Equal discs covering a disc
Cover the largest possible disc with n unit-radius discs. Maximise the covered disc radius.
Formal definition
- ContainerThe editor and certificate normalise the target disc to the closed unit disc centred at (1,1), while public scores are reported in the literature's unit-covering-disc scale.
- SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
- ConstraintsThe n distinct centres lie inside the target disc. Covering discs may overlap and protrude, and their union must cover every point of the target disc.
- PrecisionInput decimals are parsed as exact rationals, with zero floating-point tolerance in feasibility checks.
- ObjectiveCovered disc radius R = 1/r, where larger is better.
Getting a feel for it
Literature targets and coordinate certificates
Literature targets are converted at their published precision, while nine-decimal coordinate certificates are stored and displayed alongside the source values.
SourceCurrent best solutions by n
Smaller instances have dedicated visual pages for inspection and interactive editing, while larger n are consolidated in the index table below for direct comparison and submission.
Discussion (0) ↓N = 41–100 index
Each row is an independently scored sub-problem. External benchmarks retain their published precision, while arena constructions are verified on the exact nine-decimal grid.
Data and citation
All sub-problems in this family — authoritative scores, proof status, coordinates, and sources — are published at the stable URL below under CC BY 4.0. Because scores update as records fall, please cite the file's generatedAt timestamp.
GET https://minmaxarena.com/data/circles-cover-disc.json
Cite the frozen 2026-10 edition: live records change, whereas a frozen edition stays fixed and reproducible.
GET https://minmaxarena.com/data/editions/2026-10/circles-cover-disc.json
BibTeX (click to copy)
@misc{minmaxarena-circles-cover-disc-2026-10,
title = {{Equal discs covering a disc} (P87)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/circles-cover-disc.json},
license = {CC BY 4.0}
}Discussion
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