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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 = 76

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 = 76
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[0.1027791819469670048220856306785, 0.1027791819469670048220856306795]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 = 0.102779181

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 = 76 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": [
    [
      "0.969184802",
      "0.103308515"
    ],
    [
      "1.174078268",
      "0.119828453"
    ],
    [
      "0.765908806",
      "0.133855376"
    ],
    [
      "0.574920118",
      "0.209865651"
    ],
    [
      "1.512045004",
      "0.263240127"
    ],
    [
      "1.306927106",
      "0.276689624"
    ],
    [
      "1.075189611",
      "0.300037427"
    ],
    [
      "0.869931852",
      "0.311150178"
    ],
    [
      "0.406243618",
      "0.327349604"
    ],
    [
      "1.666291233",
      "0.399116327"
    ],
    [
      "0.592325956",
      "0.414685761"
    ],
    [
      "1.421133661",
      "0.447602186"
    ],
    [
      "1.20803845",
      "0.456898599"
    ],
    [
      "0.268733035",
      "0.480140574"
    ],
    [
      "0.986184991",
      "0.48532752"
    ],
    [
      "0.780927232",
      "0.496440271"
    ],
    [
      "1.785564235",
      "0.566532551"
    ],
    [
      "0.454815373",
      "0.567476732"
    ],
    [
      "1.575379891",
      "0.583478386"
    ],
    [
      "1.324158847",
      "0.628848224"
    ],
    [
      "1.119033829",
      "0.642188692"
    ],
    [
      "0.643117012",
      "0.650782338"
    ],
    [
      "0.169606218",
      "0.660218655"
    ],
    [
      "0.880478309",
      "0.676284159"
    ],
    [
      "0.355688556",
      "0.747554812"
    ],
    [
      "1.86360344",
      "0.756701222"
    ],
    [
      "1.658574779",
      "0.771448733"
    ],
    [
      "1.453719531",
      "0.788436037"
    ],
    [
      "1.233149522",
      "0.813161934"
    ],
    [
      "1.023319656",
      "0.824103612"
    ],
    [
      "0.745387431",
      "0.831218647"
    ],
    [
      "0.539963195",
      "0.838643221"
    ],
    [
      "0.114066273",
      "0.858131655"
    ],
    [
      "1.775887252",
      "0.942604722"
    ],
    [
      "0.300148611",
      "0.945467812"
    ],
    [
      "1.570858591",
      "0.957352234"
    ],
    [
      "1.362710206",
      "0.972749747"
    ],
    [
      "0.88947677",
      "0.980117466"
    ],
    [
      "1.137435349",
      "0.995076854"
    ],
    [
      "0.689847486",
      "1.029131647"
    ],
    [
      "0.48442325",
      "1.036556221"
    ],
    [
      "0.105028459",
      "1.063491239"
    ],
    [
      "1.890002945",
      "1.113577965"
    ],
    [
      "1.684974284",
      "1.128325476"
    ],
    [
      "1.479849266",
      "1.141665944"
    ],
    [
      "0.313401054",
      "1.150598535"
    ],
    [
      "1.003592463",
      "1.151090709"
    ],
    [
      "1.266996033",
      "1.154664667"
    ],
    [
      "0.779892029",
      "1.213918609"
    ],
    [
      "0.573469485",
      "1.226863207"
    ],
    [
      "0.142967166",
      "1.265518206"
    ],
    [
      "1.840795024",
      "1.313159581"
    ],
    [
      "1.143957789",
      "1.321347241"
    ],
    [
      "1.384135093",
      "1.323580864"
    ],
    [
      "1.635766363",
      "1.327907092"
    ],
    [
      "0.396324901",
      "1.338688611"
    ],
    [
      "0.937595511",
      "1.345766448"
    ],
    [
      "0.69956723",
      "1.403133216"
    ],
    [
      "0.225891013",
      "1.453608282"
    ],
    [
      "1.507156596",
      "1.488262225"
    ],
    [
      "1.266058213",
      "1.491842836"
    ],
    [
      "0.519881266",
      "1.502969059"
    ],
    [
      "1.738929325",
      "1.50713951"
    ],
    [
      "1.061297769",
      "1.509937068"
    ],
    [
      "0.857270712",
      "1.534981055"
    ],
    [
      "0.349447378",
      "1.61788873"
    ],
    [
      "0.677584749",
      "1.634816898"
    ],
    [
      "1.3841085",
      "1.660123467"
    ],
    [
      "1.588986781",
      "1.676830679"
    ],
    [
      "1.179348055",
      "1.678217699"
    ],
    [
      "0.975320999",
      "1.703261686"
    ],
    [
      "0.507150861",
      "1.749736569"
    ],
    [
      "0.6907237",
      "1.842231183"
    ],
    [
      "1.297398342",
      "1.84649833"
    ],
    [
      "0.890530266",
      "1.890517588"
    ],
    [
      "1.096549446",
      "1.891996982"
    ]
  ],
  "radius": "0.102779181"
}
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": 76
}

