Packing circles of radius 1,2,…,n into a circle
Also known aspacking circles with radii 1 to ngraduated circle packing
Pack n non-overlapping circles of radii 1, 2, …, n inside an enclosing circle to minimize the container radius.
Formal definition
- ContainerA circle of radius radius centered at (radius, radius), where the smallest circle has unit radius 1 and radius is the objective value to minimize
- SubmissionContainer radius and centers, listed in order of radii 1, 2, …, n
- ConstraintsCircle i has radius i; no two interiors overlap; every circle lies wholly inside the container circle
- ObjectiveMinimize the container radius
Getting a feel for it
Large circles and interstitial gaps
The largest few circles determine the lower bound and primary geometry of the container, while the smaller circles fit into the interstices between large circles and the boundary. Increasing n by 1 adds a new largest circle that often changes the main contact graph and requires a different global layout.
Proven optima and open frontier
Packing circles of radii 1..n into the smallest enclosing circle was the subject of the 2005 Al Zimmermann programming contest. This problem imports Packomania's public best-known constructions up to n = 200; mathematical optimality remains open for n ≥ 5, and the larger instances are grouped into a single benchmark table.
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 (1) ↓N = 31–200 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/graduated-circles-in-circle.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/graduated-circles-in-circle.json
BibTeX (click to copy)
@misc{minmaxarena-graduated-circles-in-circle-2026-10,
title = {{Packing circles of radius 1,2,…,n into a circle} (P06)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/graduated-circles-in-circle.json},
license = {CC BY 4.0}
}Discussion
Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.
Grouped by player and sub-problem, showing the latest note first. Earlier versions and replies are preserved; click any note to read or reply. Notes can be published alongside record-breaking submissions.