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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 = 34

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 = 34
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[0.1512640282467554641005694878805, 0.1512640282467554641005694878815]见来源署名packomania.com ↗
本站纪录0.151264028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 = 0.151264028

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 = 34 的当前纪录构型，0.151264028
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
    0.151264027→0.151264028
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": [
    [
      "1.788052606",
      "0.684839977"
    ],
    [
      "1.627451399",
      "0.428460420"
    ],
    [
      "1.848529095",
      "0.981261669"
    ],
    [
      "1.800247136",
      "1.281536549"
    ],
    [
      "1.485596072",
      "0.738830747"
    ],
    [
      "1.325119586",
      "0.439355367"
    ],
    [
      "1.184412361",
      "0.171540689"
    ],
    [
      "1.564269264",
      "1.084797041"
    ],
    [
      "1.650365300",
      "1.545323506"
    ],
    [
      "1.163538884",
      "0.695118737"
    ],
    [
      "1.283158656",
      "0.972993412"
    ],
    [
      "1.022831659",
      "0.427304059"
    ],
    [
      "0.882124434",
      "0.159489381"
    ],
    [
      "1.419138420",
      "1.350240563"
    ],
    [
      "1.417783646",
      "1.738789263"
    ],
    [
      "1.138027812",
      "1.238436934"
    ],
    [
      "0.861250957",
      "0.683067429"
    ],
    [
      "0.980736867",
      "0.964975536"
    ],
    [
      "0.720543732",
      "0.415252751"
    ],
    [
      "1.183766814",
      "1.540302216"
    ],
    [
      "1.132121183",
      "1.838389374"
    ],
    [
      "0.835606023",
      "1.230419058"
    ],
    [
      "0.680749732",
      "0.925848262"
    ],
    [
      "0.420038751",
      "0.380324521"
    ],
    [
      "0.540042507",
      "0.658033584"
    ],
    [
      "0.881317913",
      "1.533382155"
    ],
    [
      "0.829672282",
      "1.831469313"
    ],
    [
      "0.600509718",
      "1.420821138"
    ],
    [
      "0.239537525",
      "0.623105354"
    ],
    [
      "0.473506773",
      "1.146242411"
    ],
    [
      "0.267924167",
      "0.924298686"
    ],
    [
      "0.548864087",
      "1.718908296"
    ],
    [
      "0.305509511",
      "1.487889034"
    ],
    [
      "0.178506566",
      "1.213310308"
    ]
  ],
  "radius": "0.151264028"
}
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": 34
}

当前第一名的答案

{
  "centers": [
    [
      "1.788052606",
      "0.684839977"
    ],
    [
      "1.627451399",
      "0.428460420"
    ],
    [
      "1.848529095",
      "0.981261669"
    ],
    [
      "1.800247136",
      "1.281536549"
    ],
    [
      "1.485596072",
      "0.738830747"
    ],
    [
      "1.325119586",
      "0.439355367"
    ],
    [
      "1.184412361",
      "0.171540689"
    ],
    [
      "1.564269264",
      "1.084797041"
    ],
    [
      "1.650365300",
      "1.545323506"
    ],
    [
      "1.163538884",
      "0.695118737"
    ],
    [
      "1.283158656",
      "0.972993412"
    ],
    [
      "1.022831659",
      "0.427304059"
    ],
    [
      "0.882124434",
      "0.159489381"
    ],
    [
      "1.419138420",
      "1.350240563"
    ],
    [
      "1.417783646",
      "1.738789263"
    ],
    [
      "1.138027812",
      "1.238436934"
    ],
    [
      "0.861250957",
      "0.683067429"
    ],
    [
      "0.980736867",
      "0.964975536"
    ],
    [
      "0.720543732",
      "0.415252751"
    ],
    [
      "1.183766814",
      "1.540302216"
    ],
    [
      "1.132121183",
      "1.838389374"
    ],
    [
      "0.835606023",
      "1.230419058"
    ],
    [
      "0.680749732",
      "0.925848262"
    ],
    [
      "0.420038751",
      "0.380324521"
    ],
    [
      "0.540042507",
      "0.658033584"
    ],
    [
      "0.881317913",
      "1.533382155"
    ],
    [
      "0.829672282",
      "1.831469313"
    ],
    [
      "0.600509718",
      "1.420821138"
    ],
    [
      "0.239537525",
      "0.623105354"
    ],
    [
      "0.473506773",
      "1.146242411"
    ],
    [
      "0.267924167",
      "0.924298686"
    ],
    [
      "0.548864087",
      "1.718908296"
    ],
    [
      "0.305509511",
      "1.487889034"
    ],
    [
      "0.178506566",
      "1.213310308"
    ]
  ],
  "radius": "0.151264028"
}

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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