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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 = 67

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 = 67
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[0.1090634820231832826690615215905, 0.1090634820231832826690615215915]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 = 0.109063481

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 = 67 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": [
    [
      "0.836996022",
      "0.124101843"
    ],
    [
      "1.214294792",
      "0.135219322"
    ],
    [
      "0.629049033",
      "0.189960952"
    ],
    [
      "1.418246927",
      "0.214299278"
    ],
    [
      "1.022424368",
      "0.238974384"
    ],
    [
      "0.443337278",
      "0.304374747"
    ],
    [
      "0.830227656",
      "0.342123772"
    ],
    [
      "1.2082141",
      "0.353261514"
    ],
    [
      "1.615747305",
      "0.356087633"
    ],
    [
      "0.627168125",
      "0.421786808"
    ],
    [
      "1.411441031",
      "0.432496526"
    ],
    [
      "1.016343676",
      "0.457016576"
    ],
    [
      "0.290992536",
      "0.46048513"
    ],
    [
      "1.753755662",
      "0.525004967"
    ],
    [
      "0.824146964",
      "0.560165964"
    ],
    [
      "1.23650292",
      "0.569546315"
    ],
    [
      "0.474823383",
      "0.577897192"
    ],
    [
      "1.439729851",
      "0.648781327"
    ],
    [
      "1.044632496",
      "0.673301377"
    ],
    [
      "0.281226132",
      "0.678393345"
    ],
    [
      "0.656093635",
      "0.699225128"
    ],
    [
      "1.846583026",
      "0.722394058"
    ],
    [
      "1.630767167",
      "0.754062456"
    ],
    [
      "0.856885447",
      "0.784481677"
    ],
    [
      "1.240845445",
      "0.787630049"
    ],
    [
      "0.462496384",
      "0.799721281"
    ],
    [
      "0.125260865",
      "0.83088665"
    ],
    [
      "1.444072376",
      "0.86686506"
    ],
    [
      "1.048975021",
      "0.891385111"
    ],
    [
      "0.68265331",
      "0.915729068"
    ],
    [
      "1.888665216",
      "0.936423172"
    ],
    [
      "0.306531117",
      "0.952214586"
    ],
    [
      "1.671590061",
      "0.969634157"
    ],
    [
      "1.273839378",
      "1.003247239"
    ],
    [
      "0.873627173",
      "1.021125253"
    ],
    [
      "0.505115676",
      "1.042456181"
    ],
    [
      "1.484152733",
      "1.081278067"
    ],
    [
      "0.11350187",
      "1.087777831"
    ],
    [
      "1.073790419",
      "1.108295626"
    ],
    [
      "0.69608954",
      "1.147852366"
    ],
    [
      "1.877479776",
      "1.154263155"
    ],
    [
      "0.327671883",
      "1.169314657"
    ],
    [
      "1.664941405",
      "1.203322438"
    ],
    [
      "1.261304089",
      "1.221013718"
    ],
    [
      "0.894607256",
      "1.238240911"
    ],
    [
      "0.517843932",
      "1.276150813"
    ],
    [
      "1.465174576",
      "1.298577865"
    ],
    [
      "0.167521021",
      "1.317406094"
    ],
    [
      "1.083486076",
      "1.347347113"
    ],
    [
      "1.813697176",
      "1.362856425"
    ],
    [
      "0.717069622",
      "1.364968023"
    ],
    [
      "1.281803013",
      "1.438175331"
    ],
    [
      "0.323668599",
      "1.469712713"
    ],
    [
      "0.905948443",
      "1.474074226"
    ],
    [
      "0.540538817",
      "1.493093927"
    ],
    [
      "1.4856735",
      "1.515739479"
    ],
    [
      "1.701140617",
      "1.549699658"
    ],
    [
      "1.103853295",
      "1.565796862"
    ],
    [
      "0.720777588",
      "1.615948954"
    ],
    [
      "1.302775497",
      "1.655291723"
    ],
    [
      "0.411854982",
      "1.669218438"
    ],
    [
      "0.925466736",
      "1.691326174"
    ],
    [
      "1.506645984",
      "1.73285587"
    ],
    [
      "0.592093754",
      "1.792073465"
    ],
    [
      "1.229704953",
      "1.86081561"
    ],
    [
      "0.796782902",
      "1.867450684"
    ],
    [
      "1.013653119",
      "1.890831899"
    ]
  ],
  "radius": "0.109063481"
}
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": 67
}

