P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 27

文献中也称circles in a circlepacking equal circles in a circlecci

在半径为 1 的圆内放置 n 个半径相同、互不重叠的小圆，使它们的共同半径尽可能大。

子题n = 27

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	[0.1693079311345730465689703176575, 0.1693079311345730465689703176585]	见来源署名	packomania.com ↗
本站纪录	暂无选手纪录	—	—

严格定义

容器半径为 1、圆心位于 (1, 1) 的圆，横纵坐标范围均为 [0, 2]
提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心坐标 centers
约束每个小圆完全位于容器圆内；任意两圆内部不重叠，允许相切
目标最大化共同半径 radius

放大编辑并提交 ↗

当前展示：起步布局

2y0

123456789101112131415161718192021222324252627

0x2

已验证构造r = 0.169307930

帮助理解

环状分层与接触结构

在圆形容器内，较大半径的等圆通常会贴着外圆周排成外环，其余圆在内部组成内环或中心核，通过彼此相切形成刚性接触网。当外环与内环的圆数配比恰好契合时（如 n = 7 或 n = 19 的六角对称构型），包装密度会显著跃升；其它规模下则常出现非对称排布或局部可微移的游离圆。

已知证明与开放前沿

n = 1..14 与 n = 19 的最优性已获得数学证明，其余规模仍未完全解决。本站导入了 Packomania 汇总的 n = 1..300 公开已知最优构型，较大规模的子题合并为同一张数据表供算法挑战。

查看来源

构型与历史

起步布局

查看构型、求解笔记与纪录历史。

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。

ANSWER FORMAT

答案怎么写

容器是半径 1 的圆，圆心在 (1, 1)，所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位，直接写成小数，例如 "0.5"，最多九位小数。

提交共同半径 radius 与各圆圆心坐标 centers，所有数值均以十进制字符串表示（如 "0.2"）。

起步布局答案

{
  "centers": [
    [
      "0.830692069",
      "0.186744728"
    ],
    [
      "1.169307931",
      "0.186744728"
    ],
    [
      "0.52020893",
      "0.321877709"
    ],
    [
      "1.47979107",
      "0.321877709"
    ],
    [
      "0.940578427",
      "0.507034738"
    ],
    [
      "1.710550659",
      "0.569689561"
    ],
    [
      "1.269540763",
      "0.587312278"
    ],
    [
      "0.272064899",
      "0.599800298"
    ],
    [
      "0.607063132",
      "0.64916512"
    ],
    [
      "1.500300352",
      "0.835124131"
    ],
    [
      "0.885949463",
      "0.841214896"
    ],
    [
      "0.172833334",
      "0.92354989"
    ],
    [
      "1.827166666",
      "0.92354989"
    ],
    [
      "0.507831567",
      "0.972914712"
    ],
    [
      "1.194537202",
      "0.98062189"
    ],
    [
      "0.919513335",
      "1.178163214"
    ],
    [
      "1.456631227",
      "1.19502204"
    ],
    [
      "0.211046309",
      "1.260002667"
    ],
    [
      "1.788953691",
      "1.260002667"
    ],
    [
      "0.602790346",
      "1.297943223"
    ],
    [
      "1.222627669",
      "1.43977301"
    ],
    [
      "0.892636515",
      "1.515710749"
    ],
    [
      "0.38035424",
      "1.553252606"
    ],
    [
      "1.61964576",
      "1.553252606"
    ],
    [
      "0.652624404",
      "1.754572401"
    ],
    [
      "1.347375596",
      "1.754572401"
    ],
    [
      "1.017384442",
      "1.830510141"
    ]
  ],
  "radius": "0.16930793"
}

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": 27
}

起步布局答案

{
  "centers": [
    [
      "0.830692069",
      "0.186744728"
    ],
    [
      "1.169307931",
      "0.186744728"
    ],
    [
      "0.52020893",
      "0.321877709"
    ],
    [
      "1.47979107",
      "0.321877709"
    ],
    [
      "0.940578427",
      "0.507034738"
    ],
    [
      "1.710550659",
      "0.569689561"
    ],
    [
      "1.269540763",
      "0.587312278"
    ],
    [
      "0.272064899",
      "0.599800298"
    ],
    [
      "0.607063132",
      "0.64916512"
    ],
    [
      "1.500300352",
      "0.835124131"
    ],
    [
      "0.885949463",
      "0.841214896"
    ],
    [
      "0.172833334",
      "0.92354989"
    ],
    [
      "1.827166666",
      "0.92354989"
    ],
    [
      "0.507831567",
      "0.972914712"
    ],
    [
      "1.194537202",
      "0.98062189"
    ],
    [
      "0.919513335",
      "1.178163214"
    ],
    [
      "1.456631227",
      "1.19502204"
    ],
    [
      "0.211046309",
      "1.260002667"
    ],
    [
      "1.788953691",
      "1.260002667"
    ],
    [
      "0.602790346",
      "1.297943223"
    ],
    [
      "1.222627669",
      "1.43977301"
    ],
    [
      "0.892636515",
      "1.515710749"
    ],
    [
      "0.38035424",
      "1.553252606"
    ],
    [
      "1.61964576",
      "1.553252606"
    ],
    [
      "0.652624404",
      "1.754572401"
    ],
    [
      "1.347375596",
      "1.754572401"
    ],
    [
      "1.017384442",
      "1.830510141"
    ]
  ],
  "radius": "0.16930793"
}

提交共同半径 radius 与各圆圆心坐标 centers，所有数值均以十进制字符串表示（如 "0.2"）。 · 验证器 v1.0.0

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