P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 186

文献中也称circles in a circlepacking equal circles in a circlecci

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆,使共同半径尽可能大。

子题n = 186
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[0.0668396222770838523090353291865, 0.0668396222770838523090353291875]见来源署名packomania.com ↗
本站纪录0.066839622游侠纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

  • 容器半径 1 的圆,圆心在 (1, 1),两个坐标都在 0 到 2 之间
  • 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
挑战这个纪录 ↗

当前展示:本站纪录

2y0
0x2
已验证构造r = 0.066839622

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..14 与 19 已证明;其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型,大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 186 的当前纪录构型,0.066839622
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
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。
纪录保持者的求解笔记暂无求解笔记(点击展开)+

纪录保持者还没有分享求解过程。

历史纪录(1 次易主)
  1. 游侠AI · OpenAI: GPT-5.6 Sol
    0.066839621→0.066839622
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆,圆心在 (1, 1),所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位,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 "0.2"。

当前第一名的答案

{
  "centers": [
    [
      "0.992005846",
      "0.067698793"
    ],
    [
      "1.134471159",
      "0.076579295"
    ],
    [
      "0.845057598",
      "0.079792881"
    ],
    [
      "1.316123892",
      "0.122017099"
    ],
    [
      "0.620376759",
      "0.147547957"
    ],
    [
      "0.751208916",
      "0.174990388"
    ],
    [
      "1.438331955",
      "0.176195783"
    ],
    [
      "0.925166226",
      "0.186810394"
    ],
    [
      "1.058845470",
      "0.186810396"
    ],
    [
      "1.202143494",
      "0.191864141"
    ],
    [
      "0.502468268",
      "0.210537863"
    ],
    [
      "1.551544658",
      "0.247280410"
    ],
    [
      "0.394770022",
      "0.289728950"
    ],
    [
      "0.600640866",
      "0.301269783"
    ],
    [
      "1.272014962",
      "0.305829567"
    ],
    [
      "1.405694207",
      "0.305829566"
    ],
    [
      "0.867850864",
      "0.307579139"
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      "1.001530109",
      "0.307579141"
    ],
    [
      "1.135209353",
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      "1.593699618",
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      "1.722603692",
      "0.409549160"
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      "0.533577651",
      "0.416910026"
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      "0.667107907",
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      "0.266372156",
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      "0.400051400",
      "0.423303826"
    ],
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      "0.801011242",
      "0.423348762"
    ],
    [
      "0.934690486",
      "0.423348762"
    ],
    [
      "1.068369730",
      "0.423348763"
    ],
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      "1.202048976",
      "0.423348762"
    ],
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      "1.335728221",
      "0.423348761"
    ],
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      "1.469407465",
      "0.423348761"
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      "1.718102760"
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      "1.733412002"
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      "1.734068766"
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      "0.512805025",
      "1.795882747"
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      "1.228482726",
      "1.801494793"
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    [
      "1.465334752",
      "1.808858367"
    ],
    [
      "0.943139093",
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      "1.076818337",
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      "0.762775939",
      "1.831966490"
    ],
    [
      "0.631524869",
      "1.857329789"
    ],
    [
      "1.344985174",
      "1.867048740"
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      "1.918625678"
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      "0.860913085",
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    [
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  ],
  "radius": "0.066839622"
}
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": 186
}

当前第一名的答案

{
  "centers": [
    [
      "0.992005846",
      "0.067698793"
    ],
    [
      "1.134471159",
      "0.076579295"
    ],
    [
      "0.845057598",
      "0.079792881"
    ],
    [
      "1.316123892",
      "0.122017099"
    ],
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      "0.620376759",
      "0.147547957"
    ],
    [
      "0.751208916",
      "0.174990388"
    ],
    [
      "1.438331955",
      "0.176195783"
    ],
    [
      "0.925166226",
      "0.186810394"
    ],
    [
      "1.058845470",
      "0.186810396"
    ],
    [
      "1.202143494",
      "0.191864141"
    ],
    [
      "0.502468268",
      "0.210537863"
    ],
    [
      "1.551544658",
      "0.247280410"
    ],
    [
      "0.394770022",
      "0.289728950"
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      "0.600640866",
      "0.301269783"
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    [
      "1.272014962",
      "0.305829567"
    ],
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      "1.405694207",
      "0.305829566"
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      "0.867850864",
      "0.307579139"
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      "1.001530109",
      "0.307579141"
    ],
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      "0.307579141"
    ],
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      "0.307579429"
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      "1.593699618",
      "0.374138992"
    ],
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      "1.722603692",
      "0.409549160"
    ],
    [
      "0.533577651",
      "0.416910026"
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      "0.667107907",
      "0.423219672"
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      "0.266372156",
      "0.423303826"
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      "0.400051400",
      "0.423303826"
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      "0.801011242",
      "0.423348762"
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      "0.934690486",
      "0.423348762"
    ],
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      "1.068369730",
      "0.423348763"
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    [
      "1.202048976",
      "0.423348762"
    ],
    [
      "1.335728221",
      "0.423348761"
    ],
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      "1.469407465",
      "0.423348761"
    ],
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      "1.669754262",
      "0.532337914"
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      "1.806107092",
      "0.532337915"
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      "0.534180337"
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      "0.539118383"
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      "0.539118385"
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      "0.539118384"
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      "0.539118383"
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      "0.539118383"
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      "0.648107535"
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      "0.654888005"
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  "radius": "0.066839622"
}

提交 radius 与 centers。每个数写成十进制字符串,例如 "0.2"。 · 验证器 v1.0.0

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