极值竞技场
中/EN
P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 145

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 = 145
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[0.0428713567025, 0.0428713567035]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 = 0.042871356

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 = 145 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": [
    [
      "0.042871357",
      "0.042871357"
    ],
    [
      "0.191382093",
      "0.042871357"
    ],
    [
      "0.339892829",
      "0.042871357"
    ],
    [
      "0.488404204",
      "0.042871357"
    ],
    [
      "0.637132161",
      "0.042871357"
    ],
    [
      "0.722874875",
      "0.042871357"
    ],
    [
      "0.808617588",
      "0.042871357"
    ],
    [
      "0.957128643",
      "0.042871357"
    ],
    [
      "0.562768183",
      "0.085554041"
    ],
    [
      "0.414148516",
      "0.08574216"
    ],
    [
      "0.882873116",
      "0.085742437"
    ],
    [
      "0.117126725",
      "0.085742713"
    ],
    [
      "0.265637461",
      "0.085742713"
    ],
    [
      "0.488621904",
      "0.128613794"
    ],
    [
      "0.63713248",
      "0.12861407"
    ],
    [
      "0.722875194",
      "0.12861407"
    ],
    [
      "0.808617907",
      "0.12861407"
    ],
    [
      "0.957128643",
      "0.12861407"
    ],
    [
      "0.042871357",
      "0.12861407"
    ],
    [
      "0.191382093",
      "0.12861407"
    ],
    [
      "0.339892829",
      "0.12861407"
    ],
    [
      "0.414148516",
      "0.171484873"
    ],
    [
      "0.562877112",
      "0.171485427"
    ],
    [
      "0.882873275",
      "0.171485427"
    ],
    [
      "0.117126725",
      "0.171485427"
    ],
    [
      "0.265637461",
      "0.171485427"
    ],
    [
      "0.488403884",
      "0.21435623"
    ],
    [
      "0.63713248",
      "0.214356784"
    ],
    [
      "0.722875194",
      "0.214356784"
    ],
    [
      "0.808617907",
      "0.214356784"
    ],
    [
      "0.957128643",
      "0.214356784"
    ],
    [
      "0.042871357",
      "0.214356784"
    ],
    [
      "0.191382093",
      "0.214356784"
    ],
    [
      "0.339892829",
      "0.214356784"
    ],
    [
      "0.414148516",
      "0.257227587"
    ],
    [
      "0.562877112",
      "0.25722814"
    ],
    [
      "0.882873275",
      "0.25722814"
    ],
    [
      "0.117126725",
      "0.25722814"
    ],
    [
      "0.265637461",
      "0.25722814"
    ],
    [
      "0.488621266",
      "0.300098668"
    ],
    [
      "0.63713248",
      "0.300099497"
    ],
    [
      "0.722875194",
      "0.300099497"
    ],
    [
      "0.808617907",
      "0.300099497"
    ],
    [
      "0.957128643",
      "0.300099497"
    ],
    [
      "0.042871357",
      "0.300099497"
    ],
    [
      "0.191382093",
      "0.300099497"
    ],
    [
      "0.339892829",
      "0.300099497"
    ],
    [
      "0.414148516",
      "0.3429703"
    ],
    [
      "0.56287707",
      "0.342970854"
    ],
    [
      "0.882873275",
      "0.342970854"
    ],
    [
      "0.117126725",
      "0.342970854"
    ],
    [
      "0.265637461",
      "0.342970854"
    ],
    [
      "0.488404202",
      "0.385841106"
    ],
    [
      "0.63713248",
      "0.38584221"
    ],
    [
      "0.722875194",
      "0.38584221"
    ],
    [
      "0.808617907",
      "0.38584221"
    ],
    [
      "0.957128643",
      "0.38584221"
    ],
    [
      "0.042871357",
      "0.38584221"
    ],
    [
      "0.191382093",
      "0.38584221"
    ],
    [
      "0.339892872",
      "0.38584221"
    ],
    [
      "0.414149152",
      "0.428713013"
    ],
    [
      "0.562877112",
      "0.428713567"
    ],
    [
      "0.882873275",
      "0.428713567"
    ],
    [
      "0.117126725",
      "0.428713567"
    ],
    [
      "0.265637461",
      "0.428713567"
    ],
    [
      "0.488620949",
      "0.471583546"
    ],
    [
      "0.63713248",
      "0.471584924"
    ],
    [
      "0.722875194",
      "0.471584924"
    ],
    [
      "0.808617907",
      "0.471584924"
    ],
    [
      "0.957128643",
      "0.471584924"
