Initially, the model used heuristic methods, such as fiber analysis (grouping nodes) and simulated annealing. However, these methods provided only partial solutions, failing to lead to a comprehensive constructive approach. At one point, "Claude" noted: "Simulated annealing can find solutions but does not provide an overall construction. Pure mathematics is needed," highlighting the AI's ability to recognize the limitations of certain methods and the necessity to transition to others.
A key moment came when the model focused on searching for mathematical patterns. While studying two-dimensional serpentine functions and analyzing the graph structure, "Claude" grasped that the choice within each "fiber" depended solely on one coordinate. This discovery allowed for the formulation of a specific rule for constructing Hamiltonian cycles based on modular arithmetic.
This rule states: for three coordinates i, j, k, their sum is calculated modulo m (s = (i+j+k) mod m). Depending on the resulting value of s and the current coordinates, it is determined which coordinate needs to be increased to move to the next vertex. When systematically applied, this rule generates the three desired cycles.
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 led him to conclude that the problem was solved for odd parameters. The final stage was a rigorous mathematical proof, which Knuth described as "quite interesting." The entire process demonstrated a new capability of AI for creative mathematical exploration: formulating hypotheses, systematic investigation, discarding ineffective versions, and ultimately finding an aesthetically pleasing constructive solution with verification through calculations.
Who is Donald Knuth? He is an American scholar, emeritus professor at Stanford University, and one of the most cited authors in the field of computer science. He is the author of 19 monographs, including classic books on programming and the fundamentals of algorithms, as well as the developer of several well-known software technologies and the creator of the TeX and METAFONT systems. Knuth is also the author of the classic monograph "The Art of Computer Programming" and is considered 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 system, which specializes in mathematics, has already demonstrated its capabilities by successfully solving several open, yet unresolved mathematical problems.
The record of AI Claude showing the ability to think by solving an open mathematical problem first appeared on K-News.
