MinMax Arena
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P67 · Packing and covering · Classic · Applied frontier

Variable-radius circles in a fixed-perimeter rectangle · n = 4

Also known ascircle packing in a rectangle of fixed perimetervariable radius circle packingmaximum sum of radii

Pack n circles with disjoint interiors into a rectangle of fixed perimeter 4, choosing both the rectangle's dimensions and each circle's radius to maximize the sum of the radii.

Instancen = 4
ObjectiveMaximize sum of radii

Record comparison

sum of radii · Higher is better
RecordValue / intervalAuthor / holderSource
Proven optimum6 − 2√2 − 2√(4 − 2√2)zzzcy #308MinMax Arena ↗
Site record1.006788474邱仲普Record details ↓

The continuous optimum is proved, so competition submissions are closed. Historical records and contribution points are retained. Because finite-decimal coordinates cannot attain this value exactly, the current certificate is not claimed optimal on the nine-decimal grid.

Formal definition

  • Containerwidth > 0, height > 0, and width + height = 2; the lower-left corner is fixed at (0,0)
  • SubmissionExactly n circles, each written as [x,y,r] with strictly positive radius
  • ConstraintsEvery circle lies wholly inside the rectangle; interiors are pairwise disjoint and tangency is allowed
  • ObjectiveMaximize the exact sum of all circle radii
Read the optimality proof ↗

Layout shown: site record

1y0
0x1
VERIFIED CONSTRUCTIONscored by the sum of the radii

Getting a feel for it

Coupled aspect ratio and packing

Because the aspect ratio is free, shifting perimeter from height to width alters wall contacts and interstitial gaps throughout the box. The rectangle dimensions and the circle arrangement must be optimized together.

Combining large and small circles

Equal radii leave large gaps between curved boundaries. Good configurations use a few large circles to set the rectangle's aspect ratio and smaller circles to fill the remaining pockets; adding a single circle can shift the optimal aspect ratio to a different contact topology.

Published frontier

Dominik Kamp's 2026 computational project provides complete coordinates up to n = 60. The benchmark starts at the first non-trivial case n = 3 and includes verifiable nine-decimal certificates. In October 2026, a computer-assisted proof by zzzcy, independently verified by the platform, established the exact nine-decimal grid optima for n = 3–9; n ≥ 10 remains open.

Source
The record-holding arrangement for Variable-radius circles in a fixed-perimeter rectangle n = 4, 1.006788474
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder邱仲普
Solution methodAI · MoonshotAI: Kimi K3
Proven optimumSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.Read the published proof ↗
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (2 changes)
  1. 邱仲普AI · MoonshotAI: Kimi K3
    1.006788472→1.006788474
  2. 邱仲普AI · DeepSeek: DeepSeek V3
    1.006788468→1.006788472
ANSWER FORMAT

How to write your answer

The rectangle has lower-left corner (0,0), width width and height height, satisfying 2(width + height) = 4.

Submit width, height, and exactly n triples [x, y, r]. The sum width + height must equal 2 exactly; write every number as a decimal string with at most nine places.

The current leader's answer

{
  "circles": [
    [
      "0.789498981",
      "0.789498981",
      "0.210501019"
    ],
    [
      "0.292893218",
      "0.707106782",
      "0.292893218"
    ],
    [
      "0.707106782",
      "0.292893218",
      "0.292893218"
    ],
    [
      "0.210501019",
      "0.210501019",
      "0.210501019"
    ]
  ],
  "height": "1",
  "width": "1"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 4
}

The current leader's answer

{
  "circles": [
    [
      "0.789498981",
      "0.789498981",
      "0.210501019"
    ],
    [
      "0.292893218",
      "0.707106782",
      "0.292893218"
    ],
    [
      "0.707106782",
      "0.292893218",
      "0.292893218"
    ],
    [
      "0.210501019",
      "0.210501019",
      "0.210501019"
    ]
  ],
  "height": "1",
  "width": "1"
}

Submit width, height, and exactly n triples [x, y, r]. The sum width + height must equal 2 exactly; write every number as a decimal string with at most nine places. · Verifier v1.0.0

DISCUSSION

Discussion

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