    ],
    [
      "0.042871357",
      "0.471584924"
    ],
    [
      "0.191382093",
      "0.471584924"
    ],
    [
      "0.339892829",
      "0.471584924"
    ],
    [
      "0.414148516",
      "0.514455727"
    ],
    [
      "0.562876684",
      "0.51445628"
    ],
    [
      "0.882873275",
      "0.51445628"
    ],
    [
      "0.117126725",
      "0.51445628"
    ],
    [
      "0.265637461",
      "0.51445628"
    ],
    [
      "0.488404518",
      "0.557325986"
    ],
    [
      "0.339893468",
      "0.557327637"
    ],
    [
      "0.63713248",
      "0.557327637"
    ],
    [
      "0.722875194",
      "0.557327637"
    ],
    [
      "0.808617907",
      "0.557327637"
    ],
    [
      "0.957128643",
      "0.557327637"
    ],
    [
      "0.042871357",
      "0.557327637"
    ],
    [
      "0.191382093",
      "0.557327637"
    ],
    [
      "0.414149783",
      "0.60019844"
    ],
    [
      "0.562877112",
      "0.600198994"
    ],
    [
      "0.882873275",
      "0.600198994"
    ],
    [
      "0.117126725",
      "0.600198994"
    ],
    [
      "0.265637461",
      "0.600198994"
    ],
    [
      "0.488620634",
      "0.643068427"
    ],
    [
      "0.339892829",
      "0.643070351"
    ],
    [
      "0.63713248",
      "0.643070351"
    ],
    [
      "0.722875194",
      "0.643070351"
    ],
    [
      "0.808617907",
      "0.643070351"
    ],
    [
      "0.957128643",
      "0.643070351"
    ],
    [
      "0.042871357",
      "0.643070351"
    ],
    [
      "0.191382093",
      "0.643070351"
    ],
    [
      "0.414148516",
      "0.685941154"
    ],
    [
      "0.56287707",
      "0.685941707"
    ],
    [
      "0.882873275",
      "0.685941707"
    ],
    [
      "0.117126725",
      "0.685941707"
    ],
    [
      "0.265637461",
      "0.685941707"
    ],
    [
      "0.488404832",
      "0.728810869"
    ],
    [
      "0.339893468",
      "0.728813064"
    ],
    [
      "0.63713248",
      "0.728813064"
    ],
    [
      "0.722875194",
      "0.728813064"
    ],
    [
      "0.808617907",
      "0.728813064"
    ],
    [
      "0.957128643",
      "0.728813064"
    ],
    [
      "0.042871357",
      "0.728813064"
    ],
    [
      "0.191382093",
      "0.728813064"
    ],
    [
      "0.117126725",
      "0.771684421"
    ],
    [
      "0.259605262",
      "0.78074968"
    ],
    [
      "0.424426431",
      "0.794217488"
    ],
    [
      "0.866001405",
      "0.798504623"
    ],
    [
      "0.604933053",
      "0.8082801"
    ],
    [
      "0.690675766",
      "0.8082801"
    ],
    [
      "0.77641848",
      "0.8082801"
    ],
    [
      "0.519192139",
      "0.808835582"
    ],
    [
      "0.338407742",
      "0.814542904"
    ],
    [
      "0.957128643",
      "0.814555777"
    ],
    [
      "0.042871357",
      "0.814555777"
    ],
    [
      "0.185349894",
      "0.823621037"
    ],
    [
      "0.111094526",
      "0.866492393"
    ],
    [
      "0.268803407",
      "0.869242362"
    ],
    [
      "0.64780441",
      "0.882535468"
    ],
    [
      "0.819289664",
      "0.882535567"
    ],
    [
      "0.733547037",
      "0.882657048"
    ],
    [
      "0.905032292",
      "0.882657048"
    ],
    [
      "0.390203237",
      "0.882873275"
    ],
    [
      "0.475945951",
      "0.882873275"
    ],
    [
      "0.562063496",
      "0.88309095"
    ],
    [
      "0.190486474",
      "0.909209753"
    ],
    [
      "0.042871357",
      "0.918429009"
    ],
    [
      "0.954985075",
      "0.954985075"
    ],
    [
      "0.776795025",
      "0.957007163"
    ],
    [
      "0.119383782",
      "0.957128643"
    ],
    [
      "0.261589167",
      "0.957128643"
    ],
    [
      "0.34733188",
      "0.957128643"
    ],
    [
      "0.433074594",
      "0.957128643"
    ],
    [
      "0.518817307",
      "0.957128643"
    ],
    [
      "0.605309684",
      "0.957128643"
    ],
    [
      "0.691052398",
      "0.957128643"
    ],
    [
      "0.862537652",
      "0.957128643"
    ]
  ],
  "radius": "0.042871356"
}
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": 145
}

