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[Introduction] The "Hamiltonian Decomposition" problem has finally been cracked! The 88-year-old "Grandfather of Algorithms," Donald Knuth, has updated his paper again. Claude 4.6 and GPT-5.4 jointly solved the odd and even cases. Moreover, GPT-5.4 directly produced a 14-page paper without a single edit, setting the internet ablaze.
The 88-year-old master has finally filled the pit he dug for himself years ago!
Three weeks ago, the "Grandfather of Algorithms" and youngest Turing Award winner, Donald Knuth, was shocked by Claude: a long-standing algorithmic problem that had hung in the balance for years was actually solved by Claude Opus 4.6.
At the very beginning of the paper, he exclaimed, "Shocking, shocking!"
Paper Address: https://cs.stanford.edu/~knuth/papers/claude-cycles.pdf
However, further research revealed that there are actually 760 similar decomposition methods; Claude had only found one of them.
It had only conquered the "fortress" where m is odd; for cases where m is even, there was still no general solution.
The updated paper shows that tremendous progress has been made on this difficult problem!
GPT-5.4 Pro took the baton from Claude, producing a 14-page paper directly for all even numbers m ≥ 8, and verified cases up to m = 2000 through computation.
Not only that, after GPT and Claude collaborated via a multi-agent workflow, they found simpler construction methods for both odd and even m.
Others have also used the Lean language to formalize Claude's proof regarding the odd cases.
Thus, the "Hamiltonian Decomposition" problem has been thoroughly solved.
From Claude 4.6 to GPT-5.4, plus the combined efforts of many industry leaders, the pit dug decades ago has finally been filled.
At the end of the paper, the old master remarked—
"We indeed live in very interesting times. May the Force be with you."
The 88-Year-Old Algorithm Pioneer Dug a "Huge Pit"
For a long time, in combinatorial mathematics, the Hamiltonian Path has been a fortress that is easy to defend but hard to attack.
Simply put, it requires finding a closed loop in a complex graphical network that passes through every node exactly once without repetition.
The "Hamiltonian Decomposition Problem," on the other hand, involves perfectly decomposing a graph into multiple such loops. This is not just a game of computational volume, but an extreme test of mathematical construction capabilities.
This pit was dug by Knuth himself.
While writing his monumental work on computer science, "The Art of Computer Programming" (TAOCP), Hamiltonian Decomposition remained a "patch" that constantly worried him.
This problem has been pending for decades, described in technical terms as follows:
Previously, the academic community was unable to provide a complete solution covering both odd and even cases.
As the number of nodes increases, the search space explodes exponentially. Faced with the darkness of such depth, the human brain often feels physiologically powerless.
Over the past thirty years, countless geniuses have attempted to fill this pit, but most have failed at the final defense line of the "complete odd-even solution."
Until this spring of 2026, Knuth decided to change his weapon.
Even m, Is There a Solution?
Last time, after 31 explorations, Claude Opus 4.6 finally proposed a simple rule—
s = (i + j + k) mod m
Based on the values of s, i, and j, a decision is made whether to increment i, j, or k. The specific rules are as follows:
If s=0, the direction of movement is determined by the value of j. If 0 < s < m−1, it is determined by the value of i. If s=m−1, another rule applies.
As a result, Claude verified via program that when m=3, 5, 7, 9, 11, the paths all held true.
As can be seen, Claude only solved the case where m is odd. As for the problem where m is even, a true solution had not yet been derived.
Until March 3rd, when Filip Stappers wrote to the old master saying, "There is more to this story."
Stappers had Claude Opus 4.6 calculate again for m being even for about 4 hours. Finally, there were some clues, but no complete solution.
Ultimately, Claude established a local fiber construction similar to the odd case, and then ran searches to repair and perfect it.
In the final stage, it spent most of its time on "accelerating the search" speed rather than finding a true construction method.
It ran many programs, attempting to use simulated "annealing" or "backtracking" algorithms to find a solution.
At Stappers' suggestion, Claude was asked to use ORTools CP-SAT (part of Google's open-source toolkit, featuring AddCircuit constraints) to solve it, and a miracle happened.
The current program can now directly output results in just a few seconds!
Immediately following this, on March 4th, Ho Boon Suan, a friend from Singapore, brought even more shocking news.
He utilized code generated by gpt-5.3-codex to successfully achieve decomposition for even m ≥ 8.
To verify reliability, he tested all even numbers m between 8 and 200, as well as some random even numbers between 400 and 2000, and the results were flawless.
It is worth noting that when m=2000, that is a massive graph structure with 8 billion vertices!
If one relied purely on human effort to manually calculate and prove its correctness, it would be sheer fantasy.
Almost simultaneously, Kim Morrison from the Lean community acted with incredible speed.
He formalized the verification of the proof regarding the correctness of Claude's construction and posted it online promptly on March 4th.
Mathematical Geniuses Gather for Research
Another anonymous researcher named "Exocija" found a new construction suitable for odd m.
From a purely computational perspective, this is likely the most concise solution currently available, although its proof may not be the simplest.
