Circle packing in the unit square: new sum-of-radii layouts

Numaro Autoresearch Team

Plain language

What this result means

Circle packing is easy to check and hard to find. Circles either fit or they do not, but moving one circle changes the feasible radii of many others. Record layouts live in narrow local basins, and some margins are only around 1e-4, so a candidate has to survive exact overlap and wall checks rather than just look better.

For n=26, the result clears AlphaEvolve and ShinkaEvolve, and is effectively tied with ThetaEvolve at the same optimum-level value while using a stricter feasibility tolerance.
For n=33-40, the public table had recent automated-sweep values that heavier search could still improve; the hard part is escaping layout topologies that look stable but are not record-level.
For n=41 and n=42, the table had no entry, so the results are frontier extensions rather than beats.

Visual notes

How to read the result

Minimal black and white visualization of 36 non-overlapping circles packed in a unit square, with faint contact lines between touching circles. — **The n=36 layout**The actual coordinates for the largest-margin record in the set. Thin internal lines mark near-contact pairs; the square boundary is the constraint.

Bar chart showing record margins for n equals 26 and n equals 33 through 40, with n equals 36 having the largest margin. — **Record margins**The margin chart keeps the scale honest: n=36 is the clear outlier, while n=34 is a small but strict table beat.

Line chart comparing Numaro circle-packing sum of radii values to previous public values, plus first-known frontier points for n greater than 40. — **Frontier shape**Solid points are the new layouts. The dashed line is the prior public table where it exists; the open points are first-known n greater than 40.

Result table

Nine best-known records beaten, including the n=26 AlphaEvolve benchmark.

Cell	Baseline	Numaro	Delta	Note
n=26	2.63586276	2.6359830853	+0.000120	AlphaEvolve benchmark
n=33	2.9789	2.9872850086	+0.008385	Viquerat June 2026
n=36	3.11023	3.1210039084	+0.010774	largest margin in the set
n=40	3.28632	3.2923915726	+0.006072	Viquerat June 2026
n=41	none listed	3.3346864337	first-known	frontier extension
n=42	none listed	3.3769116039	first-known	frontier extension

Method

How it was found

The system searches circle centers, then uses a linear-program referee to assign the best feasible radii for those centers.

Reproduced the public baselines and fixed them as dated targets.
Ran GPU multistart, island genetic search, adaptive basin hopping, and analytic-Jacobian SLSQP.
Used cross-n seeding: a good n layout seeds n+1 by adding a circle, and n-1 by removing one.
Rejected higher-looking values when the LP or overlap checker showed they were tolerance artifacts.

Verification

How it was checked

Each layout is checked from its raw coordinates. Every pair of circles must be far enough apart, every circle must stay inside the square, and the final score is the sum of all radii. The reported beats are then compared against the dated public baseline values.

Scope

What is not being claimed

The records are best-known feasible packings, not proofs of optimality. Values are pinned to the dated public baselines named in the report.

References

Baseline sources

Citation

How to cite

Numaro Autoresearch Team. "Circle packing in the unit square: new sum-of-radii layouts." Numaro Research Report NUMARO-2026-004, 2026.

@techreport{numaro2026CirclePackingUnit,
  title = {Circle packing in the unit square: new sum-of-radii layouts},
  author = {Numaro Autoresearch Team},
  institution = {Numaro},
  number = {NUMARO-2026-004},
  year = {2026},
  url = {https://numaro.tech/research/circle-packing-unit-square-2026/}
}