AI-assisted proof establishes optimal packing of 11 squares
claude
| Source: HN | Original article
An AI‑assisted proof released on Sep 29 2026 confirms the optimal packing density for 11 squares at about 3.877084, validating Walter Trump’s 1979 arrangement.
A formal proof confirming that Walter Trump’s 1979 arrangement of eleven unit squares is the tightest possible has been completed with AI assistance. The proof, verified in the Lean 4 proof assistant, checks all 7,920 Lean modules without a single admission, establishing the optimal side length at approximately 3.877084. OpenAI’s Astra and Anthropic’s Claude are credited alongside human collaborators for supplying the computational geometry insights that bridged heuristic packing searches and rigorous verification.
The result settles a conjecture that has lingered for 47 years, closing a long‑standing gap between experimental packing configurations and mathematically proven optimality. By demonstrating that AI can generate and certify the intricate geometric arguments required for such a proof, the work showcases a new tier of machine‑human partnership in mathematics. It also highlights the growing maturity of formal proof environments like Lean, where large‑scale verification can be automated without sacrificing certainty.
The breakthrough follows OpenAI’s recent release of hundreds of mathematical preprints, underscoring a rapid expansion of AI‑driven research in pure mathematics. Looking ahead, the community will watch whether similar AI‑augmented methods can tackle larger square‑packing cases, higher‑dimensional analogues, or other combinatorial geometry problems that have resisted traditional proof techniques. Success could accelerate the formal verification pipeline, making AI an indispensable tool for turning conjecture into theorem across mathematics and its engineering applications.
Sources
Back to AIPULSEN