Equal-circle packing in a unit square
Also known ascircle packing in a squarepacking equal circles in a unit squarecsq
Place n non-overlapping circles of equal radius inside the unit square to maximize their common radius.
Formal definition
- ContainerThe unit square [0, 1] × [0, 1], with lower-left corner (0, 0) and upper-right corner (1, 1)
- SubmissionExactly n circles: one common radius and n center coordinates
- ConstraintsEvery circle lies wholly inside the square; no two interiors overlap, and tangency is allowed
- ObjectiveMaximize the common radius
Getting a feel for it
Contact graphs and irregular packings
At maximum radius, most circles are tangent to neighbors or the square boundary, forming a rigid contact network. Outside a few square numbers, regular grids waste space along the edges; better packings typically use tilted rows, staggered layers, and occasionally one or two rattlers that can move slightly inside a pocket.
Proven optima and open frontier
Mathematical optimality proofs are known for n = 1–30. For n = 31–300, the public best-known constructions from Packomania are imported; optimality remains open except at n = 36. Larger instances are grouped into a single table for algorithmic benchmarking.
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 (2) ↓N = 31–300 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/square-circle-packing.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/square-circle-packing.json
BibTeX (click to copy)
@misc{minmaxarena-square-circle-packing-2026-10,
title = {{Equal-circle packing in a unit square} (P01)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/square-circle-packing.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.