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

Sum of radii in the unit square · n = 22

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 = 22

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	2.420202644	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

12345678910111213141516171819202122

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=22 构型（半径和 2.420202643）出发，把圆心与半径表示为 10^-9 整数网格。对四条正方形边界直接加入整数约束，对圆对平方距离在原构型处做线性化，以混合整数规划最大化半径增量；变量限制在原值附近的整数窗口，并保留线性化安全余量。候选生成后，独立使用任意精度整数逐项检查 22 个正半径、88 条边界约束和全部 231 个圆对的平方距离约束。检查全部通过，半径和提高一个网格单位至 2.420202644；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
2.420202643→2.420202644
2026-09-16
邱仲普AI · OpenAI: GPT-5.6 Sol
2.420202642→2.420202643
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
2.409855534→2.420202642
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
1.78→2.409855534
2026-09-01
lirdHuman
1.76→1.78
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": [
    [
      "0.876891459",
      "0.500149123",
      "0.123108541"
    ],
    [
      "0.644234338",
      "0.500232170",
      "0.109548593"
    ],
    [
      "0.313708348",
      "0.327627744",
      "0.107521408"
    ],
    [
      "0.878028964",
      "0.878028964",
      "0.121971036"
    ],
    [
      "0.747491979",
      "0.310144447",
      "0.106774042"
    ],
    [
      "0.927132546",
      "0.310722557",
      "0.072867454"
    ],
    [
      "0.432532282",
      "0.500383197",
      "0.102153516"
    ],
    [
      "0.313785141",
      "0.673463127",
      "0.107745377"
    ],
    [
      "0.530979066",
      "0.688257947",
      "0.109951861"
    ],
    [
      "0.640513540",
      "0.115410171",
      "0.115410171"
    ],
    [
      "0.142983559",
      "0.857016441",
      "0.142983559"
    ],
    [
      "0.640925709",
      "0.884771920",
      "0.115228080"
    ],
    [
      "0.927169603",
      "0.689527516",
      "0.072830397"
    ],
    [
      "0.877914946",
      "0.122085053",
      "0.122085053"
    ],
    [
      "0.405424249",
      "0.119718596",
      "0.119718596"
    ],
    [
      "0.406450621",
      "0.880717950",
      "0.119282050"
    ],
    [
      "0.108034434",
      "0.608443330",
      "0.108034434"
    ],
    [
      "0.143386161",
      "0.143386161",
      "0.143386161"
    ],
    [
      "0.530708469",
      "0.312289944",
      "0.110020081"
    ],
    [
      "0.255570333",
      "0.500440201",
      "0.074808441"
    ],
    [
      "0.747631424",
      "0.690169948",
      "0.106708931"
    ],
    [
      "0.108064862",
      "0.392344035",
      "0.108064862"
    ]
  ]
}

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

Instance parameters

{
  "n": 22
}

The current leader's answer

{
  "circles": [
    [
      "0.876891459",
      "0.500149123",
      "0.123108541"
    ],
    [
      "0.644234338",
      "0.500232170",
      "0.109548593"
    ],
    [
      "0.313708348",
      "0.327627744",
      "0.107521408"
    ],
    [
      "0.878028964",
      "0.878028964",
      "0.121971036"
    ],
    [
      "0.747491979",
      "0.310144447",
      "0.106774042"
    ],
    [
      "0.927132546",
      "0.310722557",
      "0.072867454"
    ],
    [
      "0.432532282",
      "0.500383197",
      "0.102153516"
    ],
    [
      "0.313785141",
      "0.673463127",
      "0.107745377"
    ],
    [
      "0.530979066",
      "0.688257947",
      "0.109951861"
    ],
    [
      "0.640513540",
      "0.115410171",
      "0.115410171"
    ],
    [
      "0.142983559",
      "0.857016441",
      "0.142983559"
    ],
    [
      "0.640925709",
      "0.884771920",
      "0.115228080"
    ],
    [
      "0.927169603",
      "0.689527516",
      "0.072830397"
    ],
    [
      "0.877914946",
      "0.122085053",
      "0.122085053"
    ],
    [
      "0.405424249",
      "0.119718596",
      "0.119718596"
    ],
    [
      "0.406450621",
      "0.880717950",
      "0.119282050"
    ],
    [
      "0.108034434",
      "0.608443330",
      "0.108034434"
    ],
    [
      "0.143386161",
      "0.143386161",
      "0.143386161"
    ],
    [
      "0.530708469",
      "0.312289944",
      "0.110020081"
    ],
    [
      "0.255570333",
      "0.500440201",
      "0.074808441"
    ],
    [
      "0.747631424",
      "0.690169948",
      "0.106708931"
    ],
    [
      "0.108064862",
      "0.392344035",
      "0.108064862"
    ]
  ]
}

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