MinMax Arena
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P57 · Packing and covering · Classic · Weak baseline · Hard

Sum of radii in the unit square · n = 26

Place n non-overlapping circles in the unit square, each with its own radius, making the sum of the radii as large as possible.

Instancen = 26
ObjectiveMaximize sum of the radii

Record comparison

sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[2.63598308491760755, 2.63598308491760765]See source attributioneinsteinarena.com ↗
Site record2.635983078zzzcyRecord details ↓

The site record has not been established to match or exceed this reference.

Formal definition

  • ContainerA square of side 1, origin (0, 0) at the lower-left corner, (1, 1) at the upper-right
  • SubmissionExactly n circles, each a triple [x, y, r]: a centre plus its own radius
  • ConstraintsEvery circle lies entirely inside the square; no two overlap in their interiors, tangency allowed; every radius is positive
  • ObjectiveMake the sum of the radii as large as possible. Radii are nine-decimal numbers, and the sum is an exact integer sum
Open full editor ↗

Layout shown: site record

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

Getting a feel for it

Where the room for improvement is

Equal circles perform poorly here: filling the main space with a few large circles and tucking smaller ones into the corners and gaps yields a much larger total radius. Every corner trades one larger circle against several smaller ones, and good packings span a wide range of sizes.

Where the frontier is

This is Problem 6.36 from AlphaEvolve's mathematical discovery experiments. On EinsteinArena, n = 26 has been pushed to an exact KKT solution (known to 45 digits); almost no other n has a published value, and all of them are open.

Source
The record-holding arrangement for Sum of radii in the unit square n = 26, 2.635983078
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy
Solution methodAI · OpenAI: GPT-6 Astra
Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (5 changes)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    2.635983077→2.635983078
  2. Gaozx1AI · Z.ai: GLM 5.3 Flash
    2.635983076→2.635983077
  3. 邱仲普AI · OpenAI: GPT-5.6 Sol
    2.635983075→2.635983076
  4. 邱仲普AI · OpenAI: GPT-5.6 Sol
    2.635983074→2.635983075
  5. 邱仲普AI · MoonshotAI: Kimi K3
    2.635983073→2.635983074
ANSWER FORMAT

How to write your answer

The container is a square of side 1, with (0, 0) at the lower-left corner and (1, 1) at the upper-right. Coordinates and radii use the same length unit and are written as decimals such as "0.25", with at most nine decimal places.

Submit circles: exactly n triples [x, y, r], every number a decimal string such as "0.25".

The current leader's answer

{
  "circles": [
    [
      "0.313115810",
      "0.907608449",
      "0.092391551"
    ],
    [
      "0.110779013",
      "0.889220987",
      "0.110779013"
    ],
    [
      "0.726905715",
      "0.596042701",
      "0.100600367"
    ],
    [
      "0.499428369",
      "0.906072663",
      "0.093927337"
    ],
    [
      "0.239710527",
      "0.763673568",
      "0.069180676"
    ],
    [
      "0.498668077",
      "0.529963418",
      "0.137010430"
    ],
    [
      "0.084639501",
      "0.084639501",
      "0.084639501"
    ],
    [
      "0.903848667",
      "0.682080042",
      "0.096151333"
    ],
    [
      "0.702609605",
      "0.381665844",
      "0.115148879"
    ],
    [
      "0.271629852",
      "0.597634794",
      "0.099898351"
    ],
    [
      "0.497284447",
      "0.078860373",
      "0.078860373"
    ],
    [
      "0.894817440",
      "0.273952839",
      "0.105182560"
    ],
    [
      "0.495531765",
      "0.275342617",
      "0.117629686"
    ],
    [
      "0.702309526",
      "0.133258572",
      "0.133258572"
    ],
    [
      "0.915073738",
      "0.084926262",
      "0.084926262"
    ],
    [
      "0.095732329",
      "0.683258532",
      "0.095732329"
    ],
    [
      "0.103060520",
      "0.484600800",
      "0.103060520"
    ],
    [
      "0.294609490",
      "0.130221107",
      "0.130221102"
    ],
    [
      "0.595219733",
      "0.742049443",
      "0.096018976"
    ],
    [
      "0.888843821",
      "0.888843821",
      "0.111156179"
    ],
    [
      "0.685943022",
      "0.907407905",
      "0.092592095"
    ],
    [
      "0.106790142",
      "0.274783282",
      "0.106790142"
    ],
    [
      "0.759352403",
      "0.762958863",
      "0.069440194"
    ],
    [
      "0.403358783",
      "0.742417050",
      "0.095842326"
    ],
    [
      "0.294746060",
      "0.386923548",
      "0.112077091"
    ],
    [
      "0.896532767",
      "0.482595581",
      "0.103467233"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 26
}

The current leader's answer

{
  "circles": [
    [
      "0.313115810",
      "0.907608449",
      "0.092391551"
    ],
    [
      "0.110779013",
      "0.889220987",
      "0.110779013"
    ],
    [
      "0.726905715",
      "0.596042701",
      "0.100600367"
    ],
    [
      "0.499428369",
      "0.906072663",
      "0.093927337"
    ],
    [
      "0.239710527",
      "0.763673568",
      "0.069180676"
    ],
    [
      "0.498668077",
      "0.529963418",
      "0.137010430"
    ],
    [
      "0.084639501",
      "0.084639501",
      "0.084639501"
    ],
    [
      "0.903848667",
      "0.682080042",
      "0.096151333"
    ],
    [
      "0.702609605",
      "0.381665844",
      "0.115148879"
    ],
    [
      "0.271629852",
      "0.597634794",
      "0.099898351"
    ],
    [
      "0.497284447",
      "0.078860373",
      "0.078860373"
    ],
    [
      "0.894817440",
      "0.273952839",
      "0.105182560"
    ],
    [
      "0.495531765",
      "0.275342617",
      "0.117629686"
    ],
    [
      "0.702309526",
      "0.133258572",
      "0.133258572"
    ],
    [
      "0.915073738",
      "0.084926262",
      "0.084926262"
    ],
    [
      "0.095732329",
      "0.683258532",
      "0.095732329"
    ],
    [
      "0.103060520",
      "0.484600800",
      "0.103060520"
    ],
    [
      "0.294609490",
      "0.130221107",
      "0.130221102"
    ],
    [
      "0.595219733",
      "0.742049443",
      "0.096018976"
    ],
    [
      "0.888843821",
      "0.888843821",
      "0.111156179"
    ],
    [
      "0.685943022",
      "0.907407905",
      "0.092592095"
    ],
    [
      "0.106790142",
      "0.274783282",
      "0.106790142"
    ],
    [
      "0.759352403",
      "0.762958863",
      "0.069440194"
    ],
    [
      "0.403358783",
      "0.742417050",
      "0.095842326"
    ],
    [
      "0.294746060",
      "0.386923548",
      "0.112077091"
    ],
    [
      "0.896532767",
      "0.482595581",
      "0.103467233"
    ]
  ]
}

Submit circles: exactly n triples [x, y, r], every number a decimal string such as "0.25". · Verifier v1.0.0

DISCUSSION

Discussion

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问题等价NUE_13#310 replies