起步布局答案

{
  "centers": [
    [
      "0.042871357",
      "0.042871357"
    ],
    [
      "0.191382093",
      "0.042871357"
    ],
    [
      "0.339892829",
      "0.042871357"
    ],
    [
      "0.488404204",
      "0.042871357"
    ],
    [
      "0.637132161",
      "0.042871357"
    ],
    [
      "0.722874875",
      "0.042871357"
    ],
    [
      "0.808617588",
      "0.042871357"
    ],
    [
      "0.957128643",
      "0.042871357"
    ],
    [
      "0.562768183",
      "0.085554041"
    ],
    [
      "0.414148516",
      "0.08574216"
    ],
    [
      "0.882873116",
      "0.085742437"
    ],
    [
      "0.117126725",
      "0.085742713"
    ],
    [
      "0.265637461",
      "0.085742713"
    ],
    [
      "0.488621904",
      "0.128613794"
    ],
    [
      "0.63713248",
      "0.12861407"
    ],
    [
      "0.722875194",
      "0.12861407"
    ],
    [
      "0.808617907",
      "0.12861407"
    ],
    [
      "0.957128643",
      "0.12861407"
    ],
    [
      "0.042871357",
      "0.12861407"
    ],
    [
      "0.191382093",
      "0.12861407"
    ],
    [
      "0.339892829",
      "0.12861407"
    ],
    [
      "0.414148516",
      "0.171484873"
    ],
    [
      "0.562877112",
      "0.171485427"
    ],
    [
      "0.882873275",
      "0.171485427"
    ],
    [
      "0.117126725",
      "0.171485427"
    ],
    [
      "0.265637461",
      "0.171485427"
    ],
    [
      "0.488403884",
      "0.21435623"
    ],
    [
      "0.63713248",
      "0.214356784"
    ],
    [
      "0.722875194",
      "0.214356784"
    ],
    [
      "0.808617907",
      "0.214356784"
    ],
    [
      "0.957128643",
      "0.214356784"
    ],
    [
      "0.042871357",
      "0.214356784"
    ],
    [
      "0.191382093",
      "0.214356784"
    ],
    [
      "0.339892829",
      "0.214356784"
    ],
    [
      "0.414148516",
      "0.257227587"
    ],
    [
      "0.562877112",
      "0.25722814"
    ],
    [
      "0.882873275",
      "0.25722814"
    ],
    [
      "0.117126725",
      "0.25722814"
    ],
    [
      "0.265637461",
      "0.25722814"
    ],
    [
      "0.488621266",
      "0.300098668"
    ],
    [
      "0.63713248",
      "0.300099497"
    ],
    [
      "0.722875194",
      "0.300099497"
    ],
    [
      "0.808617907",
      "0.300099497"
    ],
    [
      "0.957128643",
      "0.300099497"
    ],
    [
      "0.042871357",
      "0.300099497"
    ],
    [
      "0.191382093",
      "0.300099497"
    ],
    [
      "0.339892829",
      "0.300099497"
    ],
    [
      "0.414148516",
      "0.3429703"
    ],
    [
      "0.56287707",
      "0.342970854"
    ],
    [
      "0.882873275",
      "0.342970854"
    ],
    [
      "0.117126725",
      "0.342970854"
    ],
    [
      "0.265637461",
      "0.342970854"
    ],
    [
      "0.488404202",
      "0.385841106"
    ],
    [
      "0.63713248",
      "0.38584221"
    ],
    [
      "0.722875194",
      "0.38584221"
    ],
    [
      "0.808617907",
      "0.38584221"
    ],
    [
      "0.957128643",
      "0.38584221"
    ],
    [
      "0.042871357",
      "0.38584221"
    ],
    [
      "0.191382093",
      "0.38584221"
    ],
    [
      "0.339892872",
      "0.38584221"
    ],
    [
      "0.414149152",
      "0.428713013"
    ],
    [
      "0.562877112",
      "0.428713567"
    ],
    [
      "0.882873275",
      "0.428713567"
    ],
    [
      "0.117126725",
      "0.428713567"
    ],
    [
      "0.265637461",
      "0.428713567"
    ],
    [
      "0.488620949",
      "0.471583546"
    ],
    [
      "0.63713248",
      "0.471584924"
    ],
    [
      "0.722875194",
      "0.471584924"
    ],
    [
      "0.808617907",
      "0.471584924"
    ],
    [
      "0.957128643",
      "0.471584924"
    ],
    [
      "0.042871357",
      "0.471584924"
    ],
    [
      "0.191382093",
      "0.471584924"
    ],
    [
      "0.339892829",
      "0.471584924"
    ],
    [
      "0.414148516",
      "0.514455727"
    ],