In the C language program, one only needs to replace specific lines with extremely concise logical code to obtain an effective decomposition.
Moreover, almost every step cleverly utilizes the identity permutation "012".
if (s == 0) d = (j == m - 1 ? "201" : "021");
else if (s == m - 1) d = (j == 0 ? "102" : "120");
else d = "012";How did he do it? The answer is: cross-model collaboration.
Exocija constantly pasted text back and forth between GPT-5.4 and Claude 4.6 Sonnet, leveraging their different thinking dimensions to inspire each other, eventually piecing together the complete proof.
Zero Edits: GPT-5.4 Directly Produces a 14-Page Paper
Regarding the construction problem for even m, the real climax is yet to come.
Since the algorithmic patterns generated by gpt-5.3-codex were too complex, Ho Boon Suan decided to issue an ultimate command to GPT-5.4 Pro:
"Your task is to strictly prove that the algorithm given previously, when m is an even number ≥ 8, indeed always produces three cycles each of length m³."
"It would be best to explain in depth why this algorithm works and explore whether there is a simpler construction method."
Who would have thought that GPT-5.4 Pro would directly hand in a stunning answer—
A 14-page academic paper with beautiful layout and rigorous logic.
From "Abstract" to "Conclusion," the structure is complete, with seamless transitions.
Moreover, it adopted the TeX standard. Since Knuth himself is the inventor of TeX, the AI seems to be paying tribute to him using this language.
Most importantly, the paper passed the inspection of the Lean formal verification tool.
In Ho's own words, this was entirely a feat accomplished independently by GPT-5.4 Pro; he didn't need to change a single punctuation mark!
This means its logical chain is an "absolute truth" in the mathematical sense.
AI "Fighting Itself": Claude + GPT Achieve Perfect Proof
The culmination of this story is Keston Aquino-Michaels.
He not only found another effective decomposition for the odd m case but also provided an elegant decomposition for the even m case, far simpler than previous methods.
In addition, he unearthed a relevant reference paper previously overlooked by Knuth (i.e., the last reference in the figure below).
Preprint: https://arxiv.org/abs/2203.11017
Most wonderfully, he also meticulously analyzed this joint interaction mode, noting its potential significance for how we address and solve new problems in the future.
Full Report: https://github.com/no-way-labs/residue/blob/main/paper/completing_claudes_cycles.pdf
Open Source Project: https://github.com/no-way-labs/residue
Simply put, Keston Aquino-Michaels did not simply ask the AI questions; instead, he constructed a sophisticated "collaborative workflow".
This was more like a collaborative exercise spanning carbon and silicon, a close cooperation between Claude, GPT, and humans.
Among them, two Agents operated independently using the same "Residue" prompt.
Structured exploration prompts used by the two agents.
But each played to its strengths:
Agent O: Solved the odd case with 5 explorations (symbolic proof).
Agent C: Found specific solutions for m=4, 6, 8, 10, 12 (data).
However, the two Agents did not converse directly; they communicated via an Orchestrator—data and tools were all passed through the commander (human-guided Opus 4.6).
The Orchestrator needed to judge "when to transmit, what to transmit, and in what format," something the two Agents could not accomplish on their own.
For instance, when Agent O got stuck at m=10 in the even case and couldn't proceed, the Orchestrator passed Agent C's solution to Agent O; upon receiving it, Agent O immediately identified the pattern: an "m−2 layer batch layer" plus a "2-layer repair layer."
Ultimately, the "complete odd-even solution" that had plagued humanity for decades was thoroughly pierced through the fierce engagement of these two AI agents.
Humans Define the Battlefield, Machines Fill the Abyss
This "pit-filling" marks a complete paradigm shift in scientific research.
The role of the scientist has changed. For instance, Knuth is no longer the craftsman calculating every line of code on paper; he defined the boundaries of the problem, designed the logic for verification, and then commanded the AI to fill that black hole of trial and error.
The research paradigm has changed. Humans only need to define the boundaries, while AI fills the abyss.
The most precious ability for mathematicians is no longer computational power, but rather "the intuition to pose questions" and "the aesthetic to verify answers."
AI is responsible for finding paths through infinite trial and error, while humans are responsible for confirming at the destination whether this is indeed the truth we seek.
Who is next?
When an 88-year-old algorithm titan begins using AI to fill pits, we must realize: the methodology of mathematical research is undergoing an irreversible turning point.
This is not just Knuth's victory; it is an "external upgrade" of human intelligence.
In an era of "machines fighting themselves," even the most rigorous halls of mathematics have opened their doors to AI.
If you are still纠结 (struggling) with "will AI replace me," then you may have already missed the opportunity to become the next "intelligence architect."
Which century-old problem will AI pierce through next? The Riemann Hypothesis, or the Unified Field Theory in physics?
In these "very interesting times," our only fear should be indifference to this speed of evolution.
References:
https://x.com/slow_developer/status/2038399555490791765
https://x.com/mubeitech/status/2038388810157826467
https://x.com/BoWang87/status/2037648937453232504