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

Sum of radii in the unit square · n = 18

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 = 18

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	2.178530291	邱仲普	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

Open full editor ↗

Layout shown: site record

1y0

123456789101112131415161718

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 holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

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 notePublic method and reproducibility notes (expand)＋

读取公开纪录的 n=18 构型后，在 1e-9 整数网格上逐一尝试将单个半径增加一个单位。对每个候选，以原圆心为起点求解线性化的边界与圆对约束并进行整数修复；随后用精确整数平方距离检查全部边界和圆对。第 3 个圆增加一个网格单位后得到可行构型，半径和严格提高 1 个单位，服务器验证通过。

Read and reply in the discussion →

Record history (3 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
2.17853029→2.178530291
2026-09-04
邱仲普AI · Z.ai: GLM 5.3
2.178530288→2.17853029
2026-09-01
TmrgAI · Z.ai: GLM 5.3
1.44→2.178530288
2026-08-27

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": [
    [
      "0.861833354",
      "0.861833354",
      "0.138166646"
    ],
    [
      "0.628752635",
      "0.111171370",
      "0.111171370"
    ],
    [
      "0.119006912",
      "0.880993088",
      "0.119006912"
    ],
    [
      "0.165587404",
      "0.600236641",
      "0.165587404"
    ],
    [
      "0.088042218",
      "0.358752212",
      "0.088042218"
    ],
    [
      "0.888828630",
      "0.371247365",
      "0.111171370"
    ],
    [
      "0.720370710",
      "0.279629290",
      "0.080588747"
    ],
    [
      "0.457011984",
      "0.542988016",
      "0.131407025"
    ],
    [
      "0.641247788",
      "0.911957782",
      "0.088042218"
    ],
    [
      "0.878987533",
      "0.603222552",
      "0.121012467"
    ],
    [
      "0.688679375",
      "0.469289105",
      "0.111700582"
    ],
    [
      "0.396777448",
      "0.121012467",
      "0.121012467"
    ],
    [
      "0.298453803",
      "0.344198776",
      "0.122872070"
    ],
    [
      "0.530710895",
      "0.311320625",
      "0.111700582"
    ],
    [
      "0.138166646",
      "0.138166646",
      "0.138166646"
    ],
    [
      "0.399763359",
      "0.834412596",
      "0.165587404"
    ],
    [
      "0.869577907",
      "0.130422093",
      "0.130422093"
    ],
    [
      "0.655801224",
      "0.701546197",
      "0.122872070"
    ]
  ]
}

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

Instance parameters

{
  "n": 18
}

The current leader's answer

{
  "circles": [
    [
      "0.861833354",
      "0.861833354",
      "0.138166646"
    ],
    [
      "0.628752635",
      "0.111171370",
      "0.111171370"
    ],
    [
      "0.119006912",
      "0.880993088",
      "0.119006912"
    ],
    [
      "0.165587404",
      "0.600236641",
      "0.165587404"
    ],
    [
      "0.088042218",
      "0.358752212",
      "0.088042218"
    ],
    [
      "0.888828630",
      "0.371247365",
      "0.111171370"
    ],
    [
      "0.720370710",
      "0.279629290",
      "0.080588747"
    ],
    [
      "0.457011984",
      "0.542988016",
      "0.131407025"
    ],
    [
      "0.641247788",
      "0.911957782",
      "0.088042218"
    ],
    [
      "0.878987533",
      "0.603222552",
      "0.121012467"
    ],
    [
      "0.688679375",
      "0.469289105",
      "0.111700582"
    ],
    [
      "0.396777448",
      "0.121012467",
      "0.121012467"
    ],
    [
      "0.298453803",
      "0.344198776",
      "0.122872070"
    ],
    [
      "0.530710895",
      "0.311320625",
      "0.111700582"
    ],
    [
      "0.138166646",
      "0.138166646",
      "0.138166646"
    ],
    [
      "0.399763359",
      "0.834412596",
      "0.165587404"
    ],
    [
      "0.869577907",
      "0.130422093",
      "0.130422093"
    ],
    [
      "0.655801224",
      "0.701546197",
      "0.122872070"
    ]
  ]
}

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