Donald Knuth opened a new paper on February 28, 2026 with two words: "Shock! Shock!" The 87-year-old Stanford computer scientist, author of The Art of Computer Programming and arguably the most credentialed skeptic of AI hype currently alive, had just learned that Claude Opus 4.6 solved a directed Hamiltonian cycle decomposition problem he'd been working on for weeks. He named the resulting paper Claude's Cycles. He coined the term "Claude-like decompositions" as formal mathematical nomenclature. And he closed with a pun.
That's the headline. The details are more interesting.
From "studying the task of how to fake it"
In April 2023, Knuth gave ChatGPT a 20-question exam, watched it fabricate the chapter structure of a Leon Uris novel, and called the whole exercise "studying the task of how to fake it." He told Stephen Wolfram the topic was "emphatically not for me." So the opinion reversal here isn't trivial. He isn't saying AI is good now. He's saying he'll "have to revise" his views "one of these days." That's a careful man making a careful concession, which is the most you'd expect from Knuth.
The problem itself involves a directed graph with m³ vertices and three arcs from each vertex. The question: can the arcs always be decomposed into three directed Hamiltonian cycles, for all values of m greater than 2? Knuth had solved the m=3 case. His colleague Filip Stappers found solutions empirically for m up to 16. But nobody had a general construction.
Thirty-one explorations, about an hour
Stappers fed the problem to Claude cold, using Knuth's exact original wording. What followed reads less like a chatbot session and more like a messy research notebook. The model reformulated the problem algebraically, tried linear and quadratic constructions, attempted brute-force depth-first search (abandoned as "too slow without good pruning"), and introduced a "serpentine pattern" that Knuth recognized as the classical modular m-ary Gray code. Claude didn't know that's what it was. It just found the structure and named it.
The real pivot came at exploration 15, when Claude introduced what it called a fiber decomposition, mapping the three-dimensional graph onto a layered structure using (i+j+k) mod m. That reformulation opened a new search space. Simulated annealing followed, which found specific solutions but not a general rule. After exploration 25, the model apparently concluded: "SA can find solutions but cannot give a general construction. Need pure math."
Exploration 31 was the breakthrough. Revisiting a simulated annealing solution from exploration 20, Claude noticed that the permutation choices in each fiber depended on only a single coordinate. That observation generated a concrete Python program. Stappers tested it for all odd m between 3 and 101. Every case worked. Knuth then wrote the formal proof himself, and found there are exactly 760 valid "Claude-like" decompositions total. Claude had found one.
What it didn't solve
The even case is still open. When Stappers asked Claude to continue working on even values of m, it "seemed to get stuck. In the end, it was not even able to write and run explore programs correctly anymore." Worth holding onto when reading the more triumphant takes. The model degraded. It found one valid construction out of 760 for odd m and then couldn't function well enough to make progress on even m.
Knuth is also honest about what the session required. Stappers had to repeatedly instruct the model to document its progress after each exploration. Errors interrupted the session and caused some earlier results to be lost. And the rigorous proof, the part that actually makes this mathematics, was Knuth's work, not the model's.
But "collaborative" still covers real ground here. Thirty-one systematic explorations in about an hour, finding a construction that generalizes across infinitely many cases of a problem that had no general solution. Knuth closes the paper with characteristic warmth: "I think Claude Shannon's spirit is probably proud to know that his name is now being associated with such advances. Hats off to Claude!" When the person who wrote the bible of computer science says that, after calling a previous AI system a faker to Stephen Wolfram's face, it's probably worth paying attention.
