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

twitter.com

March 28, 2026

1 min read

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60/100

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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Community Sentiment

Mixed

Positives

AI's ability to discover new mathematical concepts highlights its potential in advancing fields like mathematics, which could lead to groundbreaking discoveries.
The idea of using AlphaGo-style reinforcement learning on Lean syntax trees suggests a promising direction for AI's capability to tackle complex mathematical problems over longer timescales.

Concerns

The comment about AI winning a Fields Medal before managing a McDonald's reflects skepticism about AI's practical applications in everyday tasks, indicating a gap between theoretical and practical capabilities.