of Google's fleet-wide compute, recovered on average by a scheduling rule it foundGoogle DeepMind's own report, not independently replicated.
AutoVerus
Logos belong to their owners and show who built each result. AlgoGens is not affiliated with or endorsed by either lab.
How? All four did the same thing.
What do these four have in common?
Take a guess.
A bigger model?
Not size alone. In the same seat, suggesting points to the same engine, GPT-4 solves 15 of 30 and AlphaGeometry's own 151-million-parameter model, trained for exactly that job, solves 25. AutoVerus runs GPT-4o as it comes. AlphaEvolve does use frontier Gemini models, and its paper finds a stronger model helps, but its ablations also find that taking away the evolution loop hurts.
Sources: AlphaGeometry, AutoVerus, AlphaEvolve
A cleverer prompt?
AlphaGeometry has no prompt at all: its model was trained from scratch on synthetic proofs. FunSearch's prompt is two of its own earlier programs, picked by score, plus the header of the function to write. The loop writes the prompt.
Sources: AlphaGeometry, FunSearch
More human data?
AlphaGeometry trained on 100 million synthetic proofs and zero human ones. FunSearch used a pretrained code model that was never trained on cap sets. AutoVerus uses GPT-4o as it comes, with no extra training.
Sources: AlphaGeometry, FunSearch, AutoVerus
Something else
Yes. Each one pairs a model that proposes with an algorithm that judges, in a loop where each judgment steers the next try. The algorithm judges in one of two ways: it scores (FunSearch, AlphaEvolve: better and better answers, with no proof that one is the best) or it proves (AlphaGeometry and AutoVerus: every accepted answer is proved correct against its stated goal or spec).
Sources: AlphaGeometry, FunSearch, AlphaEvolve, AutoVerus
AlphaGeometry
The model proposes: New points and lines for the figure
An algorithm judges: A symbolic engine derives what followsJudges by: proves
The judgment steers the next try: The bigger figure goes back to the model
FunSearch
The model proposes: Small Python programs that build a solution
An algorithm judges: An evaluator runs each one and scores itJudges by: scores
The judgment steers the next try: The best programs go into the next prompt
AlphaEvolve
The model proposes: Edits to existing code, from Gemini
An algorithm judges: Evaluators run each version and score itJudges by: scores
The judgment steers the next try: High scorers are kept and shown again
AutoVerus
The model proposes: Proof notes for Rust code, from GPT-4o
An algorithm judges: The Verus verifier proves it or lists errorsJudges by: proves
The judgment steers the next try: The reported errors drive the next repair
Take one half away
- Model alone
- Solved by the engine
- Engine alone, plus what the model adds
| Setup | Problems solved, of 30 |
|---|---|
| GPT-4 alone, writing full proofs (Model alone) | 0 |
| Symbolic engine alone (Solved by the engine) | 14 |
| Engine plus GPT-4 suggesting points (Engine alone, plus what the model adds) | 15 |
| Engine plus hand-written human rules (Solved by the engine) | 18 |
| AlphaGeometry: engine plus its own small model, in a loop (Engine alone, plus what the model adds) | 25 |
average IMO gold medallist: 25.9 of 30 (Fig. 2)
Each half alone stalls. Bolting a famous chatbot onto the engine adds one problem. A small model trained to propose inside the loop adds eleven.
Thirty geometry problems is a small test, and the gold medallist average is still a hair above AlphaGeometry. A later preprint pushes the symbolic side further, to 21 with no language model, and to 27 with AlphaGeometry added, on a partly re-translated set.
The pattern has a name
The name comes from a 2024 book by the author of this site. It defines it in one line:
Introduction to Algogens, section 1.2.1
“a cutting-edge framework integrating the precision of algorithmic processes with the creativity and adaptability of generative artificial intelligence”*
- "precision of algorithmic processes" is the algorithm half, the part that checks and judges.
- "creativity and adaptability of generative AI" is the model half, the part that proposes.
The test: the algorithm does real work, searching, deducing or proving. If it only stamps pass or fail on the model's answer, that is a harness, not yet an AlgoGen.
AlgoGens are a way of thinking built to answer that call.
Big labs, big problems. Here is the same machine on a problem small enough to check by hand, in its strictest form: the model writes the rules and a solver searches every roster, so it can prove which one is best.