起步布局答案

{
  "centers": [
    [
      "0.836996022",
      "0.124101843"
    ],
    [
      "1.214294792",
      "0.135219322"
    ],
    [
      "0.629049033",
      "0.189960952"
    ],
    [
      "1.418246927",
      "0.214299278"
    ],
    [
      "1.022424368",
      "0.238974384"
    ],
    [
      "0.443337278",
      "0.304374747"
    ],
    [
      "0.830227656",
      "0.342123772"
    ],
    [
      "1.2082141",
      "0.353261514"
    ],
    [
      "1.615747305",
      "0.356087633"
    ],
    [
      "0.627168125",
      "0.421786808"
    ],
    [
      "1.411441031",
      "0.432496526"
    ],
    [
      "1.016343676",
      "0.457016576"
    ],
    [
      "0.290992536",
      "0.46048513"
    ],
    [
      "1.753755662",
      "0.525004967"
    ],
    [
      "0.824146964",
      "0.560165964"
    ],
    [
      "1.23650292",
      "0.569546315"
    ],
    [
      "0.474823383",
      "0.577897192"
    ],
    [
      "1.439729851",
      "0.648781327"
    ],
    [
      "1.044632496",
      "0.673301377"
    ],
    [
      "0.281226132",
      "0.678393345"
    ],
    [
      "0.656093635",
      "0.699225128"
    ],
    [
      "1.846583026",
      "0.722394058"
    ],
    [
      "1.630767167",
      "0.754062456"
    ],
    [
      "0.856885447",
      "0.784481677"
    ],
    [
      "1.240845445",
      "0.787630049"
    ],
    [
      "0.462496384",
      "0.799721281"
    ],
    [
      "0.125260865",
      "0.83088665"
    ],
    [
      "1.444072376",
      "0.86686506"
    ],
    [
      "1.048975021",
      "0.891385111"
    ],
    [
      "0.68265331",
      "0.915729068"
    ],
    [
      "1.888665216",
      "0.936423172"
    ],
    [
      "0.306531117",
      "0.952214586"
    ],
    [
      "1.671590061",
      "0.969634157"
    ],
    [
      "1.273839378",
      "1.003247239"
    ],
    [
      "0.873627173",
      "1.021125253"
    ],
    [
      "0.505115676",
      "1.042456181"
    ],
    [
      "1.484152733",
      "1.081278067"
    ],
    [
      "0.11350187",
      "1.087777831"
    ],
    [
      "1.073790419",
      "1.108295626"
    ],
    [
      "0.69608954",
      "1.147852366"
    ],
    [
      "1.877479776",
      "1.154263155"
    ],
    [
      "0.327671883",
      "1.169314657"
    ],
    [
      "1.664941405",
      "1.203322438"
    ],
    [
      "1.261304089",
      "1.221013718"
    ],
    [
      "0.894607256",
      "1.238240911"
    ],
    [
      "0.517843932",
      "1.276150813"
    ],
    [
      "1.465174576",
      "1.298577865"
    ],
    [
      "0.167521021",
      "1.317406094"
    ],
    [
      "1.083486076",
      "1.347347113"
    ],
    [
      "1.813697176",
      "1.362856425"
    ],
    [
      "0.717069622",
      "1.364968023"
    ],
    [
      "1.281803013",
      "1.438175331"
    ],
    [
      "0.323668599",
      "1.469712713"
    ],
    [
      "0.905948443",
      "1.474074226"
    ],
    [
      "0.540538817",
      "1.493093927"
    ],
    [
      "1.4856735",
      "1.515739479"
    ],
    [
      "1.701140617",
      "1.549699658"
    ],
    [
      "1.103853295",
      "1.565796862"
    ],
    [
      "0.720777588",
      "1.615948954"
    ],
    [
      "1.302775497",
      "1.655291723"
    ],
    [
      "0.411854982",
      "1.669218438"
    ],
    [
      "0.925466736",
      "1.691326174"
    ],
    [
      "1.506645984",
      "1.73285587"
    ],
    [
      "0.592093754",
      "1.792073465"
    ],
    [
      "1.229704953",
      "1.86081561"
    ],
    [
      "0.796782902",
      "1.867450684"
    ],
    [
      "1.013653119",
      "1.890831899"
    ]
  ],
  "radius": "0.109063481"
}

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

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。

还没有帖子。第一个聊聊这道题的思路？