Claude Opus 4.6 successfully tackled a problem that took about an hour to solve, highlighting new horizons in automated mathematical thinking. Scientists, including Knuth, were impressed by how the AI demonstrated the ability for creative problem-solving. This task was proposed by Knuth several years ago and involved breaking a directed graph into three Hamiltonian cycles—closed paths that pass through each vertex exactly once. Researcher Philip Stappers presented this task to the AI, which documented all its steps in the solving process.
According to Quantum Zeitgeist, initially, Claude used heuristic methods such as "fiber" analysis (groups of nodes) and simulated annealing. Although these methods provided partial solutions, they did not offer a general constructive approach. During the work, "Claude" made an important discovery: "Simulated annealing can find solutions but does not provide a general construction. Pure mathematics is needed," which indicates the AI's ability to recognize its limitations and the need to change its approach.
A key moment was the model's focus on identifying mathematical patterns. By studying two-dimensional serpentine functions and the structure of the graph, "Claude" noticed that the choice in each "fiber" depended on a single coordinate. This discovery led to the formulation of a rule for constructing Hamiltonian cycles based on modular arithmetic.
The rule is as follows: for three coordinates i, j, k, the sum is calculated modulo m (s = (i+j+k) mod m). Based on the value of s and the current coordinates, it is determined which of the coordinates should be changed to move to the next vertex. By systematically applying this rule, three required cycles can be obtained.
Stappers tested the program developed by "Claude" in Python for all odd values of m from 3 to 101 and confirmed that the solution works. This allowed him to conclude that the problem is solved for odd parameters. In conclusion, a rigorous mathematical proof was presented, which Knuth called "very interesting."
Who is Donald Knuth? He is an American scientist, emeritus professor at Stanford University, and one of the most cited authors in the field of computer science. Knuth is the author of 19 monographs, including well-known books on programming, as well as the developer of several popular software technologies and systems, such as TeX and METAFONT. His work "The Art of Computer Programming" is considered a classic in the field of computer science.
In February, Google DeepMind introduced the AI Aletheia, based on the Gemini Deep Think model. This system, specializing in mathematics, has already demonstrated its capabilities by successfully solving several open mathematical problems.
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