GPT-6 Astra Helps Amateur Mathematician Prove 57-Year-Old Circle Packing Problem

An amateur mathematics enthusiast known online as "nor" has used GPT-6 Astra to push a decades-old geometry problem forward, producing strict optimality proofs for packing 15, 16, 17, 18 and 20 equal-radius disks into the smallest possible enclosing circle. Previously, only the cases N<=14 and N=19 had been rigorously verified.

The first layer of the problem is finding the best arrangement. The harder layer is proving, in a mathematically strict sense, that no better arrangement exists. The American mathematician Graham proved the case N<=7 in 1968; researcher Udo Pirl extended the range to 10 disks in 1969 and studied arrangements for 11 to 19 disks. It was only around 2003 that N<=14 and N=19 were strictly proven.

According to the QbitAI report, nor worked step by step: he first asked GPT-6 Astra to handle N=15, and once the model succeeded, asked it to carry the same method over to 16, 17, and then 18 and 20, all of which the model proved. N=20 was not the model's limit, the report notes; nor simply chose to stop there.

Rather than searching directly for the optimal solution, GPT-6 Astra's core approach reduces the infinitely many possible arrangements to a finite set using geometric constraints, then uses a computer to rule them out one by one, strictly proving that the classic arrangement is optimal.

The proof proceeds in five steps: showing that, under certain conditions, disks near the outer circle can be moved outward without causing overlap; using the angular separation required between non-overlapping disks to discard any arrangement whose total required angle exceeds 360 degrees; checking the distance intervals of disks from the center and their cyclic order instead of each disk's exact coordinates; using contact constraints to show that feasible arrangements cannot undergo a qualifying small deformation, upgrading "very close" to "strictly identical" (the 15-disk case used a different odd-cycle argument); and finally proving that at least one disk in the original arrangement must touch the outer boundary, meaning no smaller enclosing circle is possible.

It is worth stressing that this was not a single prompt. In his blog post, nor described his own contributions as determining the feasibility of the problem, guiding the model, fighting a bug-ridden web interface, enduring clumsy copy, fixing subtle issues, refining the exposition, manually verifying the result, and re-checking everything with Astra and Opus 5.5.

The case matters because it shows AI moving from solving problems toward rigorous proof, compressing an optimality argument that would normally take a mathematician a long time into a finite search a machine can verify. The model here acts less as an oracle than as an amplifier: the human frames the question and validates the result, while the model does the heavy elimination across a vast space of cases.

Nor also found, after publishing, that several others had been using AI on the same problem, and some commenters said the episode made mathematics feel more approachable. What to watch next is whether this geometric-constraint-plus-finite-search approach generalizes to other combinatorial geometry problems, and how human verification and machine proof can be combined into a more reliable workflow.