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P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 50

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 = 50
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[0.0713771038645, 0.0713771038655]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 = 0.071377103

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 = 50 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": [
    [
      "0.071377104",
      "0.071377104"
    ],
    [
      "0.319943701",
      "0.071377104"
    ],
    [
      "0.584307732",
      "0.071377104"
    ],
    [
      "0.72706194",
      "0.071377104"
    ],
    [
      "0.928622896",
      "0.071377104"
    ],
    [
      "0.452125717",
      "0.125290726"
    ],
    [
      "0.195660402",
      "0.141608333"
    ],
    [
      "0.827842418",
      "0.172481303"
    ],
    [
      "0.56490729",
      "0.212806898"
    ],
    [
      "0.071377104",
      "0.214131312"
    ],
    [
      "0.318522363",
      "0.214297413"
    ],
    [
      "0.704235154",
      "0.243895644"
    ],
    [
      "0.928622896",
      "0.273585502"
    ],
    [
      "0.442151134",
      "0.285674517"
    ],
    [
      "0.194901697",
      "0.285688552"
    ],
    [
      "0.805015633",
      "0.344999843"
    ],
    [
      "0.566634421",
      "0.355550657"
    ],
    [
      "0.318530468",
      "0.357065656"
    ],
    [
      "0.071377104",
      "0.357245792"
    ],
    [
      "0.928622896",
      "0.416414185"
    ],
    [
      "0.442167341",
      "0.428428724"
    ],
    [
      "0.689291066",
      "0.428585658"
    ],
    [
      "0.195005874",
      "0.428622896"
    ],
    [
      "0.071377104",
      "0.5"
    ],
    [
      "0.565683802",
      "0.5"
    ],
    [
      "0.812898329",
      "0.5"
    ],
    [
      "0.318634677",
      "0.5"
    ],
    [
      "0.195005874",
      "0.571377104"
    ],
    [
      "0.689291066",
      "0.571414342"
    ],
    [
      "0.442167341",
      "0.571571276"
    ],
    [
      "0.928622896",
      "0.583585815"
    ],
    [
      "0.071377104",
      "0.642754208"
    ],
    [
      "0.318530468",
      "0.642934344"
    ],
    [
      "0.566634421",
      "0.644449343"
    ],
    [
      "0.805015633",
      "0.655000157"
    ],
    [
      "0.194901697",
      "0.714311448"
    ],
    [
      "0.442151134",
      "0.714325483"
    ],
    [
      "0.928622896",
      "0.726414498"
    ],
    [
      "0.704235154",
      "0.756104356"
    ],
    [
      "0.318522363",
      "0.785702587"
    ],
    [
      "0.071377104",
      "0.785868688"
    ],
    [
      "0.56490729",
      "0.787193102"
    ],
    [
      "0.827842418",
      "0.827518697"
    ],
    [
      "0.195660402",
      "0.858391667"
    ],
    [
      "0.452125717",
      "0.874709274"
    ],
    [
      "0.071377104",
      "0.928622896"
    ],
    [
      "0.319943701",
      "0.928622896"
    ],
    [
      "0.584307732",
      "0.928622896"
    ],
    [
      "0.72706194",
      "0.928622896"
    ],
    [
      "0.928622896",
      "0.928622896"
    ]
  ],
  "radius": "0.071377103"
}
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": 50
}

起步布局答案

{
  "centers": [
    [
      "0.071377104",
      "0.071377104"
    ],
    [
      "0.319943701",
      "0.071377104"
    ],
    [
      "0.584307732",
      "0.071377104"
    ],
    [
      "0.72706194",
      "0.071377104"
    ],
    [
      "0.928622896",
      "0.071377104"
    ],
    [
      "0.452125717",
      "0.125290726"
    ],
    [
      "0.195660402",
      "0.141608333"
    ],
    [
      "0.827842418",
      "0.172481303"
    ],
    [
      "0.56490729",
      "0.212806898"
    ],
    [
      "0.071377104",
      "0.214131312"
    ],
    [
      "0.318522363",
      "0.214297413"
    ],
    [
      "0.704235154",
      "0.243895644"
    ],
    [
      "0.928622896",
      "0.273585502"
    ],
    [
      "0.442151134",
      "0.285674517"
    ],
    [
      "0.194901697",
      "0.285688552"
    ],
    [
      "0.805015633",
      "0.344999843"
    ],
    [
      "0.566634421",
      "0.355550657"
    ],
    [
      "0.318530468",
      "0.357065656"
    ],
    [
      "0.071377104",
      "0.357245792"
    ],
    [
      "0.928622896",
      "0.416414185"
    ],
    [
      "0.442167341",
      "0.428428724"
    ],
    [
      "0.689291066",
      "0.428585658"
    ],
    [
      "0.195005874",
      "0.428622896"
    ],
    [
      "0.071377104",
      "0.5"
    ],
    [
      "0.565683802",
      "0.5"
    ],
    [
      "0.812898329",
      "0.5"
    ],
    [
      "0.318634677",
      "0.5"
    ],
    [
      "0.195005874",
      "0.571377104"
    ],
    [
      "0.689291066",
      "0.571414342"
    ],
    [
      "0.442167341",
      "0.571571276"
    ],
    [
      "0.928622896",
      "0.583585815"
    ],
    [
      "0.071377104",
      "0.642754208"
    ],
    [
      "0.318530468",
      "0.642934344"
    ],
    [
      "0.566634421",
      "0.644449343"
    ],
    [
      "0.805015633",
      "0.655000157"
    ],
    [
      "0.194901697",
      "0.714311448"
    ],
    [
      "0.442151134",
      "0.714325483"
    ],
    [
      "0.928622896",
      "0.726414498"
    ],
    [
      "0.704235154",
      "0.756104356"
    ],
    [
      "0.318522363",
      "0.785702587"
    ],
    [
      "0.071377104",
      "0.785868688"
    ],
    [
      "0.56490729",
      "0.787193102"
    ],
    [
      "0.827842418",
      "0.827518697"
    ],
    [
      "0.195660402",
      "0.858391667"
    ],
    [
      "0.452125717",
      "0.874709274"
    ],
    [
      "0.071377104",
      "0.928622896"
    ],
    [
      "0.319943701",
      "0.928622896"
    ],
    [
      "0.584307732",
      "0.928622896"
    ],
    [
      "0.72706194",
      "0.928622896"
    ],
    [
      "0.928622896",
      "0.928622896"
    ]
  ],
  "radius": "0.071377103"
}

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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