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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 = 49

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 = 49
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[0.1267929962622069300306416667465, 0.1267929962622069300306416667475]见来源署名packomania.com ↗
本站纪录0.126792996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 = 0.126792996

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 = 49 的当前纪录构型，0.126792996
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
    0.126792995→0.126792996
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": [
    [
      "0.938780329",
      "0.128941665"
    ],
    [
      "1.197848701",
      "0.155233596"
    ],
    [
      "1.442200725",
      "0.247039848"
    ],
    [
      "0.691080807",
      "0.183262953"
    ],
    [
      "1.639901791",
      "0.405848362"
    ],
    [
      "0.469434480",
      "0.306465069"
    ],
    [
      "1.039649263",
      "0.361603036"
    ],
    [
      "1.280128440",
      "0.442074281"
    ],
    [
      "0.792587709",
      "0.441584535"
    ],
    [
      "1.783635714",
      "0.614765606"
    ],
    [
      "1.477829506",
      "0.600882795"
    ],
    [
      "0.290733833",
      "0.499110579"
    ],
    [
      "0.543297991",
      "0.549055336"
    ],
    [
      "0.991520729",
      "0.610579942"
    ],
    [
      "1.231999905",
      "0.691051187"
    ],
    [
      "0.751824816",
      "0.693355090"
    ],
    [
      "1.610764565",
      "0.816832154"
    ],
    [
      "1.861280464",
      "0.856172210"
    ],
    [
      "0.376533677",
      "0.740093340"
    ],
    [
      "0.943358153",
      "0.859550265"
    ],
    [
      "1.372067200",
      "0.902444206"
    ],
    [
      "0.142158018",
      "0.836912276"
    ],
    [
      "0.600223104",
      "0.896635134"
    ],
    [
      "1.144393640",
      "1.014116071"
    ],
    [
      "1.594143502",
      "1.070115408"
    ],
    [
      "1.866287746",
      "1.109708761"
    ],
    [
      "0.365847446",
      "0.993454071"
    ],
    [
      "0.791756441",
      "1.062830310"
    ],
    [
      "1.354941093",
      "1.155451226"
    ],
    [
      "0.131471788",
      "1.090273007"
    ],
    [
      "0.557380783",
      "1.159649247"
    ],
    [
      "0.992791929",
      "1.217396116"
    ],
    [
      "1.546763935",
      "1.321312170"
    ],
    [
      "1.798235263",
      "1.353992848"
    ],
    [
      "0.323005125",
      "1.256468183"
    ],
    [
      "0.758416270",
      "1.314215053"
    ],
    [
      "1.202900556",
      "1.359622847"
    ],
    [
      "0.524040612",
      "1.411033989"
    ],
    [
      "0.959451758",
      "1.468780859"
    ],
    [
      "1.411391002",
      "1.535741703"
    ],
    [
      "1.662862330",
      "1.568422381"
    ],
    [
      "0.289664954",
      "1.507852925"
    ],
    [
      "0.725076099",
      "1.565599795"
    ],
    [
      "1.169284325",
      "1.611175203"
    ],
    [
      "0.926111587",
      "1.720165601"
    ],
    [
      "0.468591422",
      "1.692681214"
    ],
    [
      "1.378050831",
      "1.787126445"
    ],
    [
      "0.691735929",
      "1.816984537"
    ],
    [
      "1.135944154",
      "1.862559945"
    ]
  ],
  "radius": "0.126792996"
}
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": 49
}

当前第一名的答案

{
  "centers": [
    [
      "0.938780329",
      "0.128941665"
    ],
    [
      "1.197848701",
      "0.155233596"
    ],
    [
      "1.442200725",
      "0.247039848"
    ],
    [
      "0.691080807",
      "0.183262953"
    ],
    [
      "1.639901791",
      "0.405848362"
    ],
    [
      "0.469434480",
      "0.306465069"
    ],
    [
      "1.039649263",
      "0.361603036"
    ],
    [
      "1.280128440",
      "0.442074281"
    ],
    [
      "0.792587709",
      "0.441584535"
    ],
    [
      "1.783635714",
      "0.614765606"
    ],
    [
      "1.477829506",
      "0.600882795"
    ],
    [
      "0.290733833",
      "0.499110579"
    ],
    [
      "0.543297991",
      "0.549055336"
    ],
    [
      "0.991520729",
      "0.610579942"
    ],
    [
      "1.231999905",
      "0.691051187"
    ],
    [
      "0.751824816",
      "0.693355090"
    ],
    [
      "1.610764565",
      "0.816832154"
    ],
    [
      "1.861280464",
      "0.856172210"
    ],
    [
      "0.376533677",
      "0.740093340"
    ],
    [
      "0.943358153",
      "0.859550265"
    ],
    [
      "1.372067200",
      "0.902444206"
    ],
    [
      "0.142158018",
      "0.836912276"
    ],
    [
      "0.600223104",
      "0.896635134"
    ],
    [
      "1.144393640",
      "1.014116071"
    ],
    [
      "1.594143502",
      "1.070115408"
    ],
    [
      "1.866287746",
      "1.109708761"
    ],
    [
      "0.365847446",
      "0.993454071"
    ],
    [
      "0.791756441",
      "1.062830310"
    ],
    [
      "1.354941093",
      "1.155451226"
    ],
    [
      "0.131471788",
      "1.090273007"
    ],
    [
      "0.557380783",
      "1.159649247"
    ],
    [
      "0.992791929",
      "1.217396116"
    ],
    [
      "1.546763935",
      "1.321312170"
    ],
    [
      "1.798235263",
      "1.353992848"
    ],
    [
      "0.323005125",
      "1.256468183"
    ],
    [
      "0.758416270",
      "1.314215053"
    ],
    [
      "1.202900556",
      "1.359622847"
    ],
    [
      "0.524040612",
      "1.411033989"
    ],
    [
      "0.959451758",
      "1.468780859"
    ],
    [
      "1.411391002",
      "1.535741703"
    ],
    [
      "1.662862330",
      "1.568422381"
    ],
    [
      "0.289664954",
      "1.507852925"
    ],
    [
      "0.725076099",
      "1.565599795"
    ],
    [
      "1.169284325",
      "1.611175203"
    ],
    [
      "0.926111587",
      "1.720165601"
    ],
    [
      "0.468591422",
      "1.692681214"
    ],
    [
      "1.378050831",
      "1.787126445"
    ],
    [
      "0.691735929",
      "1.816984537"
    ],
    [
      "1.135944154",
      "1.862559945"
    ]
  ],
  "radius": "0.126792996"
}

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

DISCUSSION

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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。

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