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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 = 54

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 = 54
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[0.1218920218569649676830821100265, 0.1218920218569649676830821100275]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 = 0.121892021

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 = 54 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": [
    [
      "0.788876825",
      "0.147649939"
    ],
    [
      "1.211123175",
      "0.147649939"
    ],
    [
      "0.558987728",
      "0.242042276"
    ],
    [
      "1.441003899",
      "0.242043428"
    ],
    [
      "1",
      "0.269541961"
    ],
    [
      "0.788876825",
      "0.391433983"
    ],
    [
      "1.211123175",
      "0.391433983"
    ],
    [
      "0.366117404",
      "0.392326999"
    ],
    [
      "1.633882596",
      "0.392326999"
    ],
    [
      "0.57775365",
      "0.513326005"
    ],
    [
      "1",
      "0.513326005"
    ],
    [
      "1.42224635",
      "0.513326005"
    ],
    [
      "0.788876825",
      "0.635218027"
    ],
    [
      "1.211123175",
      "0.635218027"
    ],
    [
      "0.366117404",
      "0.636111043"
    ],
    [
      "1.633882596",
      "0.636111043"
    ],
    [
      "0.57775365",
      "0.757110048"
    ],
    [
      "1",
      "0.757110048"
    ],
    [
      "1.42224635",
      "0.757110048"
    ],
    [
      "0.155598574",
      "0.759043878"
    ],
    [
      "1.844401426",
      "0.759043878"
    ],
    [
      "0.788876825",
      "0.87900207"
    ],
    [
      "1.211123175",
      "0.87900207"
    ],
    [
      "0.36723482",
      "0.880042884"
    ],
    [
      "1.63276518",
      "0.880042884"
    ],
    [
      "1",
      "1.000894092"
    ],
    [
      "0.578357995",
      "1.001934906"
    ],
    [
      "1.421642005",
      "1.001934906"
    ],
    [
      "0.122444975",
      "1.004107874"
    ],
    [
      "1.877566507",
      "1.004111249"
    ],
    [
      "0.36723482",
      "1.123826927"
    ],
    [
      "0.78948117",
      "1.123826927"
    ],
    [
      "1.21051883",
      "1.123826927"
    ],
    [
      "1.63276518",
      "1.123826927"
    ],
    [
      "0.578357995",
      "1.245718949"
    ],
    [
      "1.421642005",
      "1.245718949"
    ],
    [
      "1",
      "1.246759763"
    ],
    [
      "0.157832915",
      "1.248652816"
    ],
    [
      "1.842167085",
      "1.248652816"
    ],
    [
      "0.36895609",
      "1.370544838"
    ],
    [
      "1.63104391",
      "1.370544838"
    ],
    [
      "0.806925027",
      "1.395597643"
    ],
    [
      "1.193074973",
      "1.395597643"
    ],
    [
      "0.581941933",
      "1.489476647"
    ],
    [
      "1.418058067",
      "1.489476647"
    ],
    [
      "1",
      "1.544435523"
    ],
    [
      "0.372540028",
      "1.614302536"
    ],
    [
      "1.627459972",
      "1.614302536"
    ],
    [
      "0.775016906",
      "1.638314528"
    ],
    [
      "1.224983094",
      "1.638314528"
    ],
    [
      "0.565615002",
      "1.763140416"
    ],
    [
      "1.434384998",
      "1.763140416"
    ],
    [
      "0.877498441",
      "1.86892374"
    ],
    [
      "1.122465566",
      "1.868928814"
    ]
  ],
  "radius": "0.121892021"
}
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": 54
}

起步布局答案

{
  "centers": [
    [
      "0.788876825",
      "0.147649939"
    ],
    [
      "1.211123175",
      "0.147649939"
    ],
    [
      "0.558987728",
      "0.242042276"
    ],
    [
      "1.441003899",
      "0.242043428"
    ],
    [
      "1",
      "0.269541961"
    ],
    [
      "0.788876825",
      "0.391433983"
    ],
    [
      "1.211123175",
      "0.391433983"
    ],
    [
      "0.366117404",
      "0.392326999"
    ],
    [
      "1.633882596",
      "0.392326999"
    ],
    [
      "0.57775365",
      "0.513326005"
    ],
    [
      "1",
      "0.513326005"
    ],
    [
      "1.42224635",
      "0.513326005"
    ],
    [
      "0.788876825",
      "0.635218027"
    ],
    [
      "1.211123175",
      "0.635218027"
    ],
    [
      "0.366117404",
      "0.636111043"
    ],
    [
      "1.633882596",
      "0.636111043"
    ],
    [
      "0.57775365",
      "0.757110048"
    ],
    [
      "1",
      "0.757110048"
    ],
    [
      "1.42224635",
      "0.757110048"
    ],
    [
      "0.155598574",
      "0.759043878"
    ],
    [
      "1.844401426",
      "0.759043878"
    ],
    [
      "0.788876825",
      "0.87900207"
    ],
    [
      "1.211123175",
      "0.87900207"
    ],
    [
      "0.36723482",
      "0.880042884"
    ],
    [
      "1.63276518",
      "0.880042884"
    ],
    [
      "1",
      "1.000894092"
    ],
    [
      "0.578357995",
      "1.001934906"
    ],
    [
      "1.421642005",
      "1.001934906"
    ],
    [
      "0.122444975",
      "1.004107874"
    ],
    [
      "1.877566507",
      "1.004111249"
    ],
    [
      "0.36723482",
      "1.123826927"
    ],
    [
      "0.78948117",
      "1.123826927"
    ],
    [
      "1.21051883",
      "1.123826927"
    ],
    [
      "1.63276518",
      "1.123826927"
    ],
    [
      "0.578357995",
      "1.245718949"
    ],
    [
      "1.421642005",
      "1.245718949"
    ],
    [
      "1",
      "1.246759763"
    ],
    [
      "0.157832915",
      "1.248652816"
    ],
    [
      "1.842167085",
      "1.248652816"
    ],
    [
      "0.36895609",
      "1.370544838"
    ],
    [
      "1.63104391",
      "1.370544838"
    ],
    [
      "0.806925027",
      "1.395597643"
    ],
    [
      "1.193074973",
      "1.395597643"
    ],
    [
      "0.581941933",
      "1.489476647"
    ],
    [
      "1.418058067",
      "1.489476647"
    ],
    [
      "1",
      "1.544435523"
    ],
    [
      "0.372540028",
      "1.614302536"
    ],
    [
      "1.627459972",
      "1.614302536"
    ],
    [
      "0.775016906",
      "1.638314528"
    ],
    [
      "1.224983094",
      "1.638314528"
    ],
    [
      "0.565615002",
      "1.763140416"
    ],
    [
      "1.434384998",
      "1.763140416"
    ],
    [
      "0.877498441",
      "1.86892374"
    ],
    [
      "1.122465566",
      "1.868928814"
    ]
  ],
  "radius": "0.121892021"
}

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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