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 = 222

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 = 222
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[0.0613546248065, 0.0613546248075]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 = 0.061354624

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 = 222 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": [
    [
      "0.976269619",
      "0.061654643"
    ],
    [
      "0.854064712",
      "0.07276862"
    ],
    [
      "1.16391431",
      "0.075777495"
    ],
    [
      "0.732044326",
      "0.100651379"
    ],
    [
      "1.380350265",
      "0.141868998"
    ],
    [
      "0.611047447",
      "0.145733618"
    ],
    [
      "1.064179989",
      "0.147266131"
    ],
    [
      "1.258265578",
      "0.15423384"
    ],
    [
      "0.949059167",
      "0.189748554"
    ],
    [
      "0.826376723",
      "0.19231332"
    ],
    [
      "1.497289708",
      "0.203910755"
    ],
    [
      "0.512754433",
      "0.219191383"
    ],
    [
      "0.707280347",
      "0.221870265"
    ],
    [
      "1.158531257",
      "0.225722476"
    ],
    [
      "1.390393506",
      "0.264166558"
    ],
    [
      "1.043410435",
      "0.268204899"
    ],
    [
      "0.40357423",
      "0.275202483"
    ],
    [
      "1.265956962",
      "0.288156904"
    ],
    [
      "1.612945356",
      "0.28912071"
    ],
    [
      "0.928289613",
      "0.310687322"
    ],
    [
      "0.80560717",
      "0.313252088"
    ],
    [
      "0.626195294",
      "0.313972242"
    ],
    [
      "1.496024664",
      "0.326613483"
    ],
    [
      "0.506671366",
      "0.341749762"
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      "1.365799158",
      "0.38438585"
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    [
      "1.041206245",
      "0.39089435"
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      "0.716237309",
      "0.397338876"
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      "1.876126552"
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      "0.815469962",
      "1.881318261"
    ],
    [
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      "1.886983395"
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      "1.210723669",
      "1.914686108"
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      "0.925479436",
      "1.935682545"
    ],
    [
      "1.048176551",
      "1.937408214"
    ]
  ],
  "radius": "0.061354624"
}
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": 222
}

起步布局答案

{
  "centers": [
    [
      "0.976269619",
      "0.061654643"
    ],
    [
      "0.854064712",
      "0.07276862"
    ],
    [
      "1.16391431",
      "0.075777495"
    ],
    [
      "0.732044326",
      "0.100651379"
    ],
    [
      "1.380350265",
      "0.141868998"
    ],
    [
      "0.611047447",
      "0.145733618"
    ],
    [
      "1.064179989",
      "0.147266131"
    ],
    [
      "1.258265578",
      "0.15423384"
    ],
    [
      "0.949059167",
      "0.189748554"
    ],
    [
      "0.826376723",
      "0.19231332"
    ],
    [
      "1.497289708",
      "0.203910755"
    ],
    [
      "0.512754433",
      "0.219191383"
    ],
    [
      "0.707280347",
      "0.221870265"
    ],
    [
      "1.158531257",
      "0.225722476"
    ],
    [
      "1.390393506",
      "0.264166558"
    ],
    [
      "1.043410435",
      "0.268204899"
    ],
    [
      "0.40357423",
      "0.275202483"
    ],
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  "radius": "0.061354624"
}

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?