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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 = 58

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 = 58
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[0.1173081931282865214310234108645, 0.1173081931282865214310234108655]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 = 0.117308192

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 = 58 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": [
    [
      "0.996202938",
      "0.11731636"
    ],
    [
      "1.228870179",
      "0.147495884"
    ],
    [
      "0.763803951",
      "0.149496471"
    ],
    [
      "1.445368251",
      "0.237902928"
    ],
    [
      "0.54809169",
      "0.241762765"
    ],
    [
      "1.026157178",
      "0.350012712"
    ],
    [
      "1.258824419",
      "0.380192236"
    ],
    [
      "1.630402047",
      "0.382150435"
    ],
    [
      "0.793758191",
      "0.382192823"
    ],
    [
      "0.357318693",
      "0.394933422"
    ],
    [
      "0.578045929",
      "0.474459117"
    ],
    [
      "1.443858214",
      "0.524439743"
    ],
    [
      "1.77089934",
      "0.570047639"
    ],
    [
      "1.123955984",
      "0.57216972"
    ],
    [
      "0.220622686",
      "0.585613914"
    ],
    [
      "0.890544337",
      "0.595915285"
    ],
    [
      "0.441349921",
      "0.665139609"
    ],
    [
      "0.674832076",
      "0.688181578"
    ],
    [
      "1.584355508",
      "0.712336947"
    ],
    [
      "1.316800411",
      "0.741697304"
    ],
    [
      "1.85693431",
      "0.78832002"
    ],
    [
      "1.086211344",
      "0.803730059"
    ],
    [
      "0.138987952",
      "0.805569862"
    ],
    [
      "0.85430563",
      "0.839291114"
    ],
    [
      "0.359715187",
      "0.885095557"
    ],
    [
      "0.605120825",
      "0.914218996"
    ],
    [
      "1.477651197",
      "0.921284401"
    ],
    [
      "1.248781218",
      "0.972892362"
    ],
    [
      "1.882428778",
      "1.021547122"
    ],
    [
      "1.021443616",
      "1.030879112"
    ],
    [
      "0.118181803",
      "1.039261866"
    ],
    [
      "0.789280116",
      "1.064716337"
    ],
    [
      "1.654177614",
      "1.07582654"
    ],
    [
      "0.338909039",
      "1.118787561"
    ],
    [
      "1.407910013",
      "1.145295597"
    ],
    [
      "0.571252131",
      "1.151368785"
    ],
    [
      "1.18057241",
      "1.203282348"
    ],
    [
      "1.845581615",
      "1.253251967"
    ],
    [
      "0.946662625",
      "1.253258636"
    ],
    [
      "0.159674149",
      "1.270180107"
    ],
    [
      "1.58443643",
      "1.299837736"
    ],
    [
      "0.711876414",
      "1.339170968"
    ],
    [
      "0.479370675",
      "1.37063624"
    ],
    [
      "1.345366627",
      "1.371422052"
    ],
    [
      "1.118029025",
      "1.429408803"
    ],
    [
      "1.748995992",
      "1.467065124"
    ],
    [
      "0.260533647",
      "1.482010724"
    ],
    [
      "0.891423415",
      "1.490193228"
    ],
    [
      "1.521893045",
      "1.525964191"
    ],
    [
      "0.624490338",
      "1.55690595"
    ],
    [
      "1.284704679",
      "1.59806048"
    ],
    [
      "1.057367077",
      "1.65604723"
    ],
    [
      "0.413634797",
      "1.659788356"
    ],
    [
      "0.830761467",
      "1.716831656"
    ],
    [
      "1.461231097",
      "1.752602618"
    ],
    [
      "0.608161345",
      "1.790953408"
    ],
    [
      "1.239796962",
      "1.846390025"
    ],
    [
      "0.996705129",
      "1.882685657"
    ]
  ],
  "radius": "0.117308192"
}
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": 58
}

起步布局答案

{
  "centers": [
    [
      "0.996202938",
      "0.11731636"
    ],
    [
      "1.228870179",
      "0.147495884"
    ],
    [
      "0.763803951",
      "0.149496471"
    ],
    [
      "1.445368251",
      "0.237902928"
    ],
    [
      "0.54809169",
      "0.241762765"
    ],
    [
      "1.026157178",
      "0.350012712"
    ],
    [
      "1.258824419",
      "0.380192236"
    ],
    [
      "1.630402047",
      "0.382150435"
    ],
    [
      "0.793758191",
      "0.382192823"
    ],
    [
      "0.357318693",
      "0.394933422"
    ],
    [
      "0.578045929",
      "0.474459117"
    ],
    [
      "1.443858214",
      "0.524439743"
    ],
    [
      "1.77089934",
      "0.570047639"
    ],
    [
      "1.123955984",
      "0.57216972"
    ],
    [
      "0.220622686",
      "0.585613914"
    ],
    [
      "0.890544337",
      "0.595915285"
    ],
    [
      "0.441349921",
      "0.665139609"
    ],
    [
      "0.674832076",
      "0.688181578"
    ],
    [
      "1.584355508",
      "0.712336947"
    ],
    [
      "1.316800411",
      "0.741697304"
    ],
    [
      "1.85693431",
      "0.78832002"
    ],
    [
      "1.086211344",
      "0.803730059"
    ],
    [
      "0.138987952",
      "0.805569862"
    ],
    [
      "0.85430563",
      "0.839291114"
    ],
    [
      "0.359715187",
      "0.885095557"
    ],
    [
      "0.605120825",
      "0.914218996"
    ],
    [
      "1.477651197",
      "0.921284401"
    ],
    [
      "1.248781218",
      "0.972892362"
    ],
    [
      "1.882428778",
      "1.021547122"
    ],
    [
      "1.021443616",
      "1.030879112"
    ],
    [
      "0.118181803",
      "1.039261866"
    ],
    [
      "0.789280116",
      "1.064716337"
    ],
    [
      "1.654177614",
      "1.07582654"
    ],
    [
      "0.338909039",
      "1.118787561"
    ],
    [
      "1.407910013",
      "1.145295597"
    ],
    [
      "0.571252131",
      "1.151368785"
    ],
    [
      "1.18057241",
      "1.203282348"
    ],
    [
      "1.845581615",
      "1.253251967"
    ],
    [
      "0.946662625",
      "1.253258636"
    ],
    [
      "0.159674149",
      "1.270180107"
    ],
    [
      "1.58443643",
      "1.299837736"
    ],
    [
      "0.711876414",
      "1.339170968"
    ],
    [
      "0.479370675",
      "1.37063624"
    ],
    [
      "1.345366627",
      "1.371422052"
    ],
    [
      "1.118029025",
      "1.429408803"
    ],
    [
      "1.748995992",
      "1.467065124"
    ],
    [
      "0.260533647",
      "1.482010724"
    ],
    [
      "0.891423415",
      "1.490193228"
    ],
    [
      "1.521893045",
      "1.525964191"
    ],
    [
      "0.624490338",
      "1.55690595"
    ],
    [
      "1.284704679",
      "1.59806048"
    ],
    [
      "1.057367077",
      "1.65604723"
    ],
    [
      "0.413634797",
      "1.659788356"
    ],
    [
      "0.830761467",
      "1.716831656"
    ],
    [
      "1.461231097",
      "1.752602618"
    ],
    [
      "0.608161345",
      "1.790953408"
    ],
    [
      "1.239796962",
      "1.846390025"
    ],
    [
      "0.996705129",
      "1.882685657"
    ]
  ],
  "radius": "0.117308192"
}

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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