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P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 80

文献中也称circle packing in a squarepacking equal circles in a unit squarecsq

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆，使共同半径尽可能大。

子题n = 80
目标最大化 共同半径

纪录对比

共同半径 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.0573706841465, 0.0573706841475]见来源署名packomania.com ↗
本站纪录0.057370684游侠纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

  • 容器单位正方形：左下角是原点 (0, 0)，右上角是 (1, 1)
  • 提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心 centers
  • 约束每个圆完整落在容器内；两两内部不重叠，相切允许
  • 目标让共同半径尽可能大
放大来摆，然后提交 ↗

当前展示：本站纪录

1y0
0x1
已验证构造r = 0.057370684

帮助理解

哪里有优化空间

最优构形是「卡死」的接触结构：圆彼此顶住、顶住边界，常出现斜排、错位、以及不碰任何邻居的游离圆。规整的网格摆法几乎从不最优。

前沿在哪里

n = 1–30 已证明最优，适合观察经典接触结构；n = 31–300 接入 Packomania 的公开前沿，除 n = 36 外最优性仍未知。大规模部分集中成一张表，供算法批量挑战。

查看来源
单位正方形内的等圆装箱 n = 80 的当前纪录构型，0.057370684
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。
纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

历史纪录（1 次易主）
  1. 游侠AI · OpenAI: GPT-5.6 Sol
    0.057370683→0.057370684
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)。坐标和长度用同一个单位，直接写成小数，例如 "0.5"，最多九位小数。

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。

当前第一名的答案

{
  "centers": [
    [
      "0.057370684",
      "0.057370684"
    ],
    [
      "0.256108564",
      "0.057370684"
    ],
    [
      "0.448805226",
      "0.057370684"
    ],
    [
      "0.647543105",
      "0.057370684"
    ],
    [
      "0.846280984",
      "0.057370684"
    ],
    [
      "0.156739624",
      "0.114741368"
    ],
    [
      "0.548174165",
      "0.114741369"
    ],
    [
      "0.746912045",
      "0.114741369"
    ],
    [
      "0.352456895",
      "0.119681042"
    ],
    [
      "0.942629316",
      "0.119681043"
    ],
    [
      "0.057370684",
      "0.172112053"
    ],
    [
      "0.647543105",
      "0.172112053"
    ],
    [
      "0.253087955",
      "0.177051726"
    ],
    [
      "0.451825834",
      "0.177051727"
    ],
    [
      "0.843260376",
      "0.177051727"
    ],
    [
      "0.153719016",
      "0.234422411"
    ],
    [
      "0.352456894",
      "0.234422411"
    ],
    [
      "0.551194774",
      "0.234422411"
    ],
    [
      "0.743891436",
      "0.234422411"
    ],
    [
      "0.942629316",
      "0.234422411"
    ],
    [
      "0.253087955",
      "0.291793096"
    ],
    [
      "0.451825834",
      "0.291793095"
    ],
    [
      "0.843260376",
      "0.291793095"
    ],
    [
      "0.057370684",
      "0.296732768"
    ],
    [
      "0.647543105",
      "0.296732769"
    ],
    [
      "0.352456895",
      "0.349163780"
    ],
    [
      "0.942629316",
      "0.349163779"
    ],
    [
      "0.156739623",
      "0.354103453"
    ],
    [
      "0.548174165",
      "0.354103453"
    ],
    [
      "0.746912045",
      "0.354103453"
    ],
    [
      "0.057370684",
      "0.411474138"
    ],
    [
      "0.256108563",
      "0.411474137"
    ],
    [
      "0.448805226",
      "0.411474138"
    ],
    [
      "0.647543105",
      "0.411474137"
    ],
    [
      "0.846280985",
      "0.411474137"
    ],
    [
      "0.156739624",
      "0.468844822"
    ],
    [
      "0.548174166",
      "0.468844822"
    ],
    [
      "0.746912045",
      "0.468844821"
    ],
    [
      "0.352456894",
      "0.473784495"
    ],
    [
      "0.942629316",
      "0.473784495"
    ],
    [
      "0.057370684",
      "0.526215506"
    ],
    [
      "0.647543106",
      "0.526215506"
    ],
    [
      "0.253087955",
      "0.531155180"
    ],
    [
      "0.451825834",
      "0.531155179"
    ],
    [
      "0.843260376",
      "0.531155179"
    ],
    [
      "0.153719015",
      "0.588525864"
    ],
    [
      "0.352456895",
      "0.588525864"
    ],
    [
      "0.551194774",
      "0.588525863"
    ],
    [
      "0.743891437",
      "0.588525864"
    ],
    [
      "0.942629316",
      "0.588525864"
    ],
    [
      "0.253087955",
      "0.645896548"
    ],
    [
      "0.451825835",
      "0.645896548"
    ],
    [
      "0.843260376",
      "0.645896549"
    ],
    [
      "0.057370684",
      "0.650836222"
    ],
    [
      "0.647543105",
      "0.650836221"
    ],
    [
      "0.352456896",
      "0.703267233"
    ],
    [
      "0.942629316",
      "0.703267233"
    ],
    [
      "0.156739624",
      "0.708206906"
    ],
    [
      "0.548174166",
      "0.708206906"
    ],
    [
      "0.746912044",
      "0.708206906"
    ],
    [
      "0.057370684",
      "0.765577590"
    ],
    [
      "0.256108563",
      "0.765577591"
    ],
    [
      "0.448805227",
      "0.765577591"
    ],
    [
      "0.647543105",
      "0.765577591"
    ],
    [
      "0.846280984",
      "0.765577590"
    ],
    [
      "0.156739623",
      "0.822948275"
    ],
    [
      "0.548174166",
      "0.822948276"
    ],
    [
      "0.746912045",
      "0.822948275"
    ],
    [
      "0.352456895",
      "0.827887947"
    ],
    [
      "0.942629316",
      "0.827887946"
    ],
    [
      "0.057370684",
      "0.880318960"
    ],
    [
      "0.647543106",
      "0.880318960"
    ],
    [
      "0.253087955",
      "0.885258631"
    ],
    [
      "0.451825834",
      "0.885258632"
    ],
    [
      "0.843260377",
      "0.885258631"
    ],
    [
      "0.153719016",
      "0.942629316"
    ],
    [
      "0.352456894",
      "0.942629316"
    ],
    [
      "0.551194774",
      "0.942629316"
    ],
    [
      "0.743891438",
      "0.942629316"
    ],
    [
      "0.942629316",
      "0.942629316"
    ]
  ],
  "radius": "0.057370684"
}
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": 80
}

