The Claude Opus 4.6 model, created by Anthropic, successfully tackled the problem of directed Hamiltonian cycles, which had remained unsolved for several years. This problem was formulated by Donald Knuth and involved decomposing a directed graph into three Hamiltonian cycles, which are closed paths that visit each vertex exactly once. Researcher Philip Stappers proposed this task for AI to solve, and Claude Opus 4.6 began documenting each of its attempts.
During the work on the task, the AI initially applied heuristic methods, such as fiber analysis (groups of nodes) and simulated annealing, as reported by Quantum Zeitgeist. These approaches allowed for finding partial solutions but did not lead to a general method. At one point, "Claude" made an important observation: "Simulated annealing helps find solutions, but does not provide a general method. Pure mathematics is needed," which demonstrated the AI's ability to recognize the limitations of its methods and the necessity of seeking new approaches.
A key moment occurred when the model began searching for mathematical patterns. By exploring two-dimensional serpentine functions and analyzing the graph, "Claude" noticed that the choice within each "fiber" depended only on one coordinate. This discovery enabled the formulation of a rule for constructing Hamiltonian cycles based on modular arithmetic.
According to this rule, for three coordinates i, j, k, the sum modulo m is calculated (s = (i+j+k) mod m). Depending on the value of s and the current coordinates, it is determined which coordinate to increase in order to move to the next vertex. By systematically applying this rule, three desired cycles can be obtained.
Stappers tested the program created by "Claude" in Python for all odd values of m from 3 to 101 and confirmed that the solution was correct. This allowed him to conclude that the problem was solved for odd parameter values. The final stage was a rigorous mathematical proof, which Knuth characterized as "quite interesting." The entire process of solving took about an hour and demonstrated a new capability of AI for creative mathematical exploration: hypothesis generation, systematic investigation, abandonment of ineffective methods, and ultimately finding an elegant constructive solution confirmed by calculations. This event marks a significant step in the development of automated mathematical thinking.
Who is Donald Knuth. An American scientist, emeritus professor at Stanford University, one of the most cited authors in the field of computer science, author of 19 monographs (including several classic books on programming and algorithm fundamentals), developer of several well-known software technologies, creator of the TeX and METAFONT publishing systems. Author of the classic monograph "The Art of Computer Programming," one of the most respected scholars in his field.
In February, Google DeepMind introduced the AI Aletheia, based on the Gemini Deep Think model. This AI specializes in mathematical problems and has already demonstrated its capabilities by solving several open mathematical problems for the first time.
The record of AI Claude showed the ability to think by solving an open mathematical problem for the first time appeared in K-News.
