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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 = 104

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 = 104
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[0.0503487928115, 0.0503487928125]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 = 0.050348792

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 = 104 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": [
    [
      "0.050348793",
      "0.050348793"
    ],
    [
      "0.223889082",
      "0.050348793"
    ],
    [
      "0.398675817",
      "0.050348793"
    ],
    [
      "0.580592508",
      "0.050348793"
    ],
    [
      "0.775237869",
      "0.050348793"
    ],
    [
      "0.949651207",
      "0.050348793"
    ],
    [
      "0.677915189",
      "0.076200285"
    ],
    [
      "0.489634163",
      "0.093552764"
    ],
    [
      "0.31128245",
      "0.100372818"
    ],
    [
      "0.862444538",
      "0.100697582"
    ],
    [
      "0.137118938",
      "0.101446207"
    ],
    [
      "0.748964577",
      "0.147558452"
    ],
    [
      "0.572529072",
      "0.150723016"
    ],
    [
      "0.39830121",
      "0.151045682"
    ],
    [
      "0.949651207",
      "0.151046378"
    ],
    [
      "0.224512305",
      "0.151470232"
    ],
    [
      "0.050348793",
      "0.152543621"
    ],
    [
      "0.836171247",
      "0.197907242"
    ],
    [
      "0.661500688",
      "0.197991255"
    ],
    [
      "0.485508682",
      "0.201393081"
    ],
    [
      "0.311531065",
      "0.202143096"
    ],
    [
      "0.13774216",
      "0.202567646"
    ],
    [
      "0.923377912",
      "0.248256037"
    ],
    [
      "0.748964581",
      "0.248256038"
    ],
    [
      "0.57290044",
      "0.251419917"
    ],
    [
      "0.398738537",
      "0.252490495"
    ],
    [
      "0.22476092",
      "0.25324051"
    ],
    [
      "0.050348793",
      "0.253241207"
    ],
    [
      "0.836171247",
      "0.298604833"
    ],
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      "0.661810487",
      "0.298695781"
    ],
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      "0.486130295",
      "0.302517331"
    ],
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      "0.311968392",
      "0.303587909"
    ],
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      "0.137555058",
      "0.303590696"
    ],
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    ],
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      "0.573767026",
      "0.352113774"
    ],
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      "0.39936015",
      "0.353614745"
    ],
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      "0.22476253",
      "0.353938096"
    ],
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      "0.050348793",
      "0.353940186"
    ],
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      "0.862444537",
      "0.395814493"
    ],
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      "0.662260496",
      "0.400164887"
    ],
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      "0.486996882",
      "0.403211188"
    ],
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      "0.404287283"
    ],
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      "0.13755546",
      "0.404288979"
    ],
    [
      "0.949651207",
      "0.446163281"
    ],
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      "0.775166129",
      "0.446287749"
    ],
    [
      "0.574633613",
      "0.452807631"
    ],
    [
      "0.400226737",
      "0.454308602"
    ],
    [
      "0.224763334",
      "0.454635681"
    ],
    [
      "0.050348793",
      "0.454637772"
    ],
    [
      "0.862444537",
      "0.496512079"
    ],
    [
      "0.687363417",
      "0.497683333"
    ],
    [
      "0.487863468",
      "0.503905045"
    ],
    [
      "0.311970001",
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    ],
    [
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    ],
    [
      "0.775166129",
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    ],
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    ],
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      "0.604598958"
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      "0.60568206"
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      "0.13755546",
      "0.60568415"
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      "0.487003524",
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    ],
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    [
      "0.050348793",
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      "0.687431841",
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      "0.510545271",
      "0.803203591"
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    [
      "0.139008905",
      "0.809596761"
    ],
    [
      "0.327115706",
      "0.825908729"
    ],
    [
      "0.949651207",
      "0.848953624"
    ],
    [
      "0.775248527",
      "0.848972074"
    ],
    [
      "0.59994268",
      "0.849551266"
    ],
    [
      "0.423743503",
      "0.854247266"
    ],
    [
      "0.231405521",
      "0.85720672"
    ],
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      "0.050348793",
      "0.857428114"
    ],
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      "0.861475383",
      "0.900981046"
    ],
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      "0.68115442",
      "0.909088289"
    ],
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      "0.490256023",
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    ],
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      "0.949651207"
    ],
    [
      "0.191480335",
      "0.949651207"
    ],
    [
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      "0.949651207"
    ],
    [
      "0.588987975",
      "0.949651207"
    ],
    [
      "0.773320864",
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    ],
    [
      "0.949629901",
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  ],
  "radius": "0.050348792"
}
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": 104
}

起步布局答案

{
  "centers": [
    [
      "0.050348793",
      "0.050348793"
    ],
    [
      "0.223889082",
      "0.050348793"
    ],
    [
      "0.398675817",
      "0.050348793"
    ],
    [
      "0.580592508",
      "0.050348793"
    ],
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      "0.050348793"
    ],
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      "0.949651207",
      "0.050348793"
    ],
    [
      "0.677915189",
      "0.076200285"
    ],
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      "0.489634163",
      "0.093552764"
    ],
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      "0.31128245",
      "0.100372818"
    ],
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      "0.862444538",
      "0.100697582"
    ],
    [
      "0.137118938",
      "0.101446207"
    ],
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      "0.748964577",
      "0.147558452"
    ],
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      "0.572529072",
      "0.150723016"
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      "0.39830121",
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    ],
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      "0.938005189"
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      "0.949651207"
    ],
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      "0.191480335",
      "0.949651207"
    ],
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      "0.39152407",
      "0.949651207"
    ],
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      "0.588987975",
      "0.949651207"
    ],
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      "0.773320864",
      "0.949651207"
    ],
    [
      "0.949629901",
      "0.949651207"
    ]
  ],
  "radius": "0.050348792"
}

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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