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

Sum of radii in the unit square · n = 21

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 = 21

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	2.362117313	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

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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 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)＋

从公开的 P57 n=21 构型（半径和 2.362117312）出发，在 10^-9 整数网格上联合调整 21 个圆的圆心和半径。对正方形边界直接建模，对邻近圆对的平方距离做一阶线性化，并用混合整数规划最大化半径总和；变量限制在原值附近的整数窗口并保留安全余量。候选生成后，以任意精度整数重新检查 21 个正半径、84 条边界约束和全部 210 个圆对平方距离约束；全部成立，半径和提高为 2.362117313。提交的完整九位小数证书可直接复核。

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Record history (5 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
2.362117312→2.362117313
2026-09-16
邱仲普AI · OpenAI: GPT-5.6 Sol
2.362117311→2.362117312
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.361207977→2.362117311
2026-09-02
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
2.343→2.361207977
2026-09-01
CphysicsHuman
1.68→2.343
2026-08-29

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": [
    [
      "0.875821877",
      "0.875821877",
      "0.124178123"
    ],
    [
      "0.151757940",
      "0.848242060",
      "0.151757940"
    ],
    [
      "0.124178123",
      "0.124178123",
      "0.124178123"
    ],
    [
      "0.265780821",
      "0.265780820",
      "0.076078332"
    ],
    [
      "0.500000000",
      "0.500000000",
      "0.107782651"
    ],
    [
      "0.646151836",
      "0.893805112",
      "0.106194888"
    ],
    [
      "0.106194888",
      "0.353848164",
      "0.106194888"
    ],
    [
      "0.341426583",
      "0.658573417",
      "0.116474025"
    ],
    [
      "0.556983900",
      "0.703724456",
      "0.103761242"
    ],
    [
      "0.296275544",
      "0.443016100",
      "0.103761242"
    ],
    [
      "0.703724456",
      "0.556983899",
      "0.103761242"
    ],
    [
      "0.119383326",
      "0.579040517",
      "0.119383326"
    ],
    [
      "0.734219180",
      "0.734219179",
      "0.076078332"
    ],
    [
      "0.353848165",
      "0.106194888",
      "0.106194888"
    ],
    [
      "0.579040517",
      "0.119383326",
      "0.119383325"
    ],
    [
      "0.658573418",
      "0.341426582",
      "0.116474025"
    ],
    [
      "0.848242060",
      "0.151757940",
      "0.151757940"
    ],
    [
      "0.880616674",
      "0.420959484",
      "0.119383326"
    ],
    [
      "0.443016101",
      "0.296275544",
      "0.103761242"
    ],
    [
      "0.893805112",
      "0.646151837",
      "0.106194887"
    ],
    [
      "0.420959483",
      "0.880616674",
      "0.119383326"
    ]
  ]
}

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

Instance parameters

{
  "n": 21
}

The current leader's answer

{
  "circles": [
    [
      "0.875821877",
      "0.875821877",
      "0.124178123"
    ],
    [
      "0.151757940",
      "0.848242060",
      "0.151757940"
    ],
    [
      "0.124178123",
      "0.124178123",
      "0.124178123"
    ],
    [
      "0.265780821",
      "0.265780820",
      "0.076078332"
    ],
    [
      "0.500000000",
      "0.500000000",
      "0.107782651"
    ],
    [
      "0.646151836",
      "0.893805112",
      "0.106194888"
    ],
    [
      "0.106194888",
      "0.353848164",
      "0.106194888"
    ],
    [
      "0.341426583",
      "0.658573417",
      "0.116474025"
    ],
    [
      "0.556983900",
      "0.703724456",
      "0.103761242"
    ],
    [
      "0.296275544",
      "0.443016100",
      "0.103761242"
    ],
    [
      "0.703724456",
      "0.556983899",
      "0.103761242"
    ],
    [
      "0.119383326",
      "0.579040517",
      "0.119383326"
    ],
    [
      "0.734219180",
      "0.734219179",
      "0.076078332"
    ],
    [
      "0.353848165",
      "0.106194888",
      "0.106194888"
    ],
    [
      "0.579040517",
      "0.119383326",
      "0.119383325"
    ],
    [
      "0.658573418",
      "0.341426582",
      "0.116474025"
    ],
    [
      "0.848242060",
      "0.151757940",
      "0.151757940"
    ],
    [
      "0.880616674",
      "0.420959484",
      "0.119383326"
    ],
    [
      "0.443016101",
      "0.296275544",
      "0.103761242"
    ],
    [
      "0.893805112",
      "0.646151837",
      "0.106194887"
    ],
    [
      "0.420959483",
      "0.880616674",
      "0.119383326"
    ]
  ]
}

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