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 = 257

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 = 257
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[0.0570842619075, 0.0570842619085]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 = 0.057084261

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 = 257 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": [
    [
      "0.938079167",
      "0.059119615"
    ],
    [
      "1.13606317",
      "0.066952894"
    ],
    [
      "0.821143462",
      "0.074364549"
    ],
    [
      "1.247832204",
      "0.090236686"
    ],
    [
      "0.704376527",
      "0.104624743"
    ],
    [
      "1.0348279",
      "0.119734254"
    ],
    [
      "0.568374994",
      "0.161674321"
    ],
    [
      "1.431835811",
      "0.161782891"
    ],
    [
      "0.892994739",
      "0.1640093"
    ],
    [
      "1.31904606",
      "0.179472545"
    ],
    [
      "0.781542788",
      "0.188766409"
    ],
    [
      "1.198020339",
      "0.194309331"
    ],
    [
      "0.670090837",
      "0.213523518"
    ],
    [
      "0.470220357",
      "0.21998486"
    ],
    [
      "1.529975782",
      "0.220118112"
    ],
    [
      "1.082244246",
      "0.223590607"
    ],
    [
      "0.970747119",
      "0.248144964"
    ],
    [
      "0.859340344",
      "0.273104825"
    ],
    [
      "1.415084748",
      "0.274715852"
    ],
    [
      "1.270862457",
      "0.282975164"
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    [
      "1.621193212",
      "0.293004814"
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  "radius": "0.057084261"
}
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": 257
}

起步布局答案

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  "radius": "0.057084261"
}

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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?