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 = 149

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 = 149
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[0.0744294137778631784047692075595, 0.0744294137778631784047692075605]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 = 0.074429413

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 = 149 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": [
    [
      "0.923629199",
      "0.077585554"
    ],
    [
      "1.076370801",
      "0.077585554"
    ],
    [
      "0.776746112",
      "0.101758044"
    ],
    [
      "1.223253888",
      "0.101758044"
    ],
    [
      "0.560450037",
      "0.185459185"
    ],
    [
      "1.439549963",
      "0.185459185"
    ],
    [
      "1",
      "0.205360556"
    ],
    [
      "0.853116913",
      "0.229533047"
    ],
    [
      "1.146883087",
      "0.229533047"
    ],
    [
      "0.702282345",
      "0.23065373"
    ],
    [
      "1.297717655",
      "0.23065373"
    ],
    [
      "0.486020623",
      "0.314374711"
    ],
    [
      "1.513979377",
      "0.314374711"
    ],
    [
      "0.341405679",
      "0.349665908"
    ],
    [
      "1.658594321",
      "0.349665908"
    ],
    [
      "0.627852931",
      "0.359569256"
    ],
    [
      "0.776711759",
      "0.359569256"
    ],
    [
      "0.925570586",
      "0.359569256"
    ],
    [
      "1.074429414",
      "0.359569256"
    ],
    [
      "1.223288241",
      "0.359569256"
    ],
    [
      "1.372147069",
      "0.359569256"
    ],
    [
      "0.245669272",
      "0.463654903"
    ],
    [
      "1.754330728",
      "0.463654903"
    ],
    [
      "0.404619766",
      "0.484435823"
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      "1.595380234",
      "0.484435823"
    ],
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      "0.553423517",
      "0.488484782"
    ],
    [
      "0.702282345",
      "0.488484782"
    ],
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      "0.851141172",
      "0.488484782"
    ],
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      "1",
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      "1.148858828",
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    ],
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    ],
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    ],
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      "0.311614382",
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      "0.111462711",
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  ],
  "radius": "0.074429413"
}
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": 149
}

起步布局答案

{
  "centers": [
    [
      "0.923629199",
      "0.077585554"
    ],
    [
      "1.076370801",
      "0.077585554"
    ],
    [
      "0.776746112",
      "0.101758044"
    ],
    [
      "1.223253888",
      "0.101758044"
    ],
    [
      "0.560450037",
      "0.185459185"
    ],
    [
      "1.439549963",
      "0.185459185"
    ],
    [
      "1",
      "0.205360556"
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    [
      "0.853116913",
      "0.229533047"
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    [
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  "radius": "0.074429413"
}

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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