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 = 269

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 = 269
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[0.0559829566065, 0.0559829566075]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 = 0.055982956

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 = 269 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": [
    [
      "0.943478526",
      "0.057676541"
    ],
    [
      "1.056877328",
      "0.057697953"
    ],
    [
      "0.809452195",
      "0.075430069"
    ],
    [
      "1.258168005",
      "0.091970563"
    ],
    [
      "0.701284451",
      "0.104490536"
    ],
    [
      "1.149806685",
      "0.120151939"
    ],
    [
      "1.363860166",
      "0.128923679"
    ],
    [
      "1.000159697",
      "0.154235386"
    ],
    [
      "0.888196699",
      "0.155043298"
    ],
    [
      "1.464433783",
      "0.178130522"
    ],
    [
      "0.780056197",
      "0.184060492"
    ],
    [
      "0.517433623",
      "0.188644425"
    ],
    [
      "0.629386961",
      "0.190322446"
    ],
    [
      "1.292836672",
      "0.215480186"
    ],
    [
      "1.181521302",
      "0.227532332"
    ],
    [
      "1.070419157",
      "0.241413216"
    ],
    [
      "0.424765708",
      "0.25148568"
    ],
    [
      "1.577882521",
      "0.253528319"
    ],
    [
      "0.959791211",
      "0.258670778"
    ],
    [
      "0.848685984",
      "0.272526972"
    ],
    [
      "0.737756871",
      "0.287728871"
    ],
    [
      "1.471969159",
      "0.289842579"
    ],
    [
      "0.626737508",
      "0.302257007"
    ],
    [
      "1.360946749",
      "0.304347408"
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    [
      "0.515718145",
      "0.316785144"
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      "1.249930438",
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      "0.34018981",
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      "1.66219795",
      "0.327198355"
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      "1.138615069",
      "0.330951"
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      "1.027887014",
      "0.34755412"
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      "0.362082256"
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      "1.540103247",
      "0.378691394"
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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": 269
}

起步布局答案

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      "1.030026576",
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  "radius": "0.055982956"
}

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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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?