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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 = 48

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

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

子题n = 48
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[0.1283487565428452425825507820865, 0.1283487565428452425825507820875]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 1 的圆，圆心在 (1, 1)，两个坐标都在 0 到 2 之间
  • 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
挑战这个纪录 ↗

当前展示：起步布局

2y0
0x2
已验证构造r = 0.128348756

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..14 与 19 已证明；其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型，大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 48 的当前构型
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 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。
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 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。

起步布局答案

{
  "centers": [
    [
      "1.114842226",
      "0.135947251"
    ],
    [
      "0.858176142",
      "0.139964022"
    ],
    [
      "1.361548308",
      "0.206867785"
    ],
    [
      "0.613810114",
      "0.218569733"
    ],
    [
      "1.576898139",
      "0.346574849"
    ],
    [
      "0.402937446",
      "0.364947091"
    ],
    [
      "1.02743304",
      "0.377304346"
    ],
    [
      "0.774149655",
      "0.419031313"
    ],
    [
      "1.274139123",
      "0.44822488"
    ],
    [
      "1.742214922",
      "0.542951972"
    ],
    [
      "0.243846638",
      "0.566401126"
    ],
    [
      "0.55007231",
      "0.575291916"
    ],
    [
      "1.489488953",
      "0.587931944"
    ],
    [
      "1.039713462",
      "0.633707943"
    ],
    [
      "0.786430077",
      "0.67543491"
    ],
    [
      "1.276917091",
      "0.731830579"
    ],
    [
      "0.390981502",
      "0.77674595"
    ],
    [
      "1.843161114",
      "0.778967818"
    ],
    [
      "1.590435145",
      "0.82394779"
    ],
    [
      "0.14275873",
      "0.842160541"
    ],
    [
      "0.620590981",
      "0.891520079"
    ],
    [
      "0.876123099",
      "0.915952632"
    ],
    [
      "1.131362707",
      "0.943272141"
    ],
    [
      "1.385607707",
      "0.9786712"
    ],
    [
      "0.382178428",
      "1.033292474"
    ],
    [
      "1.870981885",
      "1.034153276"
    ],
    [
      "1.619332952",
      "1.084813196"
    ],
    [
      "0.133955656",
      "1.098707065"
    ],
    [
      "0.665549447",
      "1.144249874"
    ],
    [
      "0.980083514",
      "1.150656372"
    ],
    [
      "1.296049395",
      "1.219239116"
    ],
    [
      "0.448485386",
      "1.281278362"
    ],
    [
      "1.823264403",
      "1.286376699"
    ],
    [
      "1.52977464",
      "1.325381112"
    ],
    [
      "0.200262614",
      "1.346692953"
    ],
    [
      "0.818685529",
      "1.350266751"
    ],
    [
      "1.095770775",
      "1.379807132"
    ],
    [
      "0.601621468",
      "1.487295239"
    ],
    [
      "1.347401989",
      "1.508230508"
    ],
    [
      "1.704147095",
      "1.51376333"
    ],
    [
      "0.335928953",
      "1.564610959"
    ],
    [
      "0.864253145",
      "1.602887426"
    ],
    [
      "1.11887534",
      "1.635462745"
    ],
    [
      "1.523960731",
      "1.696592451"
    ],
    [
      "0.529188645",
      "1.733561557"
    ],
    [
      "1.298332459",
      "1.819007713"
    ],
    [
      "0.763280722",
      "1.838892051"
    ],
    [
      "1.017902916",
      "1.871467369"
    ]
  ],
  "radius": "0.128348756"
}
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": 48
}

起步布局答案

{
  "centers": [
    [
      "1.114842226",
      "0.135947251"
    ],
    [
      "0.858176142",
      "0.139964022"
    ],
    [
      "1.361548308",
      "0.206867785"
    ],
    [
      "0.613810114",
      "0.218569733"
    ],
    [
      "1.576898139",
      "0.346574849"
    ],
    [
      "0.402937446",
      "0.364947091"
    ],
    [
      "1.02743304",
      "0.377304346"
    ],
    [
      "0.774149655",
      "0.419031313"
    ],
    [
      "1.274139123",
      "0.44822488"
    ],
    [
      "1.742214922",
      "0.542951972"
    ],
    [
      "0.243846638",
      "0.566401126"
    ],
    [
      "0.55007231",
      "0.575291916"
    ],
    [
      "1.489488953",
      "0.587931944"
    ],
    [
      "1.039713462",
      "0.633707943"
    ],
    [
      "0.786430077",
      "0.67543491"
    ],
    [
      "1.276917091",
      "0.731830579"
    ],
    [
      "0.390981502",
      "0.77674595"
    ],
    [
      "1.843161114",
      "0.778967818"
    ],
    [
      "1.590435145",
      "0.82394779"
    ],
    [
      "0.14275873",
      "0.842160541"
    ],
    [
      "0.620590981",
      "0.891520079"
    ],
    [
      "0.876123099",
      "0.915952632"
    ],
    [
      "1.131362707",
      "0.943272141"
    ],
    [
      "1.385607707",
      "0.9786712"
    ],
    [
      "0.382178428",
      "1.033292474"
    ],
    [
      "1.870981885",
      "1.034153276"
    ],
    [
      "1.619332952",
      "1.084813196"
    ],
    [
      "0.133955656",
      "1.098707065"
    ],
    [
      "0.665549447",
      "1.144249874"
    ],
    [
      "0.980083514",
      "1.150656372"
    ],
    [
      "1.296049395",
      "1.219239116"
    ],
    [
      "0.448485386",
      "1.281278362"
    ],
    [
      "1.823264403",
      "1.286376699"
    ],
    [
      "1.52977464",
      "1.325381112"
    ],
    [
      "0.200262614",
      "1.346692953"
    ],
    [
      "0.818685529",
      "1.350266751"
    ],
    [
      "1.095770775",
      "1.379807132"
    ],
    [
      "0.601621468",
      "1.487295239"
    ],
    [
      "1.347401989",
      "1.508230508"
    ],
    [
      "1.704147095",
      "1.51376333"
    ],
    [
      "0.335928953",
      "1.564610959"
    ],
    [
      "0.864253145",
      "1.602887426"
    ],
    [
      "1.11887534",
      "1.635462745"
    ],
    [
      "1.523960731",
      "1.696592451"
    ],
    [
      "0.529188645",
      "1.733561557"
    ],
    [
      "1.298332459",
      "1.819007713"
    ],
    [
      "0.763280722",
      "1.838892051"
    ],
    [
      "1.017902916",
      "1.871467369"
    ]
  ],
  "radius": "0.128348756"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。 · 验证器 v1.0.0

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