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 = 264

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 = 264
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[0.0564927900041889184807972946295, 0.0564927900041889184807972946305]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 = 0.056492789

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 = 264 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": [
    [
      "0.936329567",
      "0.058643568"
    ],
    [
      "1.063670433",
      "0.058643568"
    ],
    [
      "0.798267518",
      "0.078311408"
    ],
    [
      "1.201732482",
      "0.078311408"
    ],
    [
      "0.689539199",
      "0.109034262"
    ],
    [
      "1.310460801",
      "0.109034262"
    ],
    [
      "1",
      "0.151980685"
    ],
    [
      "0.585262953",
      "0.152533755"
    ],
    [
      "1.414737047",
      "0.152533755"
    ],
    [
      "0.879836777",
      "0.156491951"
    ],
    [
      "1.120163223",
      "0.156491951"
    ],
    [
      "1.514662868",
      "0.209232353"
    ],
    [
      "0.485337029",
      "0.20923263"
    ],
    [
      "0.687779073",
      "0.222006131"
    ],
    [
      "1.312220927",
      "0.222006131"
    ],
    [
      "0.799857108",
      "0.236297927"
    ],
    [
      "1.200142892",
      "0.236297927"
    ],
    [
      "0.94350721",
      "0.249829067"
    ],
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      "0.249829067"
    ],
    [
      "0.583502828",
      "0.265505624"
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    [
      "1.416497172",
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    [
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    [
      "1.60550403",
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    [
      "0.49255494",
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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": 264
}

起步布局答案

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    [
      "1.323888267",
      "1.886172808"
    ],
    [
      "0.831141188",
      "1.928273967"
    ],
    [
      "1.168858812",
      "1.928273967"
    ],
    [
      "0.943313875",
      "1.941802813"
    ],
    [
      "1.056686125",
      "1.941802813"
    ]
  ],
  "radius": "0.056492789"
}

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

讨论区

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?