Lean Proof Verified Eleven-Square Packing Discovery
A formal computer verification has confirmed the optimality of a square packing configuration discovered in 1979.
Updated on Oct. 7, 2026 in Mathematics
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The optimality of Walter Trump's 1979 packing of eleven unit squares has been formally verified by a Lean proof checker. The project utilized AI models Astra and Claude to finalize the geometric argument.
Why it matters
This formal proof establishes mathematical certainty for a long-standing geometry problem by ensuring every logical step is valid within a computer-verified framework. The success highlights the increasing role of automated tools in solving complex, multi-step geometric proofs.
The proof confirmed that eleven unit squares can be packed into a square with a side length of 3.87708359002281. The geometric configuration features a cluster tilted at 40.182 degrees.
The players
Walter Trump
He is the researcher who first identified the optimal eleven-unit square packing configuration in 1979.
Lean
It is an interactive theorem prover and proof assistant that verifies the logical correctness of mathematical proofs.
Astra
This is an AI model developed by OpenAI that was utilized to assist in constructing the geometric proof.
Claude
This is an AI model developed by Anthropic that contributed to the mathematical formalization process.
The details
The project involved translating the complex geometric argument into the Lean proof language, which relies on a kernel to verify each deduction. Six initial placeholders that marked unproven steps in the draft were resolved to complete the formal verification.
Timeline
Walter Trump discovered the optimal packing arrangement in 1979.
OpenAI introduced the Astra model on August 1, 2026.
The formal Lean proof was published in a repository on October 6, 2026.
The Big Picture
This verification follows the pattern set by the jlevy/squares project, which catalogues and verifies complex geometric packing proofs. The success demonstrates a paradigm shift where AI-assisted formalization replaces manual checking for long-standing mathematical conjectures.
This development ensures higher reliability for geometric optimization, which underpins various applications in structural design and material science. As AI models become more adept at formalizing logic, similar verification processes could accelerate the certification of complex mathematical models.
The takeaway
The successful use of AI to finalize a 47-year-old geometric puzzle marks a significant milestone in automated mathematics. Researchers can apply these verification techniques to ensure the absolute correctness of other complex computational optimization problems.
Further reading
For more on the latest developments in proof theory and formal systems, explore the Mathematics archive.
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