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 = 267

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 = 267
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[0.0561881370365, 0.0561881370375]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 = 0.056188136

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 = 267 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": [
    [
      "0.970685798",
      "0.056643487"
    ],
    [
      "1.133716678",
      "0.065708459"
    ],
    [
      "0.838270639",
      "0.070148742"
    ],
    [
      "1.243814256",
      "0.088224018"
    ],
    [
      "0.724814503",
      "0.097196713"
    ],
    [
      "1.350455326",
      "0.123665648"
    ],
    [
      "1.047914145",
      "0.138278097"
    ],
    [
      "0.600517668",
      "0.144900767"
    ],
    [
      "0.913684033",
      "0.15348989"
    ],
    [
      "0.803818926",
      "0.17711372"
    ],
    [
      "1.469016795",
      "0.180974994"
    ],
    [
      "1.151721307",
      "0.181318775"
    ],
    [
      "1.262697505",
      "0.1990024"
    ],
    [
      "0.695106556",
      "0.205575055"
    ],
    [
      "0.527307565",
      "0.230157483"
    ],
    [
      "1.369338575",
      "0.234444029"
    ],
    [
      "0.990912379",
      "0.235124499"
    ],
    [
      "1.565717985",
      "0.244523995"
    ],
    [
      "0.418153434",
      "0.256874569"
    ],
    [
      "0.88160056",
      "0.261188904"
    ],
    [
      "0.772288741",
      "0.287253309"
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    [
      "1.177480281",
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      "1.464738499",
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    [
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}
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": 267
}

起步布局答案

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      "0.459203507",
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      "0.731557747",
      "1.801033556"
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      "1.278208137",
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      "1.165845184",
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      "1.808879779"
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      "1.475562023",
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    [
      "0.947432971",
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    [
      "1.37140242",
      "1.867664149"
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    [
      "0.647357782",
      "1.875456623"
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    [
      "0.753910031",
      "1.911164398"
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    [
      "1.243555782",
      "1.911839756"
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      "1.131776393",
      "1.923429145"
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    [
      "0.86395105",
      "1.933954772"
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    [
      "1.021115824",
      "1.943575622"
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  ],
  "radius": "0.056188136"
}

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

DISCUSSION

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聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖;发言公开署名,与纪录使用同一个名字,署名后的 #编号是账号注册序号,冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路?