    [
      "0.562876684",
      "0.51445628"
    ],
    [
      "0.882873275",
      "0.51445628"
    ],
    [
      "0.117126725",
      "0.51445628"
    ],
    [
      "0.265637461",
      "0.51445628"
    ],
    [
      "0.488404518",
      "0.557325986"
    ],
    [
      "0.339893468",
      "0.557327637"
    ],
    [
      "0.63713248",
      "0.557327637"
    ],
    [
      "0.722875194",
      "0.557327637"
    ],
    [
      "0.808617907",
      "0.557327637"
    ],
    [
      "0.957128643",
      "0.557327637"
    ],
    [
      "0.042871357",
      "0.557327637"
    ],
    [
      "0.191382093",
      "0.557327637"
    ],
    [
      "0.414149783",
      "0.60019844"
    ],
    [
      "0.562877112",
      "0.600198994"
    ],
    [
      "0.882873275",
      "0.600198994"
    ],
    [
      "0.117126725",
      "0.600198994"
    ],
    [
      "0.265637461",
      "0.600198994"
    ],
    [
      "0.488620634",
      "0.643068427"
    ],
    [
      "0.339892829",
      "0.643070351"
    ],
    [
      "0.63713248",
      "0.643070351"
    ],
    [
      "0.722875194",
      "0.643070351"
    ],
    [
      "0.808617907",
      "0.643070351"
    ],
    [
      "0.957128643",
      "0.643070351"
    ],
    [
      "0.042871357",
      "0.643070351"
    ],
    [
      "0.191382093",
      "0.643070351"
    ],
    [
      "0.414148516",
      "0.685941154"
    ],
    [
      "0.56287707",
      "0.685941707"
    ],
    [
      "0.882873275",
      "0.685941707"
    ],
    [
      "0.117126725",
      "0.685941707"
    ],
    [
      "0.265637461",
      "0.685941707"
    ],
    [
      "0.488404832",
      "0.728810869"
    ],
    [
      "0.339893468",
      "0.728813064"
    ],
    [
      "0.63713248",
      "0.728813064"
    ],
    [
      "0.722875194",
      "0.728813064"
    ],
    [
      "0.808617907",
      "0.728813064"
    ],
    [
      "0.957128643",
      "0.728813064"
    ],
    [
      "0.042871357",
      "0.728813064"
    ],
    [
      "0.191382093",
      "0.728813064"
    ],
    [
      "0.117126725",
      "0.771684421"
    ],
    [
      "0.259605262",
      "0.78074968"
    ],
    [
      "0.424426431",
      "0.794217488"
    ],
    [
      "0.866001405",
      "0.798504623"
    ],
    [
      "0.604933053",
      "0.8082801"
    ],
    [
      "0.690675766",
      "0.8082801"
    ],
    [
      "0.77641848",
      "0.8082801"
    ],
    [
      "0.519192139",
      "0.808835582"
    ],
    [
      "0.338407742",
      "0.814542904"
    ],
    [
      "0.957128643",
      "0.814555777"
    ],
    [
      "0.042871357",
      "0.814555777"
    ],
    [
      "0.185349894",
      "0.823621037"
    ],
    [
      "0.111094526",
      "0.866492393"
    ],
    [
      "0.268803407",
      "0.869242362"
    ],
    [
      "0.64780441",
      "0.882535468"
    ],
    [
      "0.819289664",
      "0.882535567"
    ],
    [
      "0.733547037",
      "0.882657048"
    ],
    [
      "0.905032292",
      "0.882657048"
    ],
    [
      "0.390203237",
      "0.882873275"
    ],
    [
      "0.475945951",
      "0.882873275"
    ],
    [
      "0.562063496",
      "0.88309095"
    ],
    [
      "0.190486474",
      "0.909209753"
    ],
    [
      "0.042871357",
      "0.918429009"
    ],
    [
      "0.954985075",
      "0.954985075"
    ],
    [
      "0.776795025",
      "0.957007163"
    ],
    [
      "0.119383782",
      "0.957128643"
    ],
    [
      "0.261589167",
      "0.957128643"
    ],
    [
      "0.34733188",
      "0.957128643"
    ],
    [
      "0.433074594",
      "0.957128643"
    ],
    [
      "0.518817307",
      "0.957128643"
    ],
    [
      "0.605309684",
      "0.957128643"
    ],
    [
      "0.691052398",
      "0.957128643"
    ],
    [
      "0.862537652",
      "0.957128643"
    ]
  ],
  "radius": "0.042871356"
}

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

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

啊？？！NUE_13#310 条回复空悲切/.lird#282 条回复