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P67 · Packing and covering · Classic · Applied frontier · Hard

Variable-radius circles in a fixed-perimeter rectangle · n = 27

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

Place n circles with disjoint interiors in a rectangle of perimeter 4. Both the rectangle's dimensions and every radius may vary; maximize the sum of the radii.

Instancen = 27
ObjectiveMaximize sum of radii
Best known, unproven2.690155069fixed-perimeter rectangle configuration from the public repository of Berthold, Kamp, Mexi, Pokutta and Polik; its continuous sum is 2.690155081571631. The same configuration was conservatively quantized to the site's nine-decimal grid and verifies at 2.690155069; optimality is open

Formal definition

  • Containerwidth>0, height>0, and width+height=2; the lower-left corner is fixed at (0,0)
  • SubmissionExactly n circles, each written [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 integer sum of all radii
Challenge this record ↗
1.051448529y0
0x0.948551471
VERIFIED CONSTRUCTIONscored by the sum of the radii

Getting a feel for it

Optimizing container and cargo together

With a fixed square only the circles move. Here, transferring a little perimeter from height to width changes every wall contact and gap. It is a minimal model of coupled container selection and packing design.

Why uniformity loses

Equal radii usually waste the gaps between curved boundaries. Large circles determine the aspect ratio while small circles fill holes in the contact graph; adding one circle can make the optimal rectangle jump to a different structure.

Published frontier

A 2026 public computational project supplies complete coordinates through n=60. The arena starts at the first non-trivial row n=3 and exhibits verifiable nine-decimal versions; these are best-known constructions, not optimality proofs.

Source
The record-holding arrangement for Variable-radius circles in a fixed-perimeter rectangle n = 27, 2.690155069
Current leader

2.690155069

sum of radii

Matches the best known
Answer sourceTimo Berthold, Dominik Kamp, Gonzalo Mexi, Sebastian Pokutta and Ilja Polik
Solution methodPublished reference construction
Challenge this record ↗
ANSWER FORMAT

How to write your answer

The rectangle has lower-left corner (0,0), width width and height height; its perimeter is 2(width+height)=4.

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

The current leader's answer

{
  "circles": [
    [
      "0.845094457",
      "0.544247521",
      "0.103457014"
    ],
    [
      "0.47596647",
      "0.346008008",
      "0.109457708"
    ],
    [
      "0.286440093",
      "0.450222897",
      "0.106831428"
    ],
    [
      "0.834052187",
      "0.114499284",
      "0.114499283"
    ],
    [
      "0.100821352",
      "0.54331104",
      "0.100821351"
    ],
    [
      "0.399008489",
      "0.792248447",
      "0.095572472"
    ],
    [
      "0.29383781",
      "0.069466133",
      "0.069465838"
    ],
    [
      "0.47512412",
      "0.118276585",
      "0.118275172"
    ],
    [
      "0.265555118",
      "0.653164123",
      "0.097180875"
    ],
    [
      "0.656043561",
      "0.069186155",
      "0.069186061"
    ],
    [
      "0.693855188",
      "0.954363446",
      "0.097083986"
    ],
    [
      "0.842567406",
      "0.334820922",
      "0.105984064"
    ],
    [
      "0.664430858",
      "0.444208179",
      "0.103054745"
    ],
    [
      "0.292003466",
      "0.241192131",
      "0.102269955"
    ],
    [
      "0.109872392",
      "0.941576136",
      "0.109872391"
    ],
    [
      "0.869293319",
      "0.972190376",
      "0.079257548"
    ],
    [
      "0.500454966",
      "0.955130975",
      "0.096317553"
    ],
    [
      "0.479934446",
      "0.583627995",
      "0.128193855"
    ],
    [
      "0.094145304",
      "0.738164256",
      "0.094145303"
    ],
    [
      "0.11504476",
      "0.11504476",
      "0.11504476"
    ],
    [
      "0.658177193",
      "0.239804304",
      "0.10144477"
    ],
    [
      "0.311589918",
      "0.958864831",
      "0.092583697"
    ],
    [
      "0.682192621",
      "0.624555594",
      "0.078163547"
    ],
    [
      "0.106321971",
      "0.336240125",
      "0.10632197"
    ],
    [
      "0.82271063",
      "0.772452011",
      "0.125840841"
    ],
    [
      "0.595686395",
      "0.781881989",
      "0.101377315"
    ],
    [
      "0.2367535",
      "0.81627353",
      "0.068451567"
    ]
  ],
  "height": "1.051448529",
  "width": "0.948551471"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 27
}

The current leader's answer

{
  "circles": [
    [
      "0.845094457",
      "0.544247521",
      "0.103457014"
    ],
    [
      "0.47596647",
      "0.346008008",
      "0.109457708"
    ],
    [
      "0.286440093",
      "0.450222897",
      "0.106831428"
    ],
    [
      "0.834052187",
      "0.114499284",
      "0.114499283"
    ],
    [
      "0.100821352",
      "0.54331104",
      "0.100821351"
    ],
    [
      "0.399008489",
      "0.792248447",
      "0.095572472"
    ],
    [
      "0.29383781",
      "0.069466133",
      "0.069465838"
    ],
    [
      "0.47512412",
      "0.118276585",
      "0.118275172"
    ],
    [
      "0.265555118",
      "0.653164123",
      "0.097180875"
    ],
    [
      "0.656043561",
      "0.069186155",
      "0.069186061"
    ],
    [
      "0.693855188",
      "0.954363446",
      "0.097083986"
    ],
    [
      "0.842567406",
      "0.334820922",
      "0.105984064"
    ],
    [
      "0.664430858",
      "0.444208179",
      "0.103054745"
    ],
    [
      "0.292003466",
      "0.241192131",
      "0.102269955"
    ],
    [
      "0.109872392",
      "0.941576136",
      "0.109872391"
    ],
    [
      "0.869293319",
      "0.972190376",
      "0.079257548"
    ],
    [
      "0.500454966",
      "0.955130975",
      "0.096317553"
    ],
    [
      "0.479934446",
      "0.583627995",
      "0.128193855"
    ],
    [
      "0.094145304",
      "0.738164256",
      "0.094145303"
    ],
    [
      "0.11504476",
      "0.11504476",
      "0.11504476"
    ],
    [
      "0.658177193",
      "0.239804304",
      "0.10144477"
    ],
    [
      "0.311589918",
      "0.958864831",
      "0.092583697"
    ],
    [
      "0.682192621",
      "0.624555594",
      "0.078163547"
    ],
    [
      "0.106321971",
      "0.336240125",
      "0.10632197"
    ],
    [
      "0.82271063",
      "0.772452011",
      "0.125840841"
    ],
    [
      "0.595686395",
      "0.781881989",
      "0.101377315"
    ],
    [
      "0.2367535",
      "0.81627353",
      "0.068451567"
    ]
  ],
  "height": "1.051448529",
  "width": "0.948551471"
}

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

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