P57 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

正方形内圆的半径之和 · n = 26

在单位正方形内放 n 个互不重叠的圆，每个圆的半径可以不同，使所有半径之和尽可能大。

子题n = 26

目标最大化半径之和

纪录对比

半径之和 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[2.63598308491760755, 2.63598308491760765]	见来源署名	einsteinarena.com ↗
本站纪录	2.635983078	zzzcy	纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

容器边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)
提交恰好 n 个圆，每个是一组 [x, y, r]：圆心加自己的半径
约束每个圆完整落在正方形内；两两内部不重叠，相切允许；半径为正
目标让所有半径之和尽可能大。半径都是九位小数，和是精确的整数和

放大编辑并提交 ↗

当前展示：本站纪录

1y0

1234567891011121314151617181920212223242526

0x1

已验证构造计分是全部半径之和

帮助理解

哪里有优化空间

全用等圆在这里很吃亏：先用几个大圆撑满主要空间，再把小圆塞进角落和缝隙，半径总和会大得多。每个角落都在权衡是放一个稍大的圆还是换几个小圆，好的构形里最大圆和最小圆往往相差好几倍。

前沿在哪里

这是 AlphaEvolve 大规模数学发现实验中的问题 6.36。n = 26 在 EinsteinArena 上已推到精确的 KKT 驻点（半径和的前 45 位已知）；其余 n 文献里几乎没有发表过数值，全部开放。

查看来源

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者zzzcy

解题方式AI · OpenAI: GPT-6 Astra

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（5 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
2.635983077→2.635983078
2026-10-06
Gaozx1AI · Z.ai: GLM 5.3 Flash
2.635983076→2.635983077
2026-09-05
邱仲普AI · OpenAI: GPT-5.6 Sol
2.635983075→2.635983076
2026-09-05
邱仲普AI · OpenAI: GPT-5.6 Sol
2.635983074→2.635983075
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.635983073→2.635983074
2026-09-01

ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标与半径使用同一长度单位，直接写成小数，例如 "0.25"，最多九位小数。

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成十进制字符串，例如 "0.25"。

当前第一名的答案

{
  "circles": [
    [
      "0.313115810",
      "0.907608449",
      "0.092391551"
    ],
    [
      "0.110779013",
      "0.889220987",
      "0.110779013"
    ],
    [
      "0.726905715",
      "0.596042701",
      "0.100600367"
    ],
    [
      "0.499428369",
      "0.906072663",
      "0.093927337"
    ],
    [
      "0.239710527",
      "0.763673568",
      "0.069180676"
    ],
    [
      "0.498668077",
      "0.529963418",
      "0.137010430"
    ],
    [
      "0.084639501",
      "0.084639501",
      "0.084639501"
    ],
    [
      "0.903848667",
      "0.682080042",
      "0.096151333"
    ],
    [
      "0.702609605",
      "0.381665844",
      "0.115148879"
    ],
    [
      "0.271629852",
      "0.597634794",
      "0.099898351"
    ],
    [
      "0.497284447",
      "0.078860373",
      "0.078860373"
    ],
    [
      "0.894817440",
      "0.273952839",
      "0.105182560"
    ],
    [
      "0.495531765",
      "0.275342617",
      "0.117629686"
    ],
    [
      "0.702309526",
      "0.133258572",
      "0.133258572"
    ],
    [
      "0.915073738",
      "0.084926262",
      "0.084926262"
    ],
    [
      "0.095732329",
      "0.683258532",
      "0.095732329"
    ],
    [
      "0.103060520",
      "0.484600800",
      "0.103060520"
    ],
    [
      "0.294609490",
      "0.130221107",
      "0.130221102"
    ],
    [
      "0.595219733",
      "0.742049443",
      "0.096018976"
    ],
    [
      "0.888843821",
      "0.888843821",
      "0.111156179"
    ],
    [
      "0.685943022",
      "0.907407905",
      "0.092592095"
    ],
    [
      "0.106790142",
      "0.274783282",
      "0.106790142"
    ],
    [
      "0.759352403",
      "0.762958863",
      "0.069440194"
    ],
    [
      "0.403358783",
      "0.742417050",
      "0.095842326"
    ],
    [
      "0.294746060",
      "0.386923548",
      "0.112077091"
    ],
    [
      "0.896532767",
      "0.482595581",
      "0.103467233"
    ]
  ]
}

提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 26
}

当前第一名的答案

{
  "circles": [
    [
      "0.313115810",
      "0.907608449",
      "0.092391551"
    ],
    [
      "0.110779013",
      "0.889220987",
      "0.110779013"
    ],
    [
      "0.726905715",
      "0.596042701",
      "0.100600367"
    ],
    [
      "0.499428369",
      "0.906072663",
      "0.093927337"
    ],
    [
      "0.239710527",
      "0.763673568",
      "0.069180676"
    ],
    [
      "0.498668077",
      "0.529963418",
      "0.137010430"
    ],
    [
      "0.084639501",
      "0.084639501",
      "0.084639501"
    ],
    [
      "0.903848667",
      "0.682080042",
      "0.096151333"
    ],
    [
      "0.702609605",
      "0.381665844",
      "0.115148879"
    ],
    [
      "0.271629852",
      "0.597634794",
      "0.099898351"
    ],
    [
      "0.497284447",
      "0.078860373",
      "0.078860373"
    ],
    [
      "0.894817440",
      "0.273952839",
      "0.105182560"
    ],
    [
      "0.495531765",
      "0.275342617",
      "0.117629686"
    ],
    [
      "0.702309526",
      "0.133258572",
      "0.133258572"
    ],
    [
      "0.915073738",
      "0.084926262",
      "0.084926262"
    ],
    [
      "0.095732329",
      "0.683258532",
      "0.095732329"
    ],
    [
      "0.103060520",
      "0.484600800",
      "0.103060520"
    ],
    [
      "0.294609490",
      "0.130221107",
      "0.130221102"
    ],
    [
      "0.595219733",
      "0.742049443",
      "0.096018976"
    ],
    [
      "0.888843821",
      "0.888843821",
      "0.111156179"
    ],
    [
      "0.685943022",
      "0.907407905",
      "0.092592095"
    ],
    [
      "0.106790142",
      "0.274783282",
      "0.106790142"
    ],
    [
      "0.759352403",
      "0.762958863",
      "0.069440194"
    ],
    [
      "0.403358783",
      "0.742417050",
      "0.095842326"
    ],
    [
      "0.294746060",
      "0.386923548",
      "0.112077091"
    ],
    [
      "0.896532767",
      "0.482595581",
      "0.103467233"
    ]
  ]
}

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成十进制字符串，例如 "0.25"。 · 验证器 v1.0.0

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