One request, four ways
An algorithm does the sums exactly but can't read the words. A language model reads every word but does the sums by feel. A harness checks the model's guesses until one fits, so the rules hold, but a check only judges the guesses it is shown, so it can't prove no better answer exists. Pick a request and run it a few times.
Nurse shifts
Exhibit AWhat it shows: search beats luck
The request
next week's roster please. two nurses every friday night, yossi's off wednesday for his kid's play, nobody does a night and then the next morning, and keep everyone around 40 hours, no overtime, and noa and omar would love the weekend off if you can swing it
Five nurses, 12-hour day ☀ and night ☾ shifts, at most one shift a day each. Shown small; a real ward has 30 nurses and a month.
| Approach | Deterministicsame rules in, same answer out | Smartreads plain words | Keeps your rulesevery rule, every run | Proves it's the bestruled out every other option | This run |
|---|---|---|---|---|---|
 | Yes | No | No | Yes |
Why A scheduling tool fills in the fields it was built with: cover, rest, hours. "His kid's play" isn't one of them, so Yossi lands on Wednesday. Every sum is right. It answered a slightly different request. |
 | No | Yes | No | No |
Why A language model writes the likeliest next words. "Everyone's at 36 hours" sounds right, so it says it. Nothing added up Dana's shifts. Ask again and you get a new roster with a new slip. |
 | No | Yes | Yes | No |
Why Here the model guesses and a checker tests each guess against the rules, so whatever comes back fits. In this harness the checker only judges the guesses it's shown. Of the 3,923,568 rosters that fit, 246,648 give Noa and Omar the weekend off. Nothing in this loop goes looking for them. |
 | Yes | Yes | Yes | Yes |
Why In this AlgoGen, each part does the job it's good at. The model reads the request and writes it down as rules. A solver searches the rosters those rules allow, skipping whole branches that can't win, and keeps the best one. Same rules in, same roster out. The rules the model wrote
Ways to fill the week: 12,207,031,250. Keep every rule: 3,923,568. Weekend off for both: 246,648. |
Flight to a Tuesday talk
The request
talk in london tuesday 9am, flying from tel aviv monday. cheapest that works: under $600, no layover over 2 hours, and i need 4 hours in a hotel bed before the talk
The hotel is 1¼ hours from Heathrow, up at 7:30. Shown: 6 flights; a real search has hundreds.
| Approach | Deterministicsame rules in, same answer out | Smartreads plain words | Keeps your rulesevery rule, every run | Proves it's the bestruled out every other option | This run |
|---|---|---|---|---|---|
| Yes | No | No | Yes |
|
| No | Yes | No | No |
|
| No | Yes | Yes | No |
|
| Yes | Yes | Yes | Yes |
|
Stock sale for a down payment
The request
need $20k in my account after tax for a car down payment this week. sell whatever costs me the least tax, but leave my tech stocks alone, i think they're going up
Tax is 25% of the gain; a loss lowers it. Shown: 6 holdings; a real account has dozens of lots.
| Approach | Deterministicsame rules in, same answer out | Smartreads plain words | Keeps your rulesevery rule, every run | Proves it's the bestruled out every other option | This run |
|---|---|---|---|---|---|
| Yes | No | No | Yes |
|
| No | Yes | No | No |
|
| No | Yes | Yes | No |
|
| Yes | Yes | Yes | Yes |
|
Why the checks can't tell you it's the best
Checking one roster against the rules takes a computer a moment. Saying no better roster exists means ruling out every other one, and doing it without trying them all.
Check one roster
- Each nurse: at most one shift a day
- Cover
- Yossi off Wednesday
- Rest
- Hours
One look per rule.
Find the best
12,207,031,250
ways to fill the week
Watch the options explode
Shown for 5 nurses over 1 week.
12,207,031,250
ways to fill the roster
A computer checking a billion rosters a second would need about 12 seconds.
Shown: every way to staff each shift, before the other rules. The rules cut it a lot: for this week, to 3,923,568.
What the search proves
Best possible: 4 of 5
No roster that keeps the rules does better. The search accounted for all 12,207,031,250 ways to fill the week after visiting 1,214 partial rosters. A branch is dropped when it breaks a rule, or when even its best case can't beat the best roster found so far. Of the 3,923,568 rosters that fit, 2,952 reach 4.
