Equal circles in a plus sign
Place n non-overlapping circles of equal radius inside a symmetric cross (a 3 × 3 square with all four 1 × 1 corner squares removed) to maximize their common radius.
Formal definition
- ContainerA symmetric cross-shaped region formed by removing the four 1 × 1 corner squares from [0, 3] × [0, 3], with lower-left corner (0, 0) and upper-right corner (3, 3)
- SubmissionExactly n circles: one common radius and n center coordinates
- ConstraintsEvery circle lies wholly inside the cross-shaped container; no two interiors overlap, and tangency is allowed
- ObjectiveMaximize the common radius
Getting a feel for it
Five unit squares and four re-entrant corners
The cross consists of a central 1 × 1 square surrounded by four 1 × 1 arms, meeting at four re-entrant right-angle corners. When n does not divide cleanly across the five unit squares, circles bridging the center and the arms must navigate the re-entrant corners, often breaking symmetry in the optimal contact graph.
Proven small cases and open frontier
For this cross-shaped packing problem, user #46 proved that the optimal common radius for n = 3, 4, and 5 is 1/2; those instances are closed as proven archives. No optimality proofs are known yet for n = 6…18, which remain open.
SourceCurrent best solutions by n
Each parameter n is an independently scored sub-problem. Select any card to inspect the current record construction or submit a stronger solution.
Discussion (4) ↓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-in-a-cross.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-in-a-cross.json
BibTeX (click to copy)
@misc{minmaxarena-circles-in-a-cross-2026-10,
title = {{Equal circles in a plus sign} (P10)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/circles-in-a-cross.json},
license = {CC BY 4.0}
}Discussion
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