circle-packing

Metric sum-radii, maximize · Best known 2.63598 (AlphaEvolve) · Play · Code · All example problems

hillclimb problem get circle-packing      # copies the problem into hillclimb/problems/
hillclimb verify circle-packing           # scores the floor; spread 0, the verifier is exact
hillclimb run circle-packing --budget 15m

The problem

Place exactly 26 circles inside the unit square [0, 1] x [0, 1] so that the sum of all radii is as large as possible.

Constraints (all verified programmatically):

• every circle lies entirely inside the unit square: r <= x <= 1 - r and r <= y <= 1 - r
• no two circles overlap: dist(center_i, center_j) >= r_i + r_j
• all radii are non-negative
• exactly 26 rows

This is a hard continuous optimization problem. The comparison targets are 2.6359773947566274 from a reported OpenEvolve run and 2.6359830849176067 from AlphaEvolve's current published construction. Good approaches combine constructive patterns (hexagonal/greedy layouts, unequal radii), local optimization (e.g. SLSQP / projected gradient / physics-style relaxation), and restarts. numpy and scipy are available.

Submission format

Write submission.csv in the working directory with the header id,x,y,r and 26 rows (id = 0..25), like sample_submission.csv (which is a weak valid baseline).

Scoring

The orchestrator runs problem/verify.py after your script finishes. The verifier validates the constraints and prints val_score: <sum of radii> (or a score of 0.0 with a reason if the packing is invalid). Higher is better.

There is no train/test data; this is a pure optimization problem. Your script should finish well within the execution time limit — budget your optimization loop accordingly (e.g. a few minutes of solver time at most).