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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 = 125

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 = 125
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[0.0459783365425, 0.0459783365435]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 = 0.045978336

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 = 125 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": [
    [
      "0.045978337",
      "0.045978337"
    ],
    [
      "0.205251966",
      "0.045978337"
    ],
    [
      "0.364525596",
      "0.045978337"
    ],
    [
      "0.523799226",
      "0.045978337"
    ],
    [
      "0.683072856",
      "0.045978337"
    ],
    [
      "0.842346486",
      "0.045978337"
    ],
    [
      "0.944826153",
      "0.058607969"
    ],
    [
      "0.125615151",
      "0.091956673"
    ],
    [
      "0.284888781",
      "0.091956673"
    ],
    [
      "0.444162411",
      "0.091956673"
    ],
    [
      "0.603436041",
      "0.091956673"
    ],
    [
      "0.762709671",
      "0.091956673"
    ],
    [
      "0.874384849",
      "0.132173308"
    ],
    [
      "0.045978337",
      "0.13793501"
    ],
    [
      "0.205251966",
      "0.13793501"
    ],
    [
      "0.364525596",
      "0.13793501"
    ],
    [
      "0.523799226",
      "0.13793501"
    ],
    [
      "0.683072856",
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    ],
    [
      "0.794748034",
      "0.178151644"
    ],
    [
      "0.954021663",
      "0.178151644"
    ],
    [
      "0.125615151",
      "0.183913346"
    ],
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      "0.284888781",
      "0.183913346"
    ],
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      "0.444162411",
      "0.183913346"
    ],
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      "0.603436044",
      "0.183913351"
    ],
    [
      "0.715111219",
      "0.224129981"
    ],
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      "0.874384849",
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    ],
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      "0.045978337",
      "0.229891683"
    ],
    [
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      "0.229891683"
    ],
    [
      "0.364525596",
      "0.229891683"
    ],
    [
      "0.523799226",
      "0.229891683"
    ],
    [
      "0.794748034",
      "0.270108317"
    ],
    [
      "0.954021663",
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    ],
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      "0.635475641",
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      "0.321848356"
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      "0.635475641",
      "0.362067133"
    ],
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      "0.362067133"
    ],
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    ],
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    ],
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    ],
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    ],
    [
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    ],
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    ],
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      "0.413805029"
    ],
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      "0.205251966",
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    ],
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      "0.454021663"
    ],
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      "0.476200774",
      "0.454021663"
    ],
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      "0.635475641",
      "0.454023806"
    ],
    [
      "0.794749271",
      "0.454023806"
    ],
    [
      "0.954021663",
      "0.454025949"
    ],
    [
      "0.125615151",
      "0.459783365"
    ],
    [
      "0.237290329",
      "0.5"
    ],
    [
      "0.396563959",
      "0.5"
    ],
    [
      "0.555837589",
      "0.5"
    ],
    [
      "0.715112456",
      "0.500002143"
    ],
    [
      "0.874384849",
      "0.500004285"
    ],
    [
      "0.045978337",
      "0.505761702"
    ],
    [
      "0.157653514",
      "0.545978337"
    ],
    [
      "0.316927144",
      "0.545978337"
    ],
    [
      "0.476200774",
      "0.545978337"
    ],
    [
      "0.635475641",
      "0.545980479"
    ],
    [
      "0.794748034",
      "0.545982622"
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      "0.730278235"
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    ],
    [
      "0.045978337",
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    [
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    [
      "0.715333147",
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    ],
    [
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      "0.816086654"
    ],
    [
      "0.284888781",
      "0.816086654"
    ],
    [
      "0.476200774",
      "0.821848356"
    ],
    [
      "0.635696332",
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    ],
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      "0.954021663"
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      "0.954021663"
    ],
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      "0.954021663"
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  ],
  "radius": "0.045978336"
}
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": 125
}

起步布局答案

{
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    [
      "0.045978337",
      "0.045978337"
    ],
    [
      "0.205251966",
      "0.045978337"
    ],
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      "0.045978337"
    ],
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      "0.045978337"
    ],
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      "0.045978337"
    ],
    [
      "0.842346486",
      "0.045978337"
    ],
    [
      "0.944826153",
      "0.058607969"
    ],
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    ],
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    ],
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    ],
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      "0.091956673"
    ],
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    ],
    [
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    ],
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    ],
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      "0.794748034",
      "0.178151644"
    ],
    [
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    ],
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      "0.444162411",
      "0.908043327"
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    [
      "0.635696332",
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      "0.944826153",
      "0.944826153"
    ],
    [
      "0.045978337",
      "0.954021663"
    ],
    [
      "0.205251966",
      "0.954021663"
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    [
      "0.364525596",
      "0.954021663"
    ],
    [
      "0.523799226",
      "0.954021663"
    ],
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      "0.748392866",
      "0.954021663"
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    [
      "0.840349539",
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  ],
  "radius": "0.045978336"
}

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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