Your turn: be the harness
You are a guess-and-check harness. Same five nurses, same rules: one shift a day each, plus the four checks below. New wish: everyone wants their three shifts on three days in a row, for four days off in a row. Tap a cell to change it. The checks below update the moment you tap.
DanaYossiNoaOmarLea
Back to back: 2 of 5 nurses
- Cover: holdsevery shift is covered
- Yossi off Wednesday: holdsYossi is off Wednesday
- Rest: holdsno night followed by the next morning
- Hours: holdsnobody works more than 3 shifts
Is that the best? The checks can't tell you. They only judge the roster in front of them.
Chatbot, harness, AlgoGen
On every step a model proposes. What changes is the job the algorithm does in the loop.
Chatbots and skills
No automatic check. You read the answer and judge it.
ChatGPT, Claude or Gemini as plain chat. A skill is saved instructions the model reads.
Algorithm's job:none in the loop
Harnesses
Tools around the model, plus checks it can run, like tests that pass or fail.The model writes the rules. A solver searches them.
Claude Code, Codex, Cursor.Like the nurse rosters above.
Wikipedia: an agent harness is the software around a model. Agent = model + harness.
If the model writes the rules and a solver does the search, the harness becomes an AlgoGen.
Now this loop is an AlgoGen.
Algorithm's job:checkssearches or proves
AlgoGens
The algorithm does the search or the proof. The model proposes.
FunSearch and AlphaEvolve search and score. AlphaGeometry deduces and proves.
Algorithm's job:searches or proves
A model proposes on every step
- A harness can load skills.
- When the model writes the rules and a solver searches them, the harness becomes an AlgoGen, as in the nurse rosters above.
- An AlgoGen can go further: the algorithm runs the loop and calls the model, as FunSearch, AlphaGeometry and AlphaEvolve do.
Where AlgoGens go further
The systems in this library go further in 8 ways. Each comes with the systems that do it.
The generator is not an LLM
The generative part can be a diffusion model, a variational autoencoder or another model type.
6 systems do this:
AI at design time
AI is used offline to discover a solver. The deployed result runs without it.
10 systems do this:
Feedback retrains the generator
Judgment flows back into the generator's weights through reinforcement learning, fine-tuning or active learning.
11 systems do this:
The algorithm is the solver
The algorithm solves the problem. The model is one part of it, not a chat agent with tools around it.
18 systems do this:
A domain authority judges
A domain authority decides what passes: a proof assistant or verifier for formal results, a simulator or docking score for computed estimates that are not lab results.
18 systems do this:
Inverted control
The algorithm is in charge and calls the model as a subroutine, instead of the model deciding when to call tools.
23 systems do this:
Search over a population
The loop runs over many candidates at once: a population, an archive or a search tree.
26 systems do this:
The output outlives the model
What comes out is an artifact (an algorithm, a heuristic, a proof, a molecule) that is useful without the model.
35 systems do this:
Try to break it
These are the strongest objections we have heard, with straight answers.
Isn't this just a harness with tests?
A harness can be an AlgoGen, and the good ones often are. The line is what the algorithm does. In many harnesses it only stamps pass or fail on the model's answer and stops at the first one that fits. In an AlgoGen it does real work: it searches by score, like FunSearch, or through every option with a proof, like the nurse solver above.
Does the loop always win?
No. A loop is only as good as its judge. A later study found AlphaEvolve sometimes exploiting weak evaluators, and any score that can be gamed will be. The pattern pays off most where the judge is exact or hard to game: a proof checker, a verifier, a count you can trust.
Next year's model will do this alone.
Models will keep improving at proposing. But five nurses over one week already allow 12,207,031,250 possible rosters, and showing that none of them beats yours is a search, which a guess cannot settle. The labs with the biggest models are the ones building these loops.
Solvers choke on NP-hard problems.
In the worst case, yes, and nobody has a fix for that. In practice solvers prune huge parts of the search, and when they stop early they report how far from the best their answer might be.
What if the model writes the wrong rule?
This is the honest weak spot. Rules come back to you in plain words to confirm before anything runs. A verifier proves exactly what the spec says, so a wrong spec still passes; the AutoVerus authors state this limit about their own tool.
