GPT-6 Astra closes 57-year circle-packing problem, proving all remaining cases up to N=20
量子位 · wechat · 2026-10-12
Quantumbit reports that math enthusiast nor used GPT-6 Astra to rigorously prove the five outstanding cases (N=15,16,17,18,20) of the circle-packing problem, completing all cases up to N=20. The optimal arrangements were proposed back in 1969, but proofs had been missing for 57 years.
- Method: rather than searching for the optimum directly, Astra used geometric constraints to reduce infinite arrangements to finitely many cases — pushing outer disks to the boundary, ruling out arrangements needing over 360° of angular spacing, enumerating remaining distance intervals and cyclic orders by computer, then using geometric rigidity to upgrade "very close" to "exactly" the classical packing.
- Human role: nor stresses it took guided iterations — assessing feasibility, steering the model, fixing bugs, refining exposition, and cross-checking everything with Astra and Opus 5.5.
- The result sparked a mini-community of others applying AI to the same problem, which commenters call the dawn of "Math 2.0."
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