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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 = 53

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 = 53
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[0.0699472525615, 0.0699472525625]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 = 0.069947252

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 = 53 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": [
    [
      "0.069947253",
      "0.069947253"
    ],
    [
      "0.305549461",
      "0.069947253"
    ],
    [
      "0.547853852",
      "0.069947253"
    ],
    [
      "0.790158242",
      "0.069947253"
    ],
    [
      "0.930052747",
      "0.069947253"
    ],
    [
      "0.426701656",
      "0.139894505"
    ],
    [
      "0.669006047",
      "0.139894505"
    ],
    [
      "0.187748357",
      "0.145401692"
    ],
    [
      "0.547853852",
      "0.209841758"
    ],
    [
      "0.790158242",
      "0.209841758"
    ],
    [
      "0.930052747",
      "0.209841758"
    ],
    [
      "0.308900552",
      "0.215348944"
    ],
    [
      "0.069947253",
      "0.220856131"
    ],
    [
      "0.669006047",
      "0.27978901"
    ],
    [
      "0.430052747",
      "0.285296197"
    ],
    [
      "0.191099448",
      "0.290803383"
    ],
    [
      "0.790158242",
      "0.349736263"
    ],
    [
      "0.930052747",
      "0.349736263"
    ],
    [
      "0.551204943",
      "0.355243449"
    ],
    [
      "0.069947253",
      "0.360750636"
    ],
    [
      "0.312251643",
      "0.360750636"
    ],
    [
      "0.672357138",
      "0.425190702"
    ],
    [
      "0.191099448",
      "0.430697889"
    ],
    [
      "0.433403838",
      "0.430697889"
    ],
    [
      "0.930052747",
      "0.489630768"
    ],
    [
      "0.790545793",
      "0.50003664"
    ],
    [
      "0.069947253",
      "0.500645141"
    ],
    [
      "0.312251643",
      "0.500645141"
    ],
    [
      "0.554922959",
      "0.501215139"
    ],
    [
      "0.191099448",
      "0.570592394"
    ],
    [
      "0.433403838",
      "0.570592394"
    ],
    [
      "0.673111724",
      "0.576061057"
    ],
    [
      "0.930052747",
      "0.629525273"
    ],
    [
      "0.069947253",
      "0.640539646"
    ],
    [
      "0.312251643",
      "0.640539646"
    ],
    [
      "0.55161339",
      "0.645405324"
    ],
    [
      "0.791704213",
      "0.650265465"
    ],
    [
      "0.191190279",
      "0.710644161"
    ],
    [
      "0.430461194",
      "0.715352577"
    ],
    [
      "0.670205879",
      "0.719609732"
    ],
    [
      "0.930052747",
      "0.769419778"
    ],
    [
      "0.069947253",
      "0.780434151"
    ],
    [
      "0.309675034",
      "0.785930023"
    ],
    [
      "0.791704213",
      "0.79015997"
    ],
    [
      "0.548674325",
      "0.790160122"
    ],
    [
      "0.188159103",
      "0.855243449"
    ],
    [
      "0.670205879",
      "0.859504237"
    ],
    [
      "0.427886884",
      "0.860739321"
    ],
    [
      "0.929357691",
      "0.909312623"
    ],
    [
      "0.069947253",
      "0.930052747"
    ],
    [
      "0.306370953",
      "0.930052747"
    ],
    [
      "0.549402815",
      "0.930052747"
    ],
    [
      "0.791008942",
      "0.930052747"
    ]
  ],
  "radius": "0.069947252"
}
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": 53
}

起步布局答案

{
  "centers": [
    [
      "0.069947253",
      "0.069947253"
    ],
    [
      "0.305549461",
      "0.069947253"
    ],
    [
      "0.547853852",
      "0.069947253"
    ],
    [
      "0.790158242",
      "0.069947253"
    ],
    [
      "0.930052747",
      "0.069947253"
    ],
    [
      "0.426701656",
      "0.139894505"
    ],
    [
      "0.669006047",
      "0.139894505"
    ],
    [
      "0.187748357",
      "0.145401692"
    ],
    [
      "0.547853852",
      "0.209841758"
    ],
    [
      "0.790158242",
      "0.209841758"
    ],
    [
      "0.930052747",
      "0.209841758"
    ],
    [
      "0.308900552",
      "0.215348944"
    ],
    [
      "0.069947253",
      "0.220856131"
    ],
    [
      "0.669006047",
      "0.27978901"
    ],
    [
      "0.430052747",
      "0.285296197"
    ],
    [
      "0.191099448",
      "0.290803383"
    ],
    [
      "0.790158242",
      "0.349736263"
    ],
    [
      "0.930052747",
      "0.349736263"
    ],
    [
      "0.551204943",
      "0.355243449"
    ],
    [
      "0.069947253",
      "0.360750636"
    ],
    [
      "0.312251643",
      "0.360750636"
    ],
    [
      "0.672357138",
      "0.425190702"
    ],
    [
      "0.191099448",
      "0.430697889"
    ],
    [
      "0.433403838",
      "0.430697889"
    ],
    [
      "0.930052747",
      "0.489630768"
    ],
    [
      "0.790545793",
      "0.50003664"
    ],
    [
      "0.069947253",
      "0.500645141"
    ],
    [
      "0.312251643",
      "0.500645141"
    ],
    [
      "0.554922959",
      "0.501215139"
    ],
    [
      "0.191099448",
      "0.570592394"
    ],
    [
      "0.433403838",
      "0.570592394"
    ],
    [
      "0.673111724",
      "0.576061057"
    ],
    [
      "0.930052747",
      "0.629525273"
    ],
    [
      "0.069947253",
      "0.640539646"
    ],
    [
      "0.312251643",
      "0.640539646"
    ],
    [
      "0.55161339",
      "0.645405324"
    ],
    [
      "0.791704213",
      "0.650265465"
    ],
    [
      "0.191190279",
      "0.710644161"
    ],
    [
      "0.430461194",
      "0.715352577"
    ],
    [
      "0.670205879",
      "0.719609732"
    ],
    [
      "0.930052747",
      "0.769419778"
    ],
    [
      "0.069947253",
      "0.780434151"
    ],
    [
      "0.309675034",
      "0.785930023"
    ],
    [
      "0.791704213",
      "0.79015997"
    ],
    [
      "0.548674325",
      "0.790160122"
    ],
    [
      "0.188159103",
      "0.855243449"
    ],
    [
      "0.670205879",
      "0.859504237"
    ],
    [
      "0.427886884",
      "0.860739321"
    ],
    [
      "0.929357691",
      "0.909312623"
    ],
    [
      "0.069947253",
      "0.930052747"
    ],
    [
      "0.306370953",
      "0.930052747"
    ],
    [
      "0.549402815",
      "0.930052747"
    ],
    [
      "0.791008942",
      "0.930052747"
    ]
  ],
  "radius": "0.069947252"
}

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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