So it's operations research with a chatbot?
Partly, yes. Operations research has spent decades on the judging half, and the model makes it usable by someone who can describe the problem but cannot write the math.
Your examples are scripted.
The written answers are illustrations and labelled as such. Every count and the solver on this page run as real code. The four lab results link to peer-reviewed papers or, for AlphaEvolve, the developer's own report.
What Google DeepMind reports
These deployment results are Google DeepMind's own reports on its AlphaEvolve impact page. Each row links to that page. We have not reproduced them.
| Area | Partner | Result | Reported by |
|---|---|---|---|
| Genomics | PacBio (with Google Research DeepConsensus) | 30% reduction in variant detection errors (improvement to DeepConsensus, a Google Research model for correcting DNA sequencing errors) Stated as a reduction in variant detection errors; Google’s page does not give the baseline or benchmark. | Google DeepMind (source for Genomics) |
| AC Optimal Power Flow | Electricity grids (no operator named) | Feasible-solution rate of the team's trained Graph Neural Network rose from 14% to over 88% This is the network’s rate of feasible solutions, not a cost saving on the grid. | Google DeepMind (source for AC Optimal Power Flow) |
Show all 11 reportsShow fewer reports
| Area | Partner | Result | Reported by |
|---|---|---|---|
| Earth sciences | Google Earth AI | Overall accuracy of natural disaster risk prediction, aggregated across 20 categories such as wildfires, floods and tornadoes, increased by 5% Google’s page does not say whether the 5% is relative or absolute. | Google DeepMind (source for Earth sciences) |
| Quantum physics | Google Willow quantum processor | Suggested quantum circuits with 10x lower error than previous conventionally optimized baselines | Google DeepMind (source for Quantum physics) |
| Hardware design | Google (next-generation TPUs) | Used as a regular tool to optimize next-generation TPU design; a proposed circuit design was integrated directly into the silicon of next-generation TPUs No numeric result is given. | Google DeepMind (source for Hardware design) |
| Databases | Google Spanner | Reduced write amplification by 20% by refining Log-Structured Merge-tree compaction heuristics | Google DeepMind (source for Databases) |
| Compiler optimization | Google (compilers) | Provided insights for new compiler optimization strategies that reduced the storage footprint of software by nearly 9% Google says AlphaEvolve “provided insights” for these strategies; the saving comes from the strategies, not directly from AlphaEvolve. | Google DeepMind (source for Compiler optimization) |
| Financial services | Klarna | Optimized one of its largest transformer models, doubling its training speed while improving model quality No quality metric given. | Google DeepMind (source for Financial services) |
| Logistics (routing, Traveling Salesman Problem) | FM Logistic | 10.4% improvement in routing efficiency over the previous heavily optimized solutions, saving over 15,000 kilometers of distance travelled annually | Google DeepMind (source for Logistics (routing, Traveling Salesman Problem)) |
| Advertising and marketing | WPP | 10% accuracy gains over their competitive manual model optimizations (refining AI model components on campaign data) Google’s page does not define the accuracy metric. | Google DeepMind (source for Advertising and marketing) |
| Computational materials and life sciences | Schrödinger | Roughly 4x speedup in both Machine Learned Force Field (MLFF) training and inference | Google DeepMind (source for Computational materials and life sciences) |
Not just one lab
We track 40 systems from 24 lead organizations, across 11 fields. The same pairing of a model that proposes and an algorithm that judges keeps turning up.
Mathematics 3
Optimization 10
- EoHCity University of Hong Kong2024
- HSEvoHanoi University of Science and Technology2024
- ReEvoPeking University2024
- CALMCity University of Hong Kong2025
- CEoHKarlsruhe Institute of Technology2025
- OpenEvolveOpenEvolve community2025
- ShinkaEvolveSakana AI2025
- A-CEoHKarlsruhe Institute of Technology2026
- CoupleEvoKarlsruhe Institute of Technology2026
- T-GADEOsaka Metropolitan University2026
Proofs 8
Code verification 2
Chemistry 4
Materials 2
Engineering 1
Read all 40 AlgoGens
Each entry shows what the model proposes, what judges it, what steers it, its headline result and what it does not prove, checked against its sources.