heilbronn-convex-13

Metric normalized-min-triangle-area, maximize · Best known 0.0309369 (AlphaEvolve) · Play · Code · All example problems

hillclimb problem get heilbronn-convex-13      # copies the problem into hillclimb/problems/
hillclimb verify heilbronn-convex-13           # scores the floor; spread 0, the verifier is exact
hillclimb run heilbronn-convex-13 --budget 30m

The problem

Place exactly 13 points in the plane so that the smallest triangle formed by any three points is as large as possible relative to the convex hull of all the points. There are C(13,3) = 286 triangles.

The score is

minimum triangle area / convex hull area

Higher is better. The normalization makes translation and uniform scaling irrelevant, so coordinates do not have to lie in a particular box. A degenerate point set (zero-area convex hull), duplicate points, or three collinear points scores zero.

This is a continuous, non-smooth max-min optimization problem. Useful approaches include structured or random starts followed by simulated annealing, differential evolution, SLSQP on a smooth approximation, and many restarts. numpy, scipy, and pandas are available. Keep each candidate's own numerical search comfortably below a minute so the 30-minute Hillclimb run can test and improve multiple candidates.

Submission format

Write submission.csv in the working directory with this header:

id,x,y

Provide exactly 13 rows, with each id from 0 through 12 appearing once. Both coordinates must be finite numbers. See sample_submission.csv for a weak but valid starting point.

Scoring and reference lines

The verifier reports normalized-min-triangle-area; higher is better. The Hillclimb chart includes comparison lines for OpenEvolve and AdaEvolve under a 100-candidate GPT-5 evaluation, plus the published AlphaEvolve reference. The AlphaEvolve evaluation used a different, undisclosed candidate budget, so its line is a target rather than a budget-matched comparison.

There is no train/test data and network access is disabled.