circle-packing-32

Metric sum-radii, maximize · Best known 2.93957 (Georgiev et al. 2025) · Play · Code · All example problems

hillclimb problem get circle-packing-32      # copies the problem into hillclimb/problems/
hillclimb verify circle-packing-32           # scores the floor; spread 0, the verifier is exact
hillclimb run circle-packing-32 --budget 15m

The problem

Place exactly 32 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 32 rows

This is a hard continuous optimization problem. The best known value for n = 32 is about 2.939572 (Georgiev et al. 2025); AlphaEvolve 2025 reported 2.937945. Equal circles on a grid are weak (the corners and the gaps between cells are wasted). Good approaches combine constructive patterns (hexagonal / greedy layouts, unequal radii — a few large circles surrounded by small ones), local optimization of the sum under the non-overlap constraints (e.g. SLSQP, projected gradient, or a physics-style relaxation that pushes overlapping circles apart while inflating them), and restarts. numpy and scipy are available.

Submission format

Write submission.csv in the working directory with the header id,x,y,r and 32 rows (id = 0..31), like sample_submission.csv (a weak valid baseline: 32 equal circles on a 6 x 6 grid).

Scoring

The orchestrator runs problem/verify.py after your script finishes. The verifier validates the constraints (with a numerical tolerance of 1e-9) 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. Keep total runtime well within the execution time limit.