起步布局答案

{
  "centers": [
    [
      "0.969184802",
      "0.103308515"
    ],
    [
      "1.174078268",
      "0.119828453"
    ],
    [
      "0.765908806",
      "0.133855376"
    ],
    [
      "0.574920118",
      "0.209865651"
    ],
    [
      "1.512045004",
      "0.263240127"
    ],
    [
      "1.306927106",
      "0.276689624"
    ],
    [
      "1.075189611",
      "0.300037427"
    ],
    [
      "0.869931852",
      "0.311150178"
    ],
    [
      "0.406243618",
      "0.327349604"
    ],
    [
      "1.666291233",
      "0.399116327"
    ],
    [
      "0.592325956",
      "0.414685761"
    ],
    [
      "1.421133661",
      "0.447602186"
    ],
    [
      "1.20803845",
      "0.456898599"
    ],
    [
      "0.268733035",
      "0.480140574"
    ],
    [
      "0.986184991",
      "0.48532752"
    ],
    [
      "0.780927232",
      "0.496440271"
    ],
    [
      "1.785564235",
      "0.566532551"
    ],
    [
      "0.454815373",
      "0.567476732"
    ],
    [
      "1.575379891",
      "0.583478386"
    ],
    [
      "1.324158847",
      "0.628848224"
    ],
    [
      "1.119033829",
      "0.642188692"
    ],
    [
      "0.643117012",
      "0.650782338"
    ],
    [
      "0.169606218",
      "0.660218655"
    ],
    [
      "0.880478309",
      "0.676284159"
    ],
    [
      "0.355688556",
      "0.747554812"
    ],
    [
      "1.86360344",
      "0.756701222"
    ],
    [
      "1.658574779",
      "0.771448733"
    ],
    [
      "1.453719531",
      "0.788436037"
    ],
    [
      "1.233149522",
      "0.813161934"
    ],
    [
      "1.023319656",
      "0.824103612"
    ],
    [
      "0.745387431",
      "0.831218647"
    ],
    [
      "0.539963195",
      "0.838643221"
    ],
    [
      "0.114066273",
      "0.858131655"
    ],
    [
      "1.775887252",
      "0.942604722"
    ],
    [
      "0.300148611",
      "0.945467812"
    ],
    [
      "1.570858591",
      "0.957352234"
    ],
    [
      "1.362710206",
      "0.972749747"
    ],
    [
      "0.88947677",
      "0.980117466"
    ],
    [
      "1.137435349",
      "0.995076854"
    ],
    [
      "0.689847486",
      "1.029131647"
    ],
    [
      "0.48442325",
      "1.036556221"
    ],
    [
      "0.105028459",
      "1.063491239"
    ],
    [
      "1.890002945",
      "1.113577965"
    ],
    [
      "1.684974284",
      "1.128325476"
    ],
    [
      "1.479849266",
      "1.141665944"
    ],
    [
      "0.313401054",
      "1.150598535"
    ],
    [
      "1.003592463",
      "1.151090709"
    ],
    [
      "1.266996033",
      "1.154664667"
    ],
    [
      "0.779892029",
      "1.213918609"
    ],
    [
      "0.573469485",
      "1.226863207"
    ],
    [
      "0.142967166",
      "1.265518206"
    ],
    [
      "1.840795024",
      "1.313159581"
    ],
    [
      "1.143957789",
      "1.321347241"
    ],
    [
      "1.384135093",
      "1.323580864"
    ],
    [
      "1.635766363",
      "1.327907092"
    ],
    [
      "0.396324901",
      "1.338688611"
    ],
    [
      "0.937595511",
      "1.345766448"
    ],
    [
      "0.69956723",
      "1.403133216"
    ],
    [
      "0.225891013",
      "1.453608282"
    ],
    [
      "1.507156596",
      "1.488262225"
    ],
    [
      "1.266058213",
      "1.491842836"
    ],
    [
      "0.519881266",
      "1.502969059"
    ],
    [
      "1.738929325",
      "1.50713951"
    ],
    [
      "1.061297769",
      "1.509937068"
    ],
    [
      "0.857270712",
      "1.534981055"
    ],
    [
      "0.349447378",
      "1.61788873"
    ],
    [
      "0.677584749",
      "1.634816898"
    ],
    [
      "1.3841085",
      "1.660123467"
    ],
    [
      "1.588986781",
      "1.676830679"
    ],
    [
      "1.179348055",
      "1.678217699"
    ],
    [
      "0.975320999",
      "1.703261686"
    ],
    [
      "0.507150861",
      "1.749736569"
    ],
    [
      "0.6907237",
      "1.842231183"
    ],
    [
      "1.297398342",
      "1.84649833"
    ],
    [
      "0.890530266",
      "1.890517588"
    ],
    [
      "1.096549446",
      "1.891996982"
    ]
  ],
  "radius": "0.102779181"
}

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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