Yuyang reporting from Aofeisi
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AI is now truly sweeping through mathematical proofs.
Just now, anotherMath enthusiastsUse GPT-6 Astra to turn aYear 57A big step forward has been made on a hard math problem.
The problem itself doesn't sound all that complicated:
Given N disks of radius 1, how do you pack them into one big circle so that the big circle's radius is as small as possible?
In fact, mathematicians had already given an elegant packing arrangement back in 1969 — it's just that… for 57 years, no one had been able to rigorously prove that it was optimal.
So this time, math enthusiast nor pulled out GPT-6 Astra… and then in one go rigorously proved the 5 remaining cases of 15, 16, 17, 18 and 20, that is,all the remaining cases with N≤20 have been rigorously proved.……
Previously, only the cases N≤14 and N=19 had been verified.
Oh right, N=20 is not AI's limit — nor just didn't feel like going further…
A proof stuck for half a century
Let's first put the problem into a more mathematical form.
Suppose there are N disks, each of radius 1, and a big circle of radius R.
The requirements:
All the small disks must lie entirely inside the big circle; they may be tangent to the big circle's boundary but must not cross it.
No two small disks may overlap internally, though mutual tangency is allowed.
The small disks can be moved and arranged freely inside the big circle, with no restriction on the arrangement.
The question is: for a given positive integer N, what is the minimum possible radius R of the big circle?
It looks like an unremarkable olympiad-style problem, but in reality, finding the optimal arrangement is only the first layer of the problem; the real difficulty lies in rigorously proving mathematically that the arrangement has reached the limit and no better packing exists.
In 1968, the American mathematician Ronald Graham proved the case N≤7.
In 1969, mathematics researcher Udo Pirl extended the proof to 10 disks, and further studied arrangements for 11–19 disks.
But it was not until around 2003 that the cases N≤14 and N=19 were rigorously proved.
nor is a technical writer who has long studied mathematics, machine learning and optimization algorithms, and has repeatedly and publicly participated in AI-assisted mathematical research.
He himself noticed this problem more than a decade ago, and indeed there had been little progress on it over the past dozen or so years…
No need to say more — just throw it to GPT-6 Astra, perfect (jk).
nor also proceeded step by step: first let GPT-6 Astra handle the case N=15; after the model solved it, let it extend the same method to 16, 17, and 18 and 20… GPT-6 Astra successfully produced proof results in every case.
It's just that…
the proof process was, as always, obscure and hard to follow.
Simply put, GPT-6 Astra's core idea is:
rather than directly searching for the optimal solution, it uses geometric constraints to reduce the infinitely many possible arrangements to a finite set of cases, then uses a computer to eliminate them one by one, ultimately rigorously proving that the classical arrangement is already optimal.
The proof is divided into 5 steps:
Push the outer disc toward the boundary. The proof shows that when specific conditions are satisfied, the disks close to the outer circle can be moved outward without causing overlaps. This divides the disks into one group on the boundary and one group in the interior.
Using angle relationships to eliminate large numbers of permutations. Because the disks cannot overlap, a certain angular gap must be maintained between them. If a given arrangement requires an angular total exceeding 360°, it can be ruled out directly.
let the computer enumerate the remaining possibilitiesInstead of searching for the specific coordinates of each disk, we check the distance interval from each disk to the center and their encircling order.
Proving exact optimality with geometric rigidityThe preceding search only proves that all feasible arrangements are very close to the known classical arrangement. Next, using the contact constraints between the disks, it is shown that these arrangements cannot undergo admissible small deformations, thereby upgrading "very close" to "strictly identical". The case of 15 disks uses a different odd-ring argument.
Restore the original arrangement. It is finally proved that in the original arrangement, at least one disk must touch the outer circle's boundary. This means it is impossible to fit all the disks into an outer circle with a smaller radius, thereby completing the optimality proof.
It is worth emphasizing that it was not the case that Astra simply proved everything in one go after nor gave a single round of prompts.
nor wrote in a blog post that his contributions as a human to this work included:
Determining the feasibility of the problem, guiding the model, wrestling with a bug-ridden web user interface, enduring terrible copy, fixing subtle issues, refining the presentation over and over, manually verifying its validity, and re-checking everything with Astra and Opus 5.5.
One More Thing
An interesting thing is that once nor posted the results, quite a few fellow researchers turned out to be working on the same problem with AI.
Another netizen commented that AI breakthroughs have made mathematics easier to understand: "I never knew the field of mathematics could have such a strong atmosphere of exploration."
From the perspective of math enthusiasts, this is also a new atmosphere of the era of Mathematics 2.0.
Original article link:
[1] https://nor-blog.pages.dev/posts/2026-10-11-circle-in-circle-packing-15-20/
[2]https://x.com/norxornor/status/2109078411251454017
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