Knuth’s Hamiltonian Problem Solved by Human‑AI Team
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- Prof. Donald Knuth titled the paper "Claude’s Cycles" after Claude found an odd‑m construction for his Hamiltonian decomposition problem in about an hour of guided exploration.
- Claude discovered an odd‑m construction for the base case m=3, yielding exactly 11,502 Hamiltonian cycles, of which 996 generalize to all odd‑m, and Knuth identified 760 valid “Claude‑like” decompositions.
- Dr. Ho Boon Suan used GPT‑5.4 Pro to produce a 14‑page proof for all even m≥8, with computational checks up to m=2000, completing the even case that Claude couldn’t finish.
- Dr. Keston Aquino‑Michaels combined GPT and Claude in a multi‑agent workflow to find simpler constructions for both odd and even m.
- Dr. Kim Morrison formalized Knuth’s odd‑case proof in the Lean proof assistant.
- The updated paper now presents a fully resolved solution to Knuth’s problem, integrating human insight, multiple AI systems, and formal verification.
Why it matters: Mathematicians gain a complete solution to Knuth’s Hamiltonian decomposition problem, while the broader research community sees a proof‑assistant‑verified workflow that blends multiple AI systems with human insight, marking a shift toward collaborative AI‑augmented mathematics and setting a precedent for future AI‑human theorem proving collaborations.
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