MinMax Arena
中/EN
P12 · Packing and covering · Classic

Equal-circle packing in a 2:1 rectangle

Also known ascircles in a rectanglepacking equal circles in a 2:1 rectangle

1y0
0x2
VERIFIED CONSTRUCTIONr = 0.271243444
n = 6Current record · open

Place n non-overlapping equal circles inside a 2 × 1 rectangle to maximize their common radius.

Formal definition

  • ContainerA 2 × 1 rectangle [0, 2] × [0, 1], with lower-left corner (0, 0) and upper-right corner (2, 1)
  • SubmissionExactly n circles: one common radius and n center coordinates
  • ConstraintsEvery circle lies wholly inside the rectangle; no two interiors overlap, and tangency is allowed
  • ObjectiveMaximize the common radius

Getting a feel for it

Staggered zig-zag chains and row transitions

In a 2 : 1 rectangle, small numbers of circles naturally alternate between the top and bottom edges in a zig-zag chain; as n grows, configurations transition between two-row staggered chains, three-row hexagonal layers, and tilted clusters. Outside exact 2 : 1 grid numbers such as n = 2 or n = 8, regular axis-aligned grids leave unused slack.

Proven optima and open frontier

Optimality for n = 3..6 is proven by elementary zig-zag chain bounds (see each instance's reference notes). Larger n cite Packomania's table for the 1 × 0.5 rectangle (this one doubled, so its radii are doubled too); it lists best known values and marks none of them proved.

ONE LEADERBOARD PER n

Current 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) ↓
n3
CURRENT RECORD3 − √7
Optimal
Answer sourceReference answer
Solution methodPublished reference construction
n4
CURRENT RECORD(11 − 6√3)/2
Optimal
Answer sourceReference answer
Solution methodPublished reference construction
n5
CURRENT RECORD9 − 2√19
Optimal
Answer sourceReference answer
Solution methodPublished reference construction
n6
CURRENT RECORD(27 − 10√7)/2
Optimal
Answer sourceReference answer
Solution methodPublished reference construction
n7
CURRENT RECORD0.254333094best known [0.254333095030249817754744760427, 0.254333095030249817754744760429]
Hard
Record holderCphysics
Solution methodHuman
n8
CURRENT RECORD0.25best known 1/4
Matches the best known
Record holderSneakyZero
Solution methodHuman
n9
CURRENT RECORD0.221307308best known [0.221307308948735482681310120883, 0.221307308948735482681310120885]
Hard
Record holderCphysics
Solution methodHuman
n10
CURRENT RECORD0.211566334best known [0.211566335087534700036125194421, 0.211566335087534700036125194423]
Hard
Record holderCphysics
Solution methodHuman
n11
CURRENT RECORD0.200467024best known [0.200467024723625961423042296809, 0.200467024723625961423042296811]
Hard
Record holderCphysics
Solution methodHuman
n12
CURRENT RECORD0.192855328best known [0.192855328998602162163227475739, 0.192855328998602162163227475741]
Hard
Record holder邱仲普
Solution methodAI · MoonshotAI: Kimi K3
n13
CURRENT RECORD0.188284729best known [0.188284729856591172283970907079, 0.188284729856591172283970907081]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n14
CURRENT RECORD0.186731677best known [0.186731677332578248940061347889, 0.186731677332578248940061347891]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n15
CURRENT RECORD0.182181163best known [0.182181163662410411364913634551, 0.182181163662410411364913634553]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n16
CURRENT RECORD0.171910456best known [0.171910456764160004435906211827, 0.171910456764160004435906211829]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n17
CURRENT RECORD0.169304761best known [0.169304761101795540758031941835, 0.169304761101795540758031941837]
Hard
Record holderNUE_13
Solution methodHuman
n18
CURRENT RECORD0.166666666best known 1/6
Hard
Record holderNUE_13
Solution methodHuman
n19
CURRENT RECORD0.157764803best known [0.157764803325351654507860626707, 0.157764803325351654507860626709]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n20
CURRENT RECORD0.153116972best known [0.153116972779124547200887884033, 0.153116972779124547200887884035]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n21
CURRENT RECORD0.150107439best known [0.150107439312725776278415930105, 0.150107439312725776278415930107]
Hard
Record holderNUE_13
Solution methodHuman
n22
CURRENT RECORD0.146302004best known [0.146302004397457323631433191113, 0.146302004397457323631433191115]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n23
CURRENT RECORD0.143973309best known [0.143973309564928455908583019597, 0.143973309564928455908583019599]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n24
CURRENT RECORD0.142489255best known [0.142489255829756548651254207669, 0.142489255829756548651254207671]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n25
CURRENT RECORD0.140532867best known [0.140532867883790389769830743883, 0.140532867883790389769830743885]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n26
CURRENT RECORD0.139850808best known [0.139850808944050801299092440295, 0.139850808944050801299092440297]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n27
CURRENT RECORD0.136422442best known [0.136422442282869550870769174795, 0.136422442282869550870769174797]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n28
CURRENT RECORD0.134650918best known [0.134650918192363436765398755139, 0.134650918192363436765398755141]
Hard
Record holderNUE_13
Solution methodHuman
n29
CURRENT RECORD0.130527409best known [0.130527409589304980810387177361, 0.130527409589304980810387177363]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
n30
CURRENT RECORD0.12800132best known [0.128001321008206630965787420565, 0.128001321008206630965787420567]
Hard
Record holderNUE_13
Solution methodAI · DeepSeek: DeepSeek V4.1 Flash
COMPUTATIONAL FRONTIER

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.

NCURRENT RECORDKNOWN BESTRECORD SOURCESTATUS

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-rectangle.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-rectangle.json
BibTeX (click to copy)
@misc{minmaxarena-circles-in-rectangle-2026-10,
  title  = {{Equal-circle packing in a 2:1 rectangle} (P12)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/circles-in-rectangle.json},
  license = {CC BY 4.0}
}
DISCUSSION

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.

本题n=8时用抽屉原理的证明NUE_13#313 replies好！！！NUE_13#310 replies