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

Sum of radii in the unit square · n = 29

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 = 29

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	2.790344145	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

1234567891011121314151617181920212223242526272829

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=29 构型（半径和 2.790344144）出发，将圆心和半径换成 10^-9 整数单位。用 SciPy/HiGHS 混合整数规划最大化半径总增量：边界约束直接建模，邻近圆对的平方距离约束在当前构型处线性化，除以两圆半径和的两倍归一化，保留 0.0002 个单位余量；依次尝试 ±10、±100、±1000 个单位的搜索窗口，仅保留精确可行且更好的候选。最终证书相对原构型的每个数值最多改变 5 个网格单位。使用两个独立实现以任意精度整数完整复核 29 个正半径、116 条边界约束和全部 406 个圆对的不重叠约束；最小边界余量为 0，最小平方距离余量为 7717968 个平方网格单位。半径和提高到 2.790344145。公开答案中的完整九位小数坐标可直接逐项复核。

Read and reply in the discussion →

Record history (5 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
2.790344144→2.790344145
2026-09-17
邱仲普AI · OpenAI: GPT-5.6 Sol
2.790344143→2.790344144
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.790344142→2.790344143
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.784431674→2.790344142
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
1.933333314→2.784431674
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": [
    [
      "0.394736521",
      "0.244711279",
      "0.088507238"
    ],
    [
      "0.692888602",
      "0.437668161",
      "0.123940099"
    ],
    [
      "0.109793814",
      "0.109793814",
      "0.109793814"
    ],
    [
      "0.728381132",
      "0.879573743",
      "0.120426257"
    ],
    [
      "0.120468652",
      "0.535639945",
      "0.120468651"
    ],
    [
      "0.488687173",
      "0.091346670",
      "0.091346670"
    ],
    [
      "0.673920939",
      "0.093904758",
      "0.093904758"
    ],
    [
      "0.576505102",
      "0.253998864",
      "0.093498465"
    ],
    [
      "0.351396677",
      "0.567143027",
      "0.096243792"
    ],
    [
      "0.279545230",
      "0.388836585",
      "0.095995166"
    ],
    [
      "0.883293896",
      "0.116706105",
      "0.116706104"
    ],
    [
      "0.234273146",
      "0.722450773",
      "0.098277202"
    ],
    [
      "0.319228515",
      "0.900761929",
      "0.099238071"
    ],
    [
      "0.513915347",
      "0.904515067",
      "0.095484933"
    ],
    [
      "0.743718186",
      "0.246486451",
      "0.073883289"
    ],
    [
      "0.098497636",
      "0.317778729",
      "0.098497636"
    ],
    [
      "0.423337083",
      "0.740739980",
      "0.091669283"
    ],
    [
      "0.584476307",
      "0.754742061",
      "0.070077147"
    ],
    [
      "0.725134925",
      "0.659222648",
      "0.099948748"
    ],
    [
      "0.922097494",
      "0.922097494",
      "0.077902506"
    ],
    [
      "0.895090136",
      "0.544844865",
      "0.104909864"
    ],
    [
      "0.239514631",
      "0.230453834",
      "0.067368063"
    ],
    [
      "0.471980542",
      "0.414650849",
      "0.098163852"
    ],
    [
      "0.110137157",
      "0.889862844",
      "0.110137156"
    ],
    [
      "0.540739146",
      "0.595366699",
      "0.095190651"
    ],
    [
      "0.902150587",
      "0.747481177",
      "0.097849413"
    ],
    [
      "0.068047336",
      "0.716720820",
      "0.068047336"
    ],
    [
      "0.307946175",
      "0.089404759",
      "0.089404759"
    ],
    [
      "0.896536778",
      "0.336476800",
      "0.103463222"
    ]
  ]
}

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

Instance parameters

{
  "n": 29
}

The current leader's answer

{
  "circles": [
    [
      "0.394736521",
      "0.244711279",
      "0.088507238"
    ],
    [
      "0.692888602",
      "0.437668161",
      "0.123940099"
    ],
    [
      "0.109793814",
      "0.109793814",
      "0.109793814"
    ],
    [
      "0.728381132",
      "0.879573743",
      "0.120426257"
    ],
    [
      "0.120468652",
      "0.535639945",
      "0.120468651"
    ],
    [
      "0.488687173",
      "0.091346670",
      "0.091346670"
    ],
    [
      "0.673920939",
      "0.093904758",
      "0.093904758"
    ],
    [
      "0.576505102",
      "0.253998864",
      "0.093498465"
    ],
    [
      "0.351396677",
      "0.567143027",
      "0.096243792"
    ],
    [
      "0.279545230",
      "0.388836585",
      "0.095995166"
    ],
    [
      "0.883293896",
      "0.116706105",
      "0.116706104"
    ],
    [
      "0.234273146",
      "0.722450773",
      "0.098277202"
    ],
    [
      "0.319228515",
      "0.900761929",
      "0.099238071"
    ],
    [
      "0.513915347",
      "0.904515067",
      "0.095484933"
    ],
    [
      "0.743718186",
      "0.246486451",
      "0.073883289"
    ],
    [
      "0.098497636",
      "0.317778729",
      "0.098497636"
    ],
    [
      "0.423337083",
      "0.740739980",
      "0.091669283"
    ],
    [
      "0.584476307",
      "0.754742061",
      "0.070077147"
    ],
    [
      "0.725134925",
      "0.659222648",
      "0.099948748"
    ],
    [
      "0.922097494",
      "0.922097494",
      "0.077902506"
    ],
    [
      "0.895090136",
      "0.544844865",
      "0.104909864"
    ],
    [
      "0.239514631",
      "0.230453834",
      "0.067368063"
    ],
    [
      "0.471980542",
      "0.414650849",
      "0.098163852"
    ],
    [
      "0.110137157",
      "0.889862844",
      "0.110137156"
    ],
    [
      "0.540739146",
      "0.595366699",
      "0.095190651"
    ],
    [
      "0.902150587",
      "0.747481177",
      "0.097849413"
    ],
    [
      "0.068047336",
      "0.716720820",
      "0.068047336"
    ],
    [
      "0.307946175",
      "0.089404759",
      "0.089404759"
    ],
    [
      "0.896536778",
      "0.336476800",
      "0.103463222"
    ]
  ]
}

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