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 = 273

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 = 273
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[0.0555700675340554991064732806355, 0.0555700675340554991064732806365]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 = 0.055570067

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 = 273 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": [
    [
      "1.013813446",
      "0.147814696"
    ],
    [
      "1.372647018",
      "1.19301244"
    ],
    [
      "1.017481443",
      "0.686199054"
    ],
    [
      "1.583215219",
      "1.401857026"
    ],
    [
      "1.111598084",
      "1.937813289"
    ],
    [
      "1.190637977",
      "0.075010779"
    ],
    [
      "1.225909088",
      "0.893618494"
    ],
    [
      "0.951289197",
      "1.646194087"
    ],
    [
      "1.470200906",
      "1.39274165"
    ],
    [
      "1.585170096",
      "0.532611222"
    ],
    [
      "0.631580698",
      "1.431284286"
    ],
    [
      "0.928744893",
      "1.16245139"
    ],
    [
      "0.867667033",
      "1.935112765"
    ],
    [
      "1.319106897",
      "0.21994222"
    ],
    [
      "1.265868694",
      "0.317501553"
    ],
    [
      "0.325599164",
      "1.024135636"
    ],
    [
      "0.622763359",
      "0.75530274"
    ],
    [
      "0.290142088",
      "0.732232051"
    ],
    [
      "1.837079433",
      "1.437308426"
    ],
    [
      "0.376344475",
      "1.709226116"
    ],
    [
      "0.990841706",
      "1.070277015"
    ],
    [
      "0.401015845",
      "0.739922281"
    ],
    [
      "0.889192384",
      "1.738368462"
    ],
    [
      "0.982024367",
      "0.394295469"
    ],
    [
      "1.586009671",
      "0.259364757"
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    [
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      "0.985792869"
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      "0.609443491",
      "0.94734172"
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    [
      "0.760276853",
      "0.378915009"
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    [
      "1.508605862",
      "1.795781361"
    ],
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      "1.150492407",
      "1.177831849"
    ],
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      "0.822373666",
      "0.286740634"
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      "0.520706941",
      "1.423594056"
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      "1.669811213",
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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": 273
}

起步布局答案

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  "radius": "0.055570067"
}

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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还没有帖子。第一个聊聊这道题的思路?