P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 166

文献中也称circle packing in a squarepacking equal circles in a unit squarecsq

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆,使共同半径尽可能大。

子题n = 166
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[0.0402377276365, 0.0402377276375]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器单位正方形:左下角是原点 (0, 0),右上角是 (1, 1)
  • 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
放大来摆,然后提交 ↗

当前展示:起步布局

1y0
0x1
已验证构造r = 0.040237727

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 = 1–30 已证明最优,适合观察经典接触结构;n = 31–300 接入 Packomania 的公开前沿,除 n = 36 外最优性仍未知。大规模部分集中成一张表,供算法批量挑战。

查看来源
单位正方形内的等圆装箱 n = 166 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

容器是边长 1 的正方形,左下角是原点 (0, 0),右上角是 (1, 1)。坐标和长度用同一个单位,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 "0.25"。

起步布局答案

{
  "centers": [
    [
      "0.040261278",
      "0.040237728"
    ],
    [
      "0.184779336",
      "0.040237728"
    ],
    [
      "0.325943347",
      "0.040237728"
    ],
    [
      "0.475222083",
      "0.040237728"
    ],
    [
      "0.634016853",
      "0.040237728"
    ],
    [
      "0.714492309",
      "0.040237728"
    ],
    [
      "0.874061781",
      "0.040237728"
    ],
    [
      "0.794277045",
      "0.050758885"
    ],
    [
      "0.554619468",
      "0.053366098"
    ],
    [
      "0.955736527",
      "0.06840611"
    ],
    [
      "0.400582715",
      "0.070325328"
    ],
    [
      "0.112520307",
      "0.075662742"
    ],
    [
      "0.255361341",
      "0.078896224"
    ],
    [
      "0.745503325",
      "0.114770002"
    ],
    [
      "0.605687663",
      "0.115562065"
    ],
    [
      "0.887503244",
      "0.119582709"
    ],
    [
      "0.463959025",
      "0.119921116"
    ],
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      "0.324131584",
      "0.120692786"
    ],
    [
      "0.040237728",
      "0.120713179"
    ],
    [
      "0.184755785",
      "0.120713179"
    ],
    [
      "0.815222588",
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    ],
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    [
      "0.675821315",
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    ],
    [
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      "0.155799793"
    ],
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      "0.254449574",
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      "0.745526876",
      "0.195245454"
    ],
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      "0.605711213",
      "0.196037517"
    ],
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      "0.88750146",
      "0.200058164"
    ],
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      "0.463982576",
      "0.200396568"
    ],
    [
      "0.324155135",
      "0.201168238"
    ],
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      "0.040237728",
      "0.201188635"
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    ],
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    ],
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      "0.845032401"
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    ],
    [
      "0.959738722",
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  ],
  "radius": "0.040237727"
}
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": 166
}

起步布局答案

{
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    ],
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}

提交 radius 与 centers。每个数写成十进制字符串,例如 "0.25"。 · 验证器 v1.0.0

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