Updated October 12, 2026 · 6 min read
GPT-6 Astra and the 57-year circle packing problem: the last unproven cases, closed
For 57 years, the "obviously right" way to pack 15, 16, 17, 18 and 20 equal circles into one enclosing circle was a conjecture nobody could prove. On October 11, 2026, math enthusiast nor published rigorous proofs for all five cases — produced with GPT-6 Astra. Here's the problem, the proof strategy, and the honest division of labor.
The problem in plain terms
Take n equal circles. Pack them all inside one larger circle so the enclosing circle's radius is as small as possible. The circles must sit fully inside the big circle — touching the boundary is fine, poking out is not — and they may touch each other but not overlap.
Finding a good arrangement is easy. Proving that no better arrangement exists anywhere in the infinite space of possible configurations is the hard part — and that is why, for decades, many cases were recorded in textbooks as "conjectured optimal."
What was already proven
| n | Enclosing radius | Status before October 2026 |
|---|---|---|
| 7 | 3 | Proved by Graham (1968) |
| 8–10 | ≈3.304 / 3.613 / 3.813 | Proved by Pirl (1969) |
| 11 | ≈3.923 | Proved by Melissen (1994) |
| 12–13 | ≈4.029 / 4.236 | Proved by Fodor (2000, 2003) |
| 14 | ≈4.328 | Proved by Ekanayake & LaFountain (2024) |
| 15 | ≈4.521 | Conjectured — now proved |
| 16 | ≈4.615 | Conjectured — now proved |
| 17 | ≈4.792 | Conjectured — now proved |
| 18 | ≈4.863 | Conjectured — now proved |
| 19 | ≈4.863 | Proved by Fodor (1999) |
| 20 | ≈5.122 | Conjectured — now proved |
Pirl proposed the classic arrangements for n=15 through 20 back in 1969. Fifty-seven years later, nor — a technical writer who follows math, machine learning and optimization — picked the problem up again and handed it to GPT-6 Astra, one case at a time: first n=15, then the method generalized to 16, 17, 18 and 20. N=20 was not a model limit; nor simply stopped there.
How the proof works
The strategy does not search for a better packing. It eliminates the possibility of one, by shrinking an infinite space to a finite one:
- Push outward. Prove that under certain conditions, circles near the outer boundary can be moved outward without creating overlaps. This splits the packing into a boundary group and an interior group.
- Angle-sum exclusion. Non-overlap forces minimum angular separation between circles. If an arrangement's required angles sum to more than 360°, it is impossible — excluded wholesale.
- Enumerate the survivors. Instead of searching each circle's exact coordinates, check distance ranges from the center and cyclic ordering. A finite list remains.
- Rigidity lifts "near" to "exact." Contact constraints between circles rule out tiny deformations, upgrading "every surviving arrangement is close to the classic one" to "every surviving arrangement is the classic one."
- Special cases and closure. n=15 required an additional odd-ring argument. Finally, the proof shows at least one circle in the original arrangement must touch the outer boundary — so nothing can be squeezed into a smaller enclosing circle.
If you followed the Grad conjecture story, you have seen this pattern before: a frontier model's contribution is turning a continuous, infinite problem into a finite, exhaustible one — then brute-force verification takes over.
The human's share
This was not one prompt producing five proofs. By nor's own account, the human work included judging the problem's feasibility, guiding the model through the web interface, manually verifying the results, rewriting the exposition, and re-checking the entire body of work with both GPT-6 Astra and Opus 5.5. Nor also noted the proofs remain hard to read — rigor, not beauty, was the goal.
That framing matters for reading this story honestly: it is human-directed, machine-assisted, independently cross-checked mathematics. It is not "AI solved a 57-year problem by itself," and the people who care about this distinction — mathematicians — are watching exactly this distinction.
Frequently asked questions
What did GPT-6 Astra prove about circle packing?
That the classic arrangements are provably optimal for n=15, 16, 17, 18 and 20 — the remaining unproven cases up to 20, open since Pirl's 1969 arrangements. The work was published by math enthusiast nor on October 11, 2026.
Is circle packing in a circle now fully solved up to n=20?
For optimality, yes: every case from n=1 through n=20 now has a proof of optimality. The last five gaps (15, 16, 17, 18, 20) closed on October 11; 19 was proved by Fodor in 1999. Cases beyond 20 remain open.
How did GPT-6 Astra prove optimality?
By reducing the infinite configuration space to a finite one: boundary-pushing arguments, an angle-sum exclusion test, enumeration of survivors by distance range and cyclic order, and geometric rigidity arguments — plus an odd-ring argument for n=15.
Does this mean AI is doing independent mathematical research?
Not on its own, based on this case. The human chose the problem, guided the process, verified manually, and cross-checked with a second model. The honest description is human-directed, machine-assisted proof — which is still a genuinely new mode of doing mathematics.
Related reading
For the other AI-assisted math breakthrough, see GPT-6 Astra and the overturned Grad conjecture. For where Astra's benchmark claims actually stand, see our honest scorecard. For the other side of the capability story, see the stopwatch failure.