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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 = 46

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 = 46
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[0.1307158800382333870987963075885, 0.1307158800382333870987963075895]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 = 0.130715879

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 = 46 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": [
    [
      "0.969374567",
      "0.131255524"
    ],
    [
      "1.229058255",
      "0.161437422"
    ],
    [
      "0.71246085",
      "0.179648783"
    ],
    [
      "1.468024358",
      "0.267464621"
    ],
    [
      "0.48155413",
      "0.302240184"
    ],
    [
      "0.898024166",
      "0.382762354"
    ],
    [
      "1.157707855",
      "0.412944252"
    ],
    [
      "1.664659156",
      "0.4397473"
    ],
    [
      "0.660773387",
      "0.492574161"
    ],
    [
      "1.396673958",
      "0.518971451"
    ],
    [
      "0.40563275",
      "0.552405148"
    ],
    [
      "0.91401527",
      "0.646498901"
    ],
    [
      "1.185368659",
      "0.672908567"
    ],
    [
      "1.593308756",
      "0.69125413"
    ],
    [
      "0.186174839",
      "0.694478335"
    ],
    [
      "0.673443523",
      "0.753698716"
    ],
    [
      "1.840230696",
      "0.777138475"
    ],
    [
      "0.418942783",
      "0.813497868"
    ],
    [
      "1.383850086",
      "0.847694633"
    ],
    [
      "0.874269915",
      "0.921076509"
    ],
    [
      "1.135564238",
      "0.929552455"
    ],
    [
      "0.132139813",
      "0.950264928"
    ],
    [
      "1.616797206",
      "0.966363098"
    ],
    [
      "0.635978685",
      "1.028611875"
    ],
    [
      "1.868494718",
      "1.037037898"
    ],
    [
      "0.378126134",
      "1.071723673"
    ],
    [
      "1.333561541",
      "1.104244097"
    ],
    [
      "0.997576692",
      "1.151602004"
    ],
    [
      "0.156599964",
      "1.210549899"
    ],
    [
      "1.566508661",
      "1.222912562"
    ],
    [
      "0.739214715",
      "1.268796985"
    ],
    [
      "1.818206173",
      "1.293587362"
    ],
    [
      "0.478869118",
      "1.312965079"
    ],
    [
      "1.209268867",
      "1.334239524"
    ],
    [
      "0.959112328",
      "1.410188658"
    ],
    [
      "0.257342948",
      "1.451791306"
    ],
    [
      "1.442215987",
      "1.452907988"
    ],
    [
      "0.646783055",
      "1.513343409"
    ],
    [
      "1.693913499",
      "1.523582789"
    ],
    [
      "1.144759915",
      "1.594524206"
    ],
    [
      "0.425256885",
      "1.652169635"
    ],
    [
      "0.866680667",
      "1.654735082"
    ],
    [
      "1.50685857",
      "1.706221828"
    ],
    [
      "0.645154497",
      "1.793561309"
    ],
    [
      "1.273959906",
      "1.824985364"
    ],
    [
      "1.016282445",
      "1.869131614"
    ]
  ],
  "radius": "0.130715879"
}
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": 46
}

起步布局答案

{
  "centers": [
    [
      "0.969374567",
      "0.131255524"
    ],
    [
      "1.229058255",
      "0.161437422"
    ],
    [
      "0.71246085",
      "0.179648783"
    ],
    [
      "1.468024358",
      "0.267464621"
    ],
    [
      "0.48155413",
      "0.302240184"
    ],
    [
      "0.898024166",
      "0.382762354"
    ],
    [
      "1.157707855",
      "0.412944252"
    ],
    [
      "1.664659156",
      "0.4397473"
    ],
    [
      "0.660773387",
      "0.492574161"
    ],
    [
      "1.396673958",
      "0.518971451"
    ],
    [
      "0.40563275",
      "0.552405148"
    ],
    [
      "0.91401527",
      "0.646498901"
    ],
    [
      "1.185368659",
      "0.672908567"
    ],
    [
      "1.593308756",
      "0.69125413"
    ],
    [
      "0.186174839",
      "0.694478335"
    ],
    [
      "0.673443523",
      "0.753698716"
    ],
    [
      "1.840230696",
      "0.777138475"
    ],
    [
      "0.418942783",
      "0.813497868"
    ],
    [
      "1.383850086",
      "0.847694633"
    ],
    [
      "0.874269915",
      "0.921076509"
    ],
    [
      "1.135564238",
      "0.929552455"
    ],
    [
      "0.132139813",
      "0.950264928"
    ],
    [
      "1.616797206",
      "0.966363098"
    ],
    [
      "0.635978685",
      "1.028611875"
    ],
    [
      "1.868494718",
      "1.037037898"
    ],
    [
      "0.378126134",
      "1.071723673"
    ],
    [
      "1.333561541",
      "1.104244097"
    ],
    [
      "0.997576692",
      "1.151602004"
    ],
    [
      "0.156599964",
      "1.210549899"
    ],
    [
      "1.566508661",
      "1.222912562"
    ],
    [
      "0.739214715",
      "1.268796985"
    ],
    [
      "1.818206173",
      "1.293587362"
    ],
    [
      "0.478869118",
      "1.312965079"
    ],
    [
      "1.209268867",
      "1.334239524"
    ],
    [
      "0.959112328",
      "1.410188658"
    ],
    [
      "0.257342948",
      "1.451791306"
    ],
    [
      "1.442215987",
      "1.452907988"
    ],
    [
      "0.646783055",
      "1.513343409"
    ],
    [
      "1.693913499",
      "1.523582789"
    ],
    [
      "1.144759915",
      "1.594524206"
    ],
    [
      "0.425256885",
      "1.652169635"
    ],
    [
      "0.866680667",
      "1.654735082"
    ],
    [
      "1.50685857",
      "1.706221828"
    ],
    [
      "0.645154497",
      "1.793561309"
    ],
    [
      "1.273959906",
      "1.824985364"
    ],
    [
      "1.016282445",
      "1.869131614"
    ]
  ],
  "radius": "0.130715879"
}

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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