Claude's Cycles: Solving Hamiltonian Decompositions with AI

In this technical note, Don Knuth details how an advanced artificial intelligence, Claude Opus 4.6, solved a long-standing mathematical conjecture regarding Hamiltonian cycles in specific directed graphs. The problem involved decomposing the arcs of a complex multidimensional digraph into three distinct paths that visit every vertex exactly once. Through a collaborative process of prompting and iterative coding, the AI identified a successful "fiber decomposition" pattern that works for all odd values of the variable m. While the AI struggled to generalize a solution for even values, subsequent experiments with other models suggests those cases may also be solvable. Ultimately, Knuth celebrates this as a significant milestone in automated deduction and creative problem-solving within the field of computer science.

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