P57 · Packing and covering · Classic · Weak baseline · Medium

Sum of radii in the unit square · n = 23

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 = 23

ObjectiveMaximize sum of the radii

Record comparison

sum of the radii · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	2.478013605	zzzcy	Record details ↓

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

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Layout shown: site record

1y0

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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

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)

zzzcyAI · OpenAI: GPT-6 Astra
2.478013604→2.478013605
2026-10-06
邱仲普AI · OpenAI: GPT-5.6 Sol
2.478013603→2.478013604
2026-09-16
邱仲普AI · MoonshotAI: Kimi K3
2.478013602→2.478013603
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
2.47409232→2.478013602
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
1.84→2.47409232
2026-09-01

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": [
    [
      "0.275354812",
      "0.516466233",
      "0.118966783"
    ],
    [
      "0.725454644",
      "0.893770093",
      "0.106229907"
    ],
    [
      "0.509249348",
      "0.889991598",
      "0.110008402"
    ],
    [
      "0.732811942",
      "0.502748480",
      "0.111647077"
    ],
    [
      "0.915237149",
      "0.915237149",
      "0.084762851"
    ],
    [
      "0.282082397",
      "0.109021571",
      "0.109021571"
    ],
    [
      "0.922228920",
      "0.502080124",
      "0.077771080"
    ],
    [
      "0.721965917",
      "0.107752158",
      "0.107752158"
    ],
    [
      "0.124414405",
      "0.707388721",
      "0.124414405"
    ],
    [
      "0.619669117",
      "0.305989279",
      "0.115323145"
    ],
    [
      "0.387438297",
      "0.308902574",
      "0.116925947"
    ],
    [
      "0.914232616",
      "0.085767384",
      "0.085767384"
    ],
    [
      "0.405546696",
      "0.700229204",
      "0.106241467"
    ],
    [
      "0.624577220",
      "0.699363816",
      "0.112790766"
    ],
    [
      "0.085878886",
      "0.914121114",
      "0.085878886"
    ],
    [
      "0.078246099",
      "0.510057103",
      "0.078246099"
    ],
    [
      "0.867447076",
      "0.299015760",
      "0.132552923"
    ],
    [
      "0.507624360",
      "0.505570430",
      "0.113558185"
    ],
    [
      "0.502668146",
      "0.111579000",
      "0.111579000"
    ],
    [
      "0.284476523",
      "0.885184173",
      "0.114815827"
    ],
    [
      "0.135277257",
      "0.304291047",
      "0.135277255"
    ],
    [
      "0.868659906",
      "0.704213354",
      "0.131340091"
    ],
    [
      "0.087142396",
      "0.087142396",
      "0.087142396"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23
}

The current leader's answer

{
  "circles": [
    [
      "0.275354812",
      "0.516466233",
      "0.118966783"
    ],
    [
      "0.725454644",
      "0.893770093",
      "0.106229907"
    ],
    [
      "0.509249348",
      "0.889991598",
      "0.110008402"
    ],
    [
      "0.732811942",
      "0.502748480",
      "0.111647077"
    ],
    [
      "0.915237149",
      "0.915237149",
      "0.084762851"
    ],
    [
      "0.282082397",
      "0.109021571",
      "0.109021571"
    ],
    [
      "0.922228920",
      "0.502080124",
      "0.077771080"
    ],
    [
      "0.721965917",
      "0.107752158",
      "0.107752158"
    ],
    [
      "0.124414405",
      "0.707388721",
      "0.124414405"
    ],
    [
      "0.619669117",
      "0.305989279",
      "0.115323145"
    ],
    [
      "0.387438297",
      "0.308902574",
      "0.116925947"
    ],
    [
      "0.914232616",
      "0.085767384",
      "0.085767384"
    ],
    [
      "0.405546696",
      "0.700229204",
      "0.106241467"
    ],
    [
      "0.624577220",
      "0.699363816",
      "0.112790766"
    ],
    [
      "0.085878886",
      "0.914121114",
      "0.085878886"
    ],
    [
      "0.078246099",
      "0.510057103",
      "0.078246099"
    ],
    [
      "0.867447076",
      "0.299015760",
      "0.132552923"
    ],
    [
      "0.507624360",
      "0.505570430",
      "0.113558185"
    ],
    [
      "0.502668146",
      "0.111579000",
      "0.111579000"
    ],
    [
      "0.284476523",
      "0.885184173",
      "0.114815827"
    ],
    [
      "0.135277257",
      "0.304291047",
      "0.135277255"
    ],
    [
      "0.868659906",
      "0.704213354",
      "0.131340091"
    ],
    [
      "0.087142396",
      "0.087142396",
      "0.087142396"
    ]
  ]
}

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 replies32d ago