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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 = 56

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 = 56
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[0.1192814970823624272775802945125, 0.1192814970823624272775802945135]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 = 0.119281496

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 = 56 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": [
    [
      "0.925913846",
      "0.122403098"
    ],
    [
      "1.164158893",
      "0.134715689"
    ],
    [
      "0.693104668",
      "0.174481777"
    ],
    [
      "1.39035923",
      "0.210516148"
    ],
    [
      "0.482813095",
      "0.287130596"
    ],
    [
      "1.587918027",
      "0.344242823"
    ],
    [
      "0.973368891",
      "0.356198566"
    ],
    [
      "1.211613938",
      "0.368511157"
    ],
    [
      "0.740559714",
      "0.408277245"
    ],
    [
      "0.297452407",
      "0.468876699"
    ],
    [
      "1.409172736",
      "0.502237832"
    ],
    [
      "0.530268141",
      "0.520926064"
    ],
    [
      "1.742339949",
      "0.526083887"
    ],
    [
      "0.956337592",
      "0.594152841"
    ],
    [
      "1.194582639",
      "0.606465432"
    ],
    [
      "0.726206172",
      "0.657016438"
    ],
    [
      "0.180684708",
      "0.676909404"
    ],
    [
      "1.563594657",
      "0.684078896"
    ],
    [
      "0.413500441",
      "0.72895877"
    ],
    [
      "1.842294708",
      "0.742697248"
    ],
    [
      "1.34900456",
      "0.788306495"
    ],
    [
      "1.102719085",
      "0.826632163"
    ],
    [
      "0.865380955",
      "0.850775779"
    ],
    [
      "0.609438473",
      "0.865049144"
    ],
    [
      "1.653353765",
      "0.905111936"
    ],
    [
      "0.124032025",
      "0.908647998"
    ],
    [
      "0.356956982",
      "0.962903831"
    ],
    [
      "1.880448399",
      "0.978189495"
    ],
    [
      "1.438763668",
      "1.009339535"
    ],
    [
      "1.202625622",
      "1.04326777"
    ],
    [
      "0.817643734",
      "1.084513794"
    ],
    [
      "0.57970435",
      "1.101751877"
    ],
    [
      "0.131651095",
      "1.147089295"
    ],
    [
      "1.621234135",
      "1.163017221"
    ],
    [
      "1.024985385",
      "1.202504225"
    ],
    [
      "0.363611471",
      "1.202826949"
    ],
    [
      "1.854001599",
      "1.215282027"
    ],
    [
      "1.345662912",
      "1.235255894"
    ],
    [
      "0.711464949",
      "1.300627333"
    ],
    [
      "0.206683181",
      "1.382509486"
    ],
    [
      "1.532127304",
      "1.384314024"
    ],
    [
      "1.167385771",
      "1.393905459"
    ],
    [
      "0.49537207",
      "1.401702404"
    ],
    [
      "0.919899884",
      "1.416675511"
    ],
    [
      "1.764894768",
      "1.43657883"
    ],
    [
      "1.353797474",
      "1.542777817"
    ],
    [
      "0.675672546",
      "1.557920331"
    ],
    [
      "0.33844378",
      "1.581384941"
    ],
    [
      "1.619665878",
      "1.625842856"
    ],
    [
      "1.131446021",
      "1.629745737"
    ],
    [
      "0.894102837",
      "1.65383962"
    ],
    [
      "0.518744256",
      "1.737602868"
    ],
    [
      "1.4289707",
      "1.769187376"
    ],
    [
      "0.734355555",
      "1.839701203"
    ],
    [
      "1.206800972",
      "1.856094878"
    ],
    [
      "0.969457788",
      "1.880188761"
    ]
  ],
  "radius": "0.119281496"
}
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": 56
}

起步布局答案

{
  "centers": [
    [
      "0.925913846",
      "0.122403098"
    ],
    [
      "1.164158893",
      "0.134715689"
    ],
    [
      "0.693104668",
      "0.174481777"
    ],
    [
      "1.39035923",
      "0.210516148"
    ],
    [
      "0.482813095",
      "0.287130596"
    ],
    [
      "1.587918027",
      "0.344242823"
    ],
    [
      "0.973368891",
      "0.356198566"
    ],
    [
      "1.211613938",
      "0.368511157"
    ],
    [
      "0.740559714",
      "0.408277245"
    ],
    [
      "0.297452407",
      "0.468876699"
    ],
    [
      "1.409172736",
      "0.502237832"
    ],
    [
      "0.530268141",
      "0.520926064"
    ],
    [
      "1.742339949",
      "0.526083887"
    ],
    [
      "0.956337592",
      "0.594152841"
    ],
    [
      "1.194582639",
      "0.606465432"
    ],
    [
      "0.726206172",
      "0.657016438"
    ],
    [
      "0.180684708",
      "0.676909404"
    ],
    [
      "1.563594657",
      "0.684078896"
    ],
    [
      "0.413500441",
      "0.72895877"
    ],
    [
      "1.842294708",
      "0.742697248"
    ],
    [
      "1.34900456",
      "0.788306495"
    ],
    [
      "1.102719085",
      "0.826632163"
    ],
    [
      "0.865380955",
      "0.850775779"
    ],
    [
      "0.609438473",
      "0.865049144"
    ],
    [
      "1.653353765",
      "0.905111936"
    ],
    [
      "0.124032025",
      "0.908647998"
    ],
    [
      "0.356956982",
      "0.962903831"
    ],
    [
      "1.880448399",
      "0.978189495"
    ],
    [
      "1.438763668",
      "1.009339535"
    ],
    [
      "1.202625622",
      "1.04326777"
    ],
    [
      "0.817643734",
      "1.084513794"
    ],
    [
      "0.57970435",
      "1.101751877"
    ],
    [
      "0.131651095",
      "1.147089295"
    ],
    [
      "1.621234135",
      "1.163017221"
    ],
    [
      "1.024985385",
      "1.202504225"
    ],
    [
      "0.363611471",
      "1.202826949"
    ],
    [
      "1.854001599",
      "1.215282027"
    ],
    [
      "1.345662912",
      "1.235255894"
    ],
    [
      "0.711464949",
      "1.300627333"
    ],
    [
      "0.206683181",
      "1.382509486"
    ],
    [
      "1.532127304",
      "1.384314024"
    ],
    [
      "1.167385771",
      "1.393905459"
    ],
    [
      "0.49537207",
      "1.401702404"
    ],
    [
      "0.919899884",
      "1.416675511"
    ],
    [
      "1.764894768",
      "1.43657883"
    ],
    [
      "1.353797474",
      "1.542777817"
    ],
    [
      "0.675672546",
      "1.557920331"
    ],
    [
      "0.33844378",
      "1.581384941"
    ],
    [
      "1.619665878",
      "1.625842856"
    ],
    [
      "1.131446021",
      "1.629745737"
    ],
    [
      "0.894102837",
      "1.65383962"
    ],
    [
      "0.518744256",
      "1.737602868"
    ],
    [
      "1.4289707",
      "1.769187376"
    ],
    [
      "0.734355555",
      "1.839701203"
    ],
    [
      "1.206800972",
      "1.856094878"
    ],
    [
      "0.969457788",
      "1.880188761"
    ]
  ],
  "radius": "0.119281496"
}

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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