- SAT solvers crack math problems by searching billions of true-or-false combinations.
- Heule solved century-old problems like Schur Number 5 using SAT encoding.
- Some SAT proofs run millions of steps — too long for any human to read.
Marijn Heule solves math problems that have stumped mathematicians for 90 years or more. His secret weapon isn't advanced calculus or elegant geometry: it's a technique called satisfiability, or SAT, that breaks problems down into billions of true-or-false statements and lets computers search for solutions humans might never find.
As John Pavlus reports in Quanta Magazine, Heule, a computer scientist at Carnegie Mellon University's Institute for Computer-Aided Reasoning in Mathematics, has cracked stubborn problems like Schur Number 5 and Keller's conjecture in seven dimensions using this approach.
A SAT solver is a computer program that determines if a Boolean logic formula can be satisfied.
Key figure
90 years
how long some of Heule's solved problems had stumped mathematicians
The Power of Simple Logic
SAT works like an enormous sudoku puzzle. Every cell can only be true or false–one or zero. The constraints define what combinations are valid. SAT solvers don't compute with these values; they search through possibilities, ruling out huge chunks of the solution space until they find an answer or prove none exists.
The technique belongs to symbolic AI, the "good old-fashioned" kind that uses hard-coded rules instead of neural networks.
What makes it significant is that SAT has already solved several problems for which only gigantic, computer-generated proofs are currently known – far too long for any human to read, sometimes running millions of steps.
Fields Medal winner Timothy Gowers once called one of Heule's proofs "the most disgusting proof ever" because humans couldn't parse it. But Heule argues that trust matters more than comprehension.
Understanding in mathematics is highly overrated.
Marijn Heule to Quanta Magazine
When Puzzles Meet Language Models
Heule's talent lies in translating math problems into formats SAT solvers can attack. He could solve 100-piece puzzles before he could walk, according to his parents.
Now he wants to remove himself from the equation entirely by teaching large language models to handle the translations.
The collaboration could work like this: LLMs suggest how to break a big theorem into smaller pieces. SAT solvers prove or disprove each piece, returning small counterexamples when something fails–the kind that reveal immediately why a statement doesn't work.
A formal proof checker like Lean verifies that all the pieces fit together correctly.
What is Lean?
Lean is a formal proof assistant: software that checks whether a mathematical argument is logically valid, step by step. Unlike a human reviewer, it cannot be fooled by a convincing-but-flawed argument – every logical step must satisfy strict rules before the proof is accepted.
"LLMs can do all of their bullshitting, but as soon as automated reasoning is able to say, 'OK, but this part is actually correct, and here's a proof,' this is actually more trustworthy than most of the pen-and-paper proofs out there," Heule said.
The Human Element Stays
When Heule has cracked open problems, mathematicians who spent years on those questions always worked alongside him. They provided insights, he encoded them for the solver to finish. Starting from scratch alone would have led nowhere.
He expects the future will look similar. LLMs might help more mathematicians learn to use SAT themselves, eliminating the need for middlemen like Heule.
But removing humans entirely would be a mistake–the creative intuition and conceptual reframing remain uniquely human strengths.
The magic comes from collaboration between human insight, machine search, and formal verification (Editor's note: somewhat like my AI writing process here at Science Reader).
That combination might finally crack problems that have resisted human reasoning for a century.
Fact Check: Claim-by-Claim Verification Verified
All claims verified against the Quanta Magazine source article, Heule's CMU page, and primary papers. Faithful condensation of the original reporting with accurate technical descriptions. Old inline fact-check block removed.
Commentary
- The article accurately conveys that SAT proofs are verified by proof checkers, addressing the trust issue despite human unreadability.
- "No human proof exists" for some SAT-solved problems is accurate for current knowledge but a human-readable proof could theoretically be found later.
- The LLM + SAT + Lean workflow is a research aspiration, not yet a fully realized system.
Sources used for verification
Academic/Peer-reviewed:
- Schur Number 5 paper - arXiv
- Keller's conjecture in 7D - arXiv
- Keller proof paper - CMU
Other reliable sources:
- Original Quanta article - Quanta Magazine
- Marijn Heule - CMU faculty page
Fact-checked by Perplexity Sonar Pro on 2026-03-14
