AI and Mathematician Prove Existence of Magic Hexagons for All Orders Greater Than Three
New proof establishes that abnormal magic hexagons exist for every order n > 3, moving beyond the single known non-trivial normal case of order 19.
A researcher has demonstrated the existence of abnormal magic hexagons for every order greater than three, resolving a long-standing mathematical conjecture. The discovery was achieved by combining computational search with advanced AI reasoning, specifically using GPT-5.6 Sol to develop a constructive algorithm. While the only non-trivial normal magic hexagon is of order 19, the new work proves that solutions exist for all higher orders by relaxing constraints to allow consecutive numbers that do not start at one. The proof, which relies on concepts such as antisymmetry and potential fields, was generated through an iterative process involving human direction and AI assistance, though it remains pending formal verification in systems such as Lean.
The research was inspired by a conversation among YSDA alumni regarding the number 19 and the properties of magic hexagons. Prior to this work, the largest known abnormal magic hexagon was of order n=9, found by Klaus Meffert in 2024. Unlike magic squares, which have formulaic constructions, finding magic hexagons required wandering through a large search space. The researcher focused on reducing this space by restricting numbers to a symmetric interval and imposing antisymmetry, where cells opposite each other under a 180-degree rotation contain opposite values.
Around the same time, the researcher participated in the Midnight Code Cup 2026, where they observed the efficacy of LLMs in developing domain-specific solvers. Instead of using generic constraint solvers like Z3, the researcher utilised GPT-5.6 Sol to develop a custom simulated annealing solver. The model identified connections to Heffter arrays, which suggested better ways to organize values. The resulting program, optimised with Numba, allowed the researcher to discover magic hexagons of every order up to n=21 through brute force.
The existence of these solutions led to a conjecture that abnormal magic hexagons exist for all orders greater than three. The researcher engaged GPT-5.6 Sol (high) to prove this claim. The AI exhibited "tunnel vision" but was effective when guided by a human arbiter. Through an iterative process, the AI developed a constructive argument for orders divisible by 16, which was then generalised to all orders greater than 114. The final constructive argument covers every order n > 3 by combining the AI-generated algorithm with the brute-force findings up to n=21.
The proof is currently not formalised in Lean or independently verified, which is cited as the next step. The researcher noted that while the AI did much of the creative work, human direction was essential to steer the process and interpret the results. The work establishes a theoretical threshold of n=114, below which the construction’s inequalities are easy to justify, though the method succeeds in practice for lower orders. The proof remains pending formal verification in systems such as Lean.
