P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 187

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 = 187
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[0.0382010645445, 0.0382010645455]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 = 0.038201064

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 = 187 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": [
    [
      "0.170827723",
      "0.038201065"
    ],
    [
      "0.961766714",
      "0.038201065"
    ],
    [
      "0.455713702",
      "0.038201065"
    ],
    [
      "0.03820172",
      "0.038201065"
    ],
    [
      "0.82741782",
      "0.038201065"
    ],
    [
      "0.608517101",
      "0.038201065"
    ],
    [
      "0.307406583",
      "0.038201065"
    ],
    [
      "0.68491923",
      "0.038201065"
    ],
    [
      "0.532115401",
      "0.038457284"
    ],
    [
      "0.381560142",
      "0.056600383"
    ],
    [
      "0.756168525",
      "0.06578409"
    ],
    [
      "0.239117153",
      "0.0724624"
    ],
    [
      "0.894592267",
      "0.074599951"
    ],
    [
      "0.104514722",
      "0.0761467"
    ],
    [
      "0.570538144",
      "0.104494987"
    ],
    [
      "0.434571199",
      "0.11161959"
    ],
    [
      "0.696656276",
      "0.113696277"
    ],
    [
      "0.302933054",
      "0.114472113"
    ],
    [
      "0.829482683",
      "0.114575286"
    ],
    [
      "0.170533106",
      "0.114602626"
    ],
    [
      "0.961798935",
      "0.114603187"
    ],
    [
      "0.038201065",
      "0.114603194"
    ],
    [
      "0.504369503",
      "0.142691798"
    ],
    [
      "0.36959205",
      "0.151806596"
    ],
    [
      "0.630461267",
      "0.151892294"
    ],
    [
      "0.76281955",
      "0.151902382"
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      "0.236770679",
      "0.152679775"
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      "0.895632751",
      "0.152804251"
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    [
      "0.104367249",
      "0.152804258"
    ],
    [
      "0.436960995",
      "0.18806309"
    ],
    [
      "0.564289261",
      "0.190093358"
    ],
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      "0.696624541",
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      "0.302940672",
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    ],
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      "0.267407452"
    ],
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      "0.267407452"
    ],
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      "0.566166187",
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    ],
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      "0.433833817",
      "0.269576095"
    ],
    [
      "0.762787815",
      "0.304706634"
    ],
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      "0.236771333",
      "0.305484033"
    ],
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      "0.895632751",
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    ],
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      "0.500000002",
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    [
      "0.895632751",
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      "0.104367249",
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      "0.961798935"
    ],
    [
      "0.698467769",
      "0.961798935"
    ],
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      "0.433809161",
      "0.961798935"
    ],
    [
      "0.169199857",
      "0.961798935"
    ],
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      "0.566141221",
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    ],
    [
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  ],
  "radius": "0.038201064"
}
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": 187
}

起步布局答案

{
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      "0.170827723",
      "0.038201065"
    ],
    [
      "0.961766714",
      "0.038201065"
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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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