Packing circles of radius 1,2,…,n into a square
Also known aspacking circles with radii 1 to n in a squaregraduated circle packing
Pack n non-overlapping circles of radii 1, 2, …, n inside an enclosing square to minimize its side length.
Formal definition
- ContainerA square [0, side] × [0, side] with the origin at its lower-left corner, where the smallest circle has unit radius 1 and side is the objective value to minimize
- SubmissionSquare side 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 square container
- ObjectiveMinimize the side length of the enclosing square
Getting a feel for it
Corner placement and interstitial gaps
Because a square has four right-angle corners, seating the largest few circles into the corners and staggering them across the diagonals is key to minimizing the side length, while smaller circles fit into the triangular pockets left behind. Increasing n by 1 introduces a new largest circle that often forces a reassignment of the four corners.
Proven optima and open frontier
Optimality for n ≤ 4 follows from elementary center-distance projections among the largest circles (see the instance notes). Packomania's csqn table lists best known values up to n = 100, and its ratio column is this problem's side. The table marks none of them proved, so n ≥ 5 remains open. The circular-container counterpart was a Zimmermann contest problem, also on Packomania.
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 = 31–100 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/graduated-circles-in-square.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-square.json
BibTeX (click to copy)
@misc{minmaxarena-graduated-circles-in-square-2026-10,
title = {{Packing circles of radius 1,2,…,n into a square} (P13)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/graduated-circles-in-square.json},
license = {CC BY 4.0}
}Discussion
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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.