Circles with radii i in a square

# Circles with radii i in a square ## Goal `pack.py` exposes `pack(n: int, time_budget: float, seed: int) -> list[tuple[float, ...]]`: the centres of `n` circles inside the unit square [0, 1]² (2 coordinates each). Object `i` (1-based) has radius `((i) · s)`, where the eval derives the largest feasible scale `s`; the score is the largest object's radius. You do not return a radius; the eval derives the largest feasible one from your centres, so there is nothing to fudge. Make it as large as possible for every `n` you are handed. ## Metric The eval runs your `pack` on a fixed set of values, `n = 5, 8, 12, 16, 20, 25, 30, 40, 50, 60, 80, 100`, each with the given time budget (12 s by default), validates the result, and reports metric = mean over n of value(n) / record(n) where `record(n)` is the best-known value on Packomania (Eckard Specht's table, maintained since 2011; table `csqn`, fetched 2026-09-06). `1.0` matches the record; above `1.0` is a new record candidate, listed under `records_beaten`. A hub-verified one is worth reporting to Packomania with your ledger entry as provenance. The hub verifies with a different `seed`, so your method must be robust to its starting point. The full records table (n up to 100) is in `eval.py`. ## Constraints - Standard library only. No numpy, no scipy, no subprocess. The eval rejects other imports. - Respect `time_budget` (seconds, per call). The eval kills the run if the whole set overruns. - Deterministic given `seed`: use `random.Random(seed)`, not the global RNG. - Every centre must lie inside the container. Objects may touch; they may not overlap. ## Where the frontier is only small n are proven. Above that every entry is "best known", found by numerical search, and Packomania's history shows improvements landing mostly at larger `n`. Budget your time per `n` deliberately; the O(n²) checks and the number of local optima both grow. ## Ideas that are known to matter (check the journal before repeating one) - Energy minimisation: treat objects as repelling points, minimise a soft overlap penalty with gradient descent, then polish by maximising the minimum scaled distance directly. - Basin hopping / perturb-and-repolish from the current best; keep a small population. - Start from structured arrangements (lattices, rings, shells) as well as random. - Identify the binding contacts and solve the equal-distance conditions exactly for the last digits. - Spend more of the budget on the `n` values whose ratio is lowest. Write one honest line in `NOTES.md`: the idea, and which `n` it helped. Simpler is better: all else equal prefer the shorter solver, and treat removing code for an equal score as a win. Log every experiment, including discards, in your results.tsv.