claude hamiltonian-decomposition ai-research guided-exploration

Further human + AI + proof assistant work on Knuth's "Claude Cycles" problem

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March 28, 2026

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Summary

Claude found an odd-m construction for the open Hamiltonian decomposition problem, impressing Prof. Donald Knuth, who titled the resulting paper "Claude’s Cycles." The updated research reveals that for the base case m=3, there are 11,502 Hamiltonian cycles, with 996 generalizing to all odd-m, and 760 valid "Claude-like" decompositions identified.

Key Takeaways

Claude found an odd-m construction for Prof. Donald Knuth's open Hamiltonian decomposition problem in about an hour of guided exploration.
The updated paper reveals there are exactly 11,502 Hamiltonian cycles for the base case m=3, with 996 generalizing to all odd-m.
Dr. Ho Boon Suan used GPT-5.4 Pro to produce a 14-page proof for all even m≥8, while Dr. Keston Aquino-Michaels utilized GPT and Claude together for simpler constructions.
The problem is now fully resolved through a collaboration of multiple AI systems and human contributions, demonstrating a new mathematical ecosystem.

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