Reported by New Intelligence Yuan
Editors: KingHZ, Taozi
[New Intelligence Yuan Digest] The "Hamiltonian Decomposition" problem has finally been cracked! At 88, the "Grandfather of Algorithms," Donald Knuth, has updated his paper again. Claude 4.6 and GPT-5.4 jointly solved the odd and even number cases. Even more astonishingly, GPT-5.4 directly generated 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, Donald Knuth, the "Grandfather of Algorithms" and the youngest Turing Award winner in history, was shocked by Claude: A long-standing algorithmic puzzle was actually solved by Claude Opus 4.6.
At the 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难题 (difficult problem)!
GPT-5.4 Pro took the baton from Claude, directly producing a 14-page paper for all even numbers m ≥ 8, and computationally verified scenarios up to m = 2000.
Moreover, after GPT and Claude linked up, they utilized a multi-agent workflow to find simpler construction methods for both odd and even m.
Others have also used the Lean language to formalize Claude's proof regarding the odd-number scenario.
Thus, the "Hamiltonian Decomposition" problem has been thoroughly solved.
From Claude 4.6 to GPT-5.4, combined with the joint efforts of numerous industry leaders, the pit dug decades ago has finally been filled.
At the end of the paper, the old master remarked emotionally—
"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拆解 (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 magnum opus of computer science, "The Art of Computer Programming" (TAOCP), Hamiltonian Decomposition remained a "patch" that constantly worried him.
This problem has been suspended for decades, described in terminology as follows:
Previously, the academic community was unable to provide a complete solution covering both odd and even scenarios.
As the number of nodes increases, the search space explodes exponentially. Faced with such depth of darkness, the human brain often feels physiologically powerless.
Over the past thirty years, countless geniuses attempted to fill this pit, but most stumbled at the final defense line of the "odd-even complete 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, it decides whether to increment i, j, or k. The specific rules are as follows:
If s=0, the direction of movement is decided based on the value of j. If 0 < s < m−1, it is decided based on the value of i. If s=m−1, another rule is applied.
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 work on m being even for about 4 hours. It finally had 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 refine 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 made 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, news even more shocking arrived from his friend in Singapore, Ho Boon Suan.
He utilized gpt-5.3-codex to generate a piece of code that successfully achieved 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. The results were flawless.
It is worth noting that when m=2000, that is a massive graph structure possessing 8 billion vertices!
If one were to rely purely on human power to manually calculate and prove its correctness, it would be sheer fantasy.
At almost the same time, Kim Morrison from the Lean community moved with incredible speed.
He formalized the verification of the proof regarding Claude's correct construction and posted it online promptly on March 4th.
Math Geniuses Gather to 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 might not be the simplest.
In the C language program, one only needs to replace specific lines with extremely streamlined 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, two top-tier models, leveraging their different thinking dimensions to inspire each other, eventually piecing together the complete proof.
Zero Edits: GPT-5.4 Directly Outputs 14-Page Paper
Regarding the construction problem for even m, the true climax is yet to come.
Since the algorithmic patterns generated by gpt-5.3-codex were too complex, Ho Boon Suan decided to give GPT-5.4 Pro an ultimate instruction:
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 beautifully typeset, logically rigorous, 14-page academic paper.
From "Abstract" to "Conclusion," the structure is complete, with seamless transitions and rigorous logic.
Moreover, it adopted the TeX standard. Since Donald 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 an achievement accomplished independently by GPT-5.4 Pro; he didn't need to modify a single punctuation mark!
This means its logical chain is an "absolute truth" in the mathematical sense.
AI "Left Hand Fights Right Hand": Claude + GPT Achieve Perfect Proof
The mastermind behind this story is Keston Aquino-Michaels.
He not only found another effective decomposition for the odd m scenario but also provided an elegant decomposition for the even m scenario, far simpler than previous methods.
Additionally, he unearthed a relevant reference paper that Knuth had previously missed (the last reference in the figure below).
Preprint: https://arxiv.org/abs/2203.11017
Most wonderfully, he meticulously analyzed this joint interaction mode, noting its potential significance for how to 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 an ingenious "collaborative workflow".
This was more like a collaborative exercise spanning carbon and silicon bases, a close cooperation between Claude, GPT, and humans.
Among them, two Agents run independently, using the same "Residue" prompt.
Structured exploration prompts used by the two intelligent agents
But each plays to its strengths:
Agent O: Solved the odd case in 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; communication was relayed through an Orchestrator—data and tools were all passed via the commander (Opus 4.6 guided by humans).
The Orchestrator needed to judge "when to transmit, what to transmit, and in what format to transmit," 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: m−2 layers of "bulk layers" + 2 layers of "repair layers."
Ultimately, that "complete solution for odd and even scenarios" which had troubled humanity for decades was thoroughly 凿穿 (pierced through) in the fierce clash of the two AI agents.
Humans Define the Battlefield, Machines Fill the Abyss
This act of "filling the pit" marks a complete turning point in the paradigm of scientific research.
The identity of scientists has changed. For instance, Donald Knuth is no longer the craftsman calculating every line of code on paper; he defined the boundaries of the problem, designed the logic of verification, and then commanded the AI to fill that black hole of trial and error.
The paradigm of research has changed. Humans only need to define the boundaries, while AI fills the abyss.
The most precious ability of mathematicians is no longer computational power, but rather "the intuition to propose problems" and "the aesthetics to verify answers".
AI is responsible for finding paths amidst 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 转折 (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 tangled up in "will AI replace me," then you may have already missed the opportunity to become the next "intellectual architect."
Which century-old problem will be the next to be 凿穿 (pierced through) by AI? 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: