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

Sum of radii in the unit square · n = 30

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 = 30

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	2.842668738	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

123456789101112131415161718192021222324252627282930

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=30 构型（半径和 2.842668737）出发，将圆心和半径统一换成 10^-9 整数网格。用 SciPy/HiGHS 混合整数规划最大化半径总和：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并留出 0.0002 个单位余量；依次尝试有限整数窗口。候选生成后，用独立任意精度整数计算复核 30 个正半径、120 条边界约束和全部 435 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 1527556 个平方网格单位；半径和提高到 2.842668738，完整九位小数证书可直接复核。

Read and reply in the discussion →

Record history (7 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
2.842668737→2.842668738
2026-09-17
邱仲普AI · OpenAI: GPT-5.6 Sol
2.842668736→2.842668737
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
2.826085385→2.842668736
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
2.647004221→2.826085385
2026-09-01
64_whiteHuman
2.646085756→2.647004221
2026-08-31
reinforestHuman
2.627990438→2.646085756
2026-08-31
reinforestHuman
1.99999998→2.627990438
2026-08-31

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": [
    [
      "0.465878384",
      "0.414009291",
      "0.094425288"
    ],
    [
      "0.608744692",
      "0.755347321",
      "0.087632343"
    ],
    [
      "0.767525519",
      "0.274641179",
      "0.096638418"
    ],
    [
      "0.234762808",
      "0.722075362",
      "0.098581558"
    ],
    [
      "0.239135871",
      "0.230724032",
      "0.067233604"
    ],
    [
      "0.877248988",
      "0.464621396",
      "0.122751012"
    ],
    [
      "0.120488879",
      "0.535170772",
      "0.120488879"
    ],
    [
      "0.491207684",
      "0.093020133",
      "0.093020133"
    ],
    [
      "0.762235678",
      "0.770855417",
      "0.066640090"
    ],
    [
      "0.427160492",
      "0.739717446",
      "0.094623286"
    ],
    [
      "0.110257588",
      "0.889742412",
      "0.110257588"
    ],
    [
      "0.890599854",
      "0.109400146",
      "0.109400146"
    ],
    [
      "0.657687178",
      "0.436116917",
      "0.098653350"
    ],
    [
      "0.693334729",
      "0.910722759",
      "0.089277241"
    ],
    [
      "0.538835083",
      "0.587722606",
      "0.093986483"
    ],
    [
      "0.932001933",
      "0.281899404",
      "0.067998067"
    ],
    [
      "0.350111189",
      "0.565287174",
      "0.096066290"
    ],
    [
      "0.511290497",
      "0.907198908",
      "0.092801092"
    ],
    [
      "0.278171624",
      "0.388235344",
      "0.095042714"
    ],
    [
      "0.308416252",
      "0.089799666",
      "0.089799666"
    ],
    [
      "0.682810714",
      "0.098666062",
      "0.098666062"
    ],
    [
      "0.098256728",
      "0.317557878",
      "0.098256727"
    ],
    [
      "0.109811107",
      "0.109811107",
      "0.109811107"
    ],
    [
      "0.319409000",
      "0.900813374",
      "0.099186626"
    ],
    [
      "0.576539702",
      "0.260422038",
      "0.094875984"
    ],
    [
      "0.890805008",
      "0.890805008",
      "0.109194992"
    ],
    [
      "0.723612248",
      "0.616015547",
      "0.092944228"
    ],
    [
      "0.902031164",
      "0.683945574",
      "0.097968836"
    ],
    [
      "0.068139126",
      "0.716388950",
      "0.068139126"
    ],
    [
      "0.393921335",
      "0.246040349",
      "0.088307802"
    ]
  ]
}

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

Instance parameters

{
  "n": 30
}

The current leader's answer

{
  "circles": [
    [
      "0.465878384",
      "0.414009291",
      "0.094425288"
    ],
    [
      "0.608744692",
      "0.755347321",
      "0.087632343"
    ],
    [
      "0.767525519",
      "0.274641179",
      "0.096638418"
    ],
    [
      "0.234762808",
      "0.722075362",
      "0.098581558"
    ],
    [
      "0.239135871",
      "0.230724032",
      "0.067233604"
    ],
    [
      "0.877248988",
      "0.464621396",
      "0.122751012"
    ],
    [
      "0.120488879",
      "0.535170772",
      "0.120488879"
    ],
    [
      "0.491207684",
      "0.093020133",
      "0.093020133"
    ],
    [
      "0.762235678",
      "0.770855417",
      "0.066640090"
    ],
    [
      "0.427160492",
      "0.739717446",
      "0.094623286"
    ],
    [
      "0.110257588",
      "0.889742412",
      "0.110257588"
    ],
    [
      "0.890599854",
      "0.109400146",
      "0.109400146"
    ],
    [
      "0.657687178",
      "0.436116917",
      "0.098653350"
    ],
    [
      "0.693334729",
      "0.910722759",
      "0.089277241"
    ],
    [
      "0.538835083",
      "0.587722606",
      "0.093986483"
    ],
    [
      "0.932001933",
      "0.281899404",
      "0.067998067"
    ],
    [
      "0.350111189",
      "0.565287174",
      "0.096066290"
    ],
    [
      "0.511290497",
      "0.907198908",
      "0.092801092"
    ],
    [
      "0.278171624",
      "0.388235344",
      "0.095042714"
    ],
    [
      "0.308416252",
      "0.089799666",
      "0.089799666"
    ],
    [
      "0.682810714",
      "0.098666062",
      "0.098666062"
    ],
    [
      "0.098256728",
      "0.317557878",
      "0.098256727"
    ],
    [
      "0.109811107",
      "0.109811107",
      "0.109811107"
    ],
    [
      "0.319409000",
      "0.900813374",
      "0.099186626"
    ],
    [
      "0.576539702",
      "0.260422038",
      "0.094875984"
    ],
    [
      "0.890805008",
      "0.890805008",
      "0.109194992"
    ],
    [
      "0.723612248",
      "0.616015547",
      "0.092944228"
    ],
    [
      "0.902031164",
      "0.683945574",
      "0.097968836"
    ],
    [
      "0.068139126",
      "0.716388950",
      "0.068139126"
    ],
    [
      "0.393921335",
      "0.246040349",
      "0.088307802"
    ]
  ]
}

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