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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 = 39

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 = 39
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[0.1416855217454031705065705883095, 0.1416855217454031705065705883105]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 = 0.141685521

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 = 39 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": [
    [
      "0.900604705",
      "0.147460078"
    ],
    [
      "1.18362486",
      "0.161557602"
    ],
    [
      "0.628418427",
      "0.2262876"
    ],
    [
      "1.446630294",
      "0.267043573"
    ],
    [
      "0.396733754",
      "0.389448135"
    ],
    [
      "1.088460392",
      "0.428471207"
    ],
    [
      "0.805718219",
      "0.447339476"
    ],
    [
      "1.660953972",
      "0.452420242"
    ],
    [
      "1.351465826",
      "0.533957178"
    ],
    [
      "0.574033545",
      "0.610500012"
    ],
    [
      "0.230803824",
      "0.619157531"
    ],
    [
      "0.982623906",
      "0.691335785"
    ],
    [
      "1.803235066",
      "0.697481946"
    ],
    [
      "1.526194986",
      "0.757046553"
    ],
    [
      "1.24562934",
      "0.796821756"
    ],
    [
      "0.741032734",
      "0.839433232"
    ],
    [
      "0.42979429",
      "0.854414311"
    ],
    [
      "0.148714648",
      "0.89037795"
    ],
    [
      "1.857965238",
      "0.975517477"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.580925158",
      "1.035082084"
    ],
    [
      "1.300359512",
      "1.074857287"
    ],
    [
      "0.596793479",
      "1.083347532"
    ],
    [
      "0.315713837",
      "1.119311171"
    ],
    [
      "0.829006108",
      "1.245755792"
    ],
    [
      "1.819179016",
      "1.256221551"
    ],
    [
      "1.109714777",
      "1.284508744"
    ],
    [
      "1.494478464",
      "1.304945229"
    ],
    [
      "0.517126126",
      "1.355289182"
    ],
    [
      "0.236046485",
      "1.391252821"
    ],
    [
      "1.29107459",
      "1.502241991"
    ],
    [
      "1.691104016",
      "1.508998019"
    ],
    [
      "0.749338756",
      "1.517697442"
    ],
    [
      "1.022570392",
      "1.592821482"
    ],
    [
      "0.405080504",
      "1.618687754"
    ],
    [
      "1.487700141",
      "1.70629478"
    ],
    [
      "0.638959489",
      "1.778687031"
    ],
    [
      "1.208836116",
      "1.810696067"
    ],
    [
      "0.912191125",
      "1.853811071"
    ]
  ],
  "radius": "0.141685521"
}
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": 39
}

起步布局答案

{
  "centers": [
    [
      "0.900604705",
      "0.147460078"
    ],
    [
      "1.18362486",
      "0.161557602"
    ],
    [
      "0.628418427",
      "0.2262876"
    ],
    [
      "1.446630294",
      "0.267043573"
    ],
    [
      "0.396733754",
      "0.389448135"
    ],
    [
      "1.088460392",
      "0.428471207"
    ],
    [
      "0.805718219",
      "0.447339476"
    ],
    [
      "1.660953972",
      "0.452420242"
    ],
    [
      "1.351465826",
      "0.533957178"
    ],
    [
      "0.574033545",
      "0.610500012"
    ],
    [
      "0.230803824",
      "0.619157531"
    ],
    [
      "0.982623906",
      "0.691335785"
    ],
    [
      "1.803235066",
      "0.697481946"
    ],
    [
      "1.526194986",
      "0.757046553"
    ],
    [
      "1.24562934",
      "0.796821756"
    ],
    [
      "0.741032734",
      "0.839433232"
    ],
    [
      "0.42979429",
      "0.854414311"
    ],
    [
      "0.148714648",
      "0.89037795"
    ],
    [
      "1.857965238",
      "0.975517477"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.580925158",
      "1.035082084"
    ],
    [
      "1.300359512",
      "1.074857287"
    ],
    [
      "0.596793479",
      "1.083347532"
    ],
    [
      "0.315713837",
      "1.119311171"
    ],
    [
      "0.829006108",
      "1.245755792"
    ],
    [
      "1.819179016",
      "1.256221551"
    ],
    [
      "1.109714777",
      "1.284508744"
    ],
    [
      "1.494478464",
      "1.304945229"
    ],
    [
      "0.517126126",
      "1.355289182"
    ],
    [
      "0.236046485",
      "1.391252821"
    ],
    [
      "1.29107459",
      "1.502241991"
    ],
    [
      "1.691104016",
      "1.508998019"
    ],
    [
      "0.749338756",
      "1.517697442"
    ],
    [
      "1.022570392",
      "1.592821482"
    ],
    [
      "0.405080504",
      "1.618687754"
    ],
    [
      "1.487700141",
      "1.70629478"
    ],
    [
      "0.638959489",
      "1.778687031"
    ],
    [
      "1.208836116",
      "1.810696067"
    ],
    [
      "0.912191125",
      "1.853811071"
    ]
  ],
  "radius": "0.141685521"
}

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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