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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 = 70

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 = 70
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[0.1070016166057629853058569493515, 0.1070016166057629853058569493525]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 = 0.107001616

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 = 70 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": [
    [
      "1.055261098",
      "0.108713108"
    ],
    [
      "0.841620069",
      "0.121158769"
    ],
    [
      "0.63707481",
      "0.18407634"
    ],
    [
      "1.41379125",
      "0.208657524"
    ],
    [
      "1.206153779",
      "0.260465481"
    ],
    [
      "0.453372369",
      "0.293852462"
    ],
    [
      "0.959218842",
      "0.299954497"
    ],
    [
      "1.590467761",
      "0.331822341"
    ],
    [
      "0.751990013",
      "0.364628111"
    ],
    [
      "1.354839521",
      "0.414380827"
    ],
    [
      "0.301062789",
      "0.444182685"
    ],
    [
      "1.11803349",
      "0.455542363"
    ],
    [
      "0.507407371",
      "0.500921538"
    ],
    [
      "1.741209613",
      "0.501949378"
    ],
    [
      "0.910643058",
      "0.508371812"
    ],
    [
      "1.530122029",
      "0.537154711"
    ],
    [
      "0.707701615",
      "0.576284656"
    ],
    [
      "1.267078613",
      "0.609561274"
    ],
    [
      "0.188893221",
      "0.626433533"
    ],
    [
      "1.059604652",
      "0.662020201"
    ],
    [
      "0.395237802",
      "0.683172387"
    ],
    [
      "1.838421527",
      "0.692598867"
    ],
    [
      "1.627333942",
      "0.7278042"
    ],
    [
      "0.856663208",
      "0.729933045"
    ],
    [
      "0.595532046",
      "0.758535505"
    ],
    [
      "1.416246357",
      "0.763009533"
    ],
    [
      "1.208772396",
      "0.815468459"
    ],
    [
      "0.123305569",
      "0.830138329"
    ],
    [
      "1.005830952",
      "0.883381304"
    ],
    [
      "0.32965015",
      "0.886877182"
    ],
    [
      "1.763366447",
      "0.893008744"
    ],
    [
      "0.74449364",
      "0.912183894"
    ],
    [
      "1.552278862",
      "0.928214077"
    ],
    [
      "0.529944395",
      "0.9622403"
    ],
    [
      "1.3448049",
      "0.980673003"
    ],
    [
      "0.108066537",
      "1.043598291"
    ],
    [
      "1.145041547",
      "1.057432279"
    ],
    [
      "1.890625022",
      "1.065062919"
    ],
    [
      "0.932759959",
      "1.084523079"
    ],
    [
      "1.679537437",
      "1.100268251"
    ],
    [
      "0.314411119",
      "1.100337145"
    ],
    [
      "0.722632558",
      "1.125067611"
    ],
    [
      "1.472063475",
      "1.152727178"
    ],
    [
      "0.514705363",
      "1.175700263"
    ],
    [
      "1.272300122",
      "1.229486453"
    ],
    [
      "0.144051304",
      "1.254554401"
    ],
    [
      "1.060018534",
      "1.256577254"
    ],
    [
      "1.849570981",
      "1.275091368"
    ],
    [
      "0.849891133",
      "1.297121785"
    ],
    [
      "1.621231219",
      "1.306175436"
    ],
    [
      "0.641963938",
      "1.347754437"
    ],
    [
      "0.429100509",
      "1.373531807"
    ],
    [
      "1.421467866",
      "1.382934712"
    ],
    [
      "1.200555286",
      "1.43110506"
    ],
    [
      "0.229193263",
      "1.450891435"
    ],
    [
      "0.988531416",
      "1.46014404"
    ],
    [
      "1.759726022",
      "1.469321302"
    ],
    [
      "0.780604221",
      "1.510776693"
    ],
    [
      "0.558448258",
      "1.544788737"
    ],
    [
      "1.559962668",
      "1.546080577"
    ],
    [
      "1.349487918",
      "1.584781522"
    ],
    [
      "0.358602703",
      "1.621333743"
    ],
    [
      "0.94380707",
      "1.669421647"
    ],
    [
      "1.157574125",
      "1.679473021"
    ],
    [
      "0.735879875",
      "1.720054299"
    ],
    [
      "1.488241294",
      "1.747707531"
    ],
    [
      "0.524943209",
      "1.756152867"
    ],
    [
      "1.296327501",
      "1.842399029"
    ],
    [
      "0.873444564",
      "1.8839852"
    ],
    [
      "1.0873956",
      "1.888711495"
    ]
  ],
  "radius": "0.107001616"
}
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": 70
}