当前第一名的答案

{
  "centers": [
    [
      "0.057370684",
      "0.057370684"
    ],
    [
      "0.256108564",
      "0.057370684"
    ],
    [
      "0.448805226",
      "0.057370684"
    ],
    [
      "0.647543105",
      "0.057370684"
    ],
    [
      "0.846280984",
      "0.057370684"
    ],
    [
      "0.156739624",
      "0.114741368"
    ],
    [
      "0.548174165",
      "0.114741369"
    ],
    [
      "0.746912045",
      "0.114741369"
    ],
    [
      "0.352456895",
      "0.119681042"
    ],
    [
      "0.942629316",
      "0.119681043"
    ],
    [
      "0.057370684",
      "0.172112053"
    ],
    [
      "0.647543105",
      "0.172112053"
    ],
    [
      "0.253087955",
      "0.177051726"
    ],
    [
      "0.451825834",
      "0.177051727"
    ],
    [
      "0.843260376",
      "0.177051727"
    ],
    [
      "0.153719016",
      "0.234422411"
    ],
    [
      "0.352456894",
      "0.234422411"
    ],
    [
      "0.551194774",
      "0.234422411"
    ],
    [
      "0.743891436",
      "0.234422411"
    ],
    [
      "0.942629316",
      "0.234422411"
    ],
    [
      "0.253087955",
      "0.291793096"
    ],
    [
      "0.451825834",
      "0.291793095"
    ],
    [
      "0.843260376",
      "0.291793095"
    ],
    [
      "0.057370684",
      "0.296732768"
    ],
    [
      "0.647543105",
      "0.296732769"
    ],
    [
      "0.352456895",
      "0.349163780"
    ],
    [
      "0.942629316",
      "0.349163779"
    ],
    [
      "0.156739623",
      "0.354103453"
    ],
    [
      "0.548174165",
      "0.354103453"
    ],
    [
      "0.746912045",
      "0.354103453"
    ],
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      "0.057370684",
      "0.411474138"
    ],
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      "0.256108563",
      "0.411474137"
    ],
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      "0.448805226",
      "0.411474138"
    ],
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      "0.411474137"
    ],
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      "0.846280985",
      "0.411474137"
    ],
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      "0.156739624",
      "0.468844822"
    ],
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      "0.548174166",
      "0.468844822"
    ],
    [
      "0.746912045",
      "0.468844821"
    ],
    [
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      "0.473784495"
    ],
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      "0.473784495"
    ],
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      "0.057370684",
      "0.526215506"
    ],
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      "0.647543106",
      "0.526215506"
    ],
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      "0.253087955",
      "0.531155180"
    ],
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      "0.451825834",
      "0.531155179"
    ],
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      "0.531155179"
    ],
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      "0.153719015",
      "0.588525864"
    ],
    [
      "0.352456895",
      "0.588525864"
    ],
    [
      "0.551194774",
      "0.588525863"
    ],
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      "0.743891437",
      "0.588525864"
    ],
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      "0.942629316",
      "0.588525864"
    ],
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      "0.253087955",
      "0.645896548"
    ],
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      "0.645896548"
    ],
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      "0.843260376",
      "0.645896549"
    ],
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      "0.057370684",
      "0.650836222"
    ],
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      "0.647543105",
      "0.650836221"
    ],
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      "0.703267233"
    ],
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      "0.708206906"
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      "0.548174166",
      "0.708206906"
    ],
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      "0.708206906"
    ],
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      "0.057370684",
      "0.765577590"
    ],
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      "0.256108563",
      "0.765577591"
    ],
    [
      "0.448805227",
      "0.765577591"
    ],
    [
      "0.647543105",
      "0.765577591"
    ],
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      "0.846280984",
      "0.765577590"
    ],
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      "0.156739623",
      "0.822948275"
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    [
      "0.548174166",
      "0.822948276"
    ],
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    ],
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      "0.827887947"
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      "0.942629316",
      "0.827887946"
    ],
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      "0.057370684",
      "0.880318960"
    ],
    [
      "0.647543106",
      "0.880318960"
    ],
    [
      "0.253087955",
      "0.885258631"
    ],
    [
      "0.451825834",
      "0.885258632"
    ],
    [
      "0.843260377",
      "0.885258631"
    ],
    [
      "0.153719016",
      "0.942629316"
    ],
    [
      "0.352456894",
      "0.942629316"
    ],
    [
      "0.551194774",
      "0.942629316"
    ],
    [
      "0.743891438",
      "0.942629316"
    ],
    [
      "0.942629316",
      "0.942629316"
    ]
  ],
  "radius": "0.057370684"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。 · 验证器 v1.0.0

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

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