Magic Hexagons of Every Order: One Human, One Model, One Broken Record

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A magic hexagon fills a honeycomb grid so every straight line in all three directions sums to the same total. Exactly one non-trivial normal version exists, the famous 19-cell specimen; a divisibility argument kills every other order. Konstantin Gukov’s new post asks what happens if you relax one rule, and answers it with AI: abnormal magic hexagons exist for every order above 3.

The record just shattered. Before this project, Wikipedia listed order 9 as the largest known abnormal magic hexagon, found by brute force in 2024. Gukov’s solver found solutions up to order 21 within days, and his constructive proof now builds them for any order greater than 3.
His first move was human: shrink the search space instead of speeding up the search. Restrict numbers to a symmetric range around zero, put zero in the center, and require opposite cells to hold opposite values. Suddenly half the line-sum constraints vanish by symmetry.
Then came the elegant idea. Alternating plus-one/minus-one rings around each interior cell leave every line sum untouched, so any zero-sum hexagon decomposes into these “potential fields,” like stacking transparent terrain maps. Search inside that space and every line already sums to zero before you start.
The AI wrote the solver, not the answer at first. Pointed at the problem, GPT-5.6 Sol connected it to Heffter arrays from the literature, abandoned generic constraint solvers for custom simulated annealing, then optimized its own code with Numba and perf profiling. Days on a home server did the rest.
The proof attempt is the honest part of the story. Two AI systems ground against the full conjecture and stalled; even a max-effort run failed, but its discarded ideas later got recombined into a breakthrough: an explicit construction for orders divisible by 16 above 800, generalized stepwise down to every order past 114.
Gukov’s verdict on working with the model deserves quoting in spirit: it reasons deeply but tunnels hard, so keep an arbiter in the loop to notice when progress has stopped. Also, internet access mattered; the model surfaced mathematical machinery no human would have fed it in advance.
The verification gap is the field’s next bottleneck. Code review became the bottleneck once LLMs made generation cheap; mathematics now has candidate proofs arriving faster than humans can responsibly check them. This construction is not yet formalized in Lean, and the author flags that openly.
Start to finish, this is what discovery looks like now: a human observation that reframes the problem, a model-written specialized solver breaking records, stalled proofs rescued by recombined fragments, and an open conversation log so you can watch every turn yourself.

The takeaway: Read the post twice, once for the hexagons and once for the workflow. Then note the pattern worth stealing for your own hard problems: shrink the search space with your own insight before asking a model to search it.

Magic Hexagons of Every Order: One Human, One Model, One Broken Record

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