起步布局答案

{
  "centers": [
    [
      "1.055261098",
      "0.108713108"
    ],
    [
      "0.841620069",
      "0.121158769"
    ],
    [
      "0.63707481",
      "0.18407634"
    ],
    [
      "1.41379125",
      "0.208657524"
    ],
    [
      "1.206153779",
      "0.260465481"
    ],
    [
      "0.453372369",
      "0.293852462"
    ],
    [
      "0.959218842",
      "0.299954497"
    ],
    [
      "1.590467761",
      "0.331822341"
    ],
    [
      "0.751990013",
      "0.364628111"
    ],
    [
      "1.354839521",
      "0.414380827"
    ],
    [
      "0.301062789",
      "0.444182685"
    ],
    [
      "1.11803349",
      "0.455542363"
    ],
    [
      "0.507407371",
      "0.500921538"
    ],
    [
      "1.741209613",
      "0.501949378"
    ],
    [
      "0.910643058",
      "0.508371812"
    ],
    [
      "1.530122029",
      "0.537154711"
    ],
    [
      "0.707701615",
      "0.576284656"
    ],
    [
      "1.267078613",
      "0.609561274"
    ],
    [
      "0.188893221",
      "0.626433533"
    ],
    [
      "1.059604652",
      "0.662020201"
    ],
    [
      "0.395237802",
      "0.683172387"
    ],
    [
      "1.838421527",
      "0.692598867"
    ],
    [
      "1.627333942",
      "0.7278042"
    ],
    [
      "0.856663208",
      "0.729933045"
    ],
    [
      "0.595532046",
      "0.758535505"
    ],
    [
      "1.416246357",
      "0.763009533"
    ],
    [
      "1.208772396",
      "0.815468459"
    ],
    [
      "0.123305569",
      "0.830138329"
    ],
    [
      "1.005830952",
      "0.883381304"
    ],
    [
      "0.32965015",
      "0.886877182"
    ],
    [
      "1.763366447",
      "0.893008744"
    ],
    [
      "0.74449364",
      "0.912183894"
    ],
    [
      "1.552278862",
      "0.928214077"
    ],
    [
      "0.529944395",
      "0.9622403"
    ],
    [
      "1.3448049",
      "0.980673003"
    ],
    [
      "0.108066537",
      "1.043598291"
    ],
    [
      "1.145041547",
      "1.057432279"
    ],
    [
      "1.890625022",
      "1.065062919"
    ],
    [
      "0.932759959",
      "1.084523079"
    ],
    [
      "1.679537437",
      "1.100268251"
    ],
    [
      "0.314411119",
      "1.100337145"
    ],
    [
      "0.722632558",
      "1.125067611"
    ],
    [
      "1.472063475",
      "1.152727178"
    ],
    [
      "0.514705363",
      "1.175700263"
    ],
    [
      "1.272300122",
      "1.229486453"
    ],
    [
      "0.144051304",
      "1.254554401"
    ],
    [
      "1.060018534",
      "1.256577254"
    ],
    [
      "1.849570981",
      "1.275091368"
    ],
    [
      "0.849891133",
      "1.297121785"
    ],
    [
      "1.621231219",
      "1.306175436"
    ],
    [
      "0.641963938",
      "1.347754437"
    ],
    [
      "0.429100509",
      "1.373531807"
    ],
    [
      "1.421467866",
      "1.382934712"
    ],
    [
      "1.200555286",
      "1.43110506"
    ],
    [
      "0.229193263",
      "1.450891435"
    ],
    [
      "0.988531416",
      "1.46014404"
    ],
    [
      "1.759726022",
      "1.469321302"
    ],
    [
      "0.780604221",
      "1.510776693"
    ],
    [
      "0.558448258",
      "1.544788737"
    ],
    [
      "1.559962668",
      "1.546080577"
    ],
    [
      "1.349487918",
      "1.584781522"
    ],
    [
      "0.358602703",
      "1.621333743"
    ],
    [
      "0.94380707",
      "1.669421647"
    ],
    [
      "1.157574125",
      "1.679473021"
    ],
    [
      "0.735879875",
      "1.720054299"
    ],
    [
      "1.488241294",
      "1.747707531"
    ],
    [
      "0.524943209",
      "1.756152867"
    ],
    [
      "1.296327501",
      "1.842399029"
    ],
    [
      "0.873444564",
      "1.8839852"
    ],
    [
      "1.0873956",
      "1.888711495"
    ]
  ],
  "radius": "0.107001616"
}

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？