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 = 165

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 = 165
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[0.0706996614922320881047366044635, 0.0706996614922320881047366044645]见来源署名packomania.com ↗
本站纪录0.070699661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 = 0.070699661

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 = 165 的当前纪录构型,0.070699661
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
    0.070699660→0.070699661
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": [
    [
      "1.599021359",
      "1.710473455"
    ],
    [
      "1.484297151",
      "1.793130121"
    ],
    [
      "1.724000041",
      "1.582600258"
    ],
    [
      "1.358360650",
      "1.857424494"
    ],
    [
      "1.819586933",
      "1.438036961"
    ],
    [
      "1.224127500",
      "1.901868052"
    ],
    [
      "1.583166307",
      "1.569965854"
    ],
    [
      "1.678589566",
      "1.448691119"
    ],
    [
      "1.366341237",
      "1.715154670"
    ],
    [
      "1.084705422",
      "1.925431851"
    ],
    [
      "1.807510630",
      "1.297154274"
    ],
    [
      "1.908026530",
      "1.197704174"
    ],
    [
      "1.444428819",
      "1.597272958"
    ],
    [
      "1.160526291",
      "1.775580068"
    ],
    [
      "1.537755833",
      "1.436056716"
    ],
    [
      "0.936770478",
      "1.926561010"
    ],
    [
      "1.237475936",
      "1.656952413"
    ],
    [
      "1.927510204",
      "1.057653629"
    ],
    [
      "1.619114414",
      "1.320408327"
    ],
    [
      "1.021104213",
      "1.799143867"
    ],
    [
      "1.315563519",
      "1.539070701"
    ],
    [
      "1.700472995",
      "1.204759938"
    ],
    [
      "1.791589590",
      "1.096632538"
    ],
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      "1.396922100",
      "1.423422312"
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      "1.096566544",
      "1.645191814"
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      "0.792309549",
      "1.905794566"
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      "1.478280681",
      "1.307773923"
    ],
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      "1.925520460",
      "0.916268307"
    ],
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      "0.879774960",
      "1.794692943"
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      "1.559639262",
      "1.192125535"
    ],
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      "1.173909130",
      "1.526819972"
    ],
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      "0.957863795",
      "1.672674827"
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      "1.640997843",
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      "1.255267711",
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      "0.590886657"
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      "0.153678717"
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      "0.492194919",
      "0.221711417"
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    [
      "0.351782794",
      "0.334264402"
    ]
  ],
  "radius": "0.070699661"
}
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": 165
}

当前第一名的答案

{
  "centers": [
    [
      "1.599021359",
      "1.710473455"
    ],
    [
      "1.484297151",
      "1.793130121"
    ],
    [
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      "1.582600258"
    ],
    [
      "1.358360650",
      "1.857424494"
    ],
    [
      "1.819586933",
      "1.438036961"
    ],
    [
      "1.224127500",
      "1.901868052"
    ],
    [
      "1.583166307",
      "1.569965854"
    ],
    [
      "1.678589566",
      "1.448691119"
    ],
    [
      "1.366341237",
      "1.715154670"
    ],
    [
      "1.084705422",
      "1.925431851"
    ],
    [
      "1.807510630",
      "1.297154274"
    ],
    [
      "1.908026530",
      "1.197704174"
    ],
    [
      "1.444428819",
      "1.597272958"
    ],
    [
      "1.160526291",
      "1.775580068"
    ],
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      "1.436056716"
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      "1.320408327"
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      "0.589498281"
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  "radius": "0.070699661"
}

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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