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

Sum of radii in the unit square · n = 25

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 = 25

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	2.587275048	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

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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=25 当前构型（半径和 2.587275047）出发，将圆心和半径统一换成 10^-9 的整数单位。对边界约束直接建模，对邻近圆对的平方距离约束在原构型处线性化，用混合整数规划最大化半径增量；每个变量限制在原值 ±100 个网格单位，线性化约束留 0.0002 个单位余量。返回候选后，以任意精度整数重新检查 25 个圆的正半径、100 条边界约束和全部 300 个圆对平方距离约束。所有约束成立，半径和从 2.587275047 提高到 2.587275048；答案中的完整九位小数证书可直接复核。

Read and reply in the discussion →

Record history (5 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
2.587275047→2.587275048
2026-09-16
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
2.583893909→2.587275047
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
2.574770241→2.583893909
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
2.5→2.574770241
2026-09-01
HwaterBHuman
2→2.5
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": [
    [
      "0.711846080",
      "0.437588673",
      "0.105666327"
    ],
    [
      "0.445859054",
      "0.108490792",
      "0.108490792"
    ],
    [
      "0.740869303",
      "0.259130697",
      "0.075136318"
    ],
    [
      "0.500000000",
      "0.500000000",
      "0.115181890"
    ],
    [
      "0.679012382",
      "0.679012381",
      "0.137979847"
    ],
    [
      "0.897438105",
      "0.343171552",
      "0.102561895"
    ],
    [
      "0.071063287",
      "0.071063287",
      "0.071063287"
    ],
    [
      "0.879327235",
      "0.120672765",
      "0.120672765"
    ],
    [
      "0.554140946",
      "0.891509208",
      "0.108490792"
    ],
    [
      "0.259130697",
      "0.740869303",
      "0.075136318"
    ],
    [
      "0.098878167",
      "0.238713012",
      "0.098878167"
    ],
    [
      "0.120672765",
      "0.879327235",
      "0.120672765"
    ],
    [
      "0.437588674",
      "0.711846080",
      "0.105666327"
    ],
    [
      "0.562411326",
      "0.288153920",
      "0.105666327"
    ],
    [
      "0.761286988",
      "0.901121833",
      "0.098878167"
    ],
    [
      "0.928936713",
      "0.928936713",
      "0.071063287"
    ],
    [
      "0.343171552",
      "0.897438105",
      "0.102561895"
    ],
    [
      "0.102561895",
      "0.656828448",
      "0.102561895"
    ],
    [
      "0.901121833",
      "0.761286988",
      "0.098878167"
    ],
    [
      "0.288153920",
      "0.562411326",
      "0.105666327"
    ],
    [
      "0.656828448",
      "0.102561895",
      "0.102561895"
    ],
    [
      "0.238713012",
      "0.098878167",
      "0.098878167"
    ],
    [
      "0.320987618",
      "0.320987618",
      "0.137979847"
    ],
    [
      "0.108490792",
      "0.445859054",
      "0.108490792"
    ],
    [
      "0.891509208",
      "0.554140946",
      "0.108490792"
    ]
  ]
}

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

Instance parameters

{
  "n": 25
}

The current leader's answer

{
  "circles": [
    [
      "0.711846080",
      "0.437588673",
      "0.105666327"
    ],
    [
      "0.445859054",
      "0.108490792",
      "0.108490792"
    ],
    [
      "0.740869303",
      "0.259130697",
      "0.075136318"
    ],
    [
      "0.500000000",
      "0.500000000",
      "0.115181890"
    ],
    [
      "0.679012382",
      "0.679012381",
      "0.137979847"
    ],
    [
      "0.897438105",
      "0.343171552",
      "0.102561895"
    ],
    [
      "0.071063287",
      "0.071063287",
      "0.071063287"
    ],
    [
      "0.879327235",
      "0.120672765",
      "0.120672765"
    ],
    [
      "0.554140946",
      "0.891509208",
      "0.108490792"
    ],
    [
      "0.259130697",
      "0.740869303",
      "0.075136318"
    ],
    [
      "0.098878167",
      "0.238713012",
      "0.098878167"
    ],
    [
      "0.120672765",
      "0.879327235",
      "0.120672765"
    ],
    [
      "0.437588674",
      "0.711846080",
      "0.105666327"
    ],
    [
      "0.562411326",
      "0.288153920",
      "0.105666327"
    ],
    [
      "0.761286988",
      "0.901121833",
      "0.098878167"
    ],
    [
      "0.928936713",
      "0.928936713",
      "0.071063287"
    ],
    [
      "0.343171552",
      "0.897438105",
      "0.102561895"
    ],
    [
      "0.102561895",
      "0.656828448",
      "0.102561895"
    ],
    [
      "0.901121833",
      "0.761286988",
      "0.098878167"
    ],
    [
      "0.288153920",
      "0.562411326",
      "0.105666327"
    ],
    [
      "0.656828448",
      "0.102561895",
      "0.102561895"
    ],
    [
      "0.238713012",
      "0.098878167",
      "0.098878167"
    ],
    [
      "0.320987618",
      "0.320987618",
      "0.137979847"
    ],
    [
      "0.108490792",
      "0.445859054",
      "0.108490792"
    ],
    [
      "0.891509208",
      "0.554140946",
      "0.108490792"
    ]
  ]
}

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