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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 = 45

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 = 45
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[0.1320495942516301792957150882985, 0.1320495942516301792957150882995]见来源署名packomania.com ↗
本站纪录0.132049594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 = 0.132049594

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 = 45 的当前纪录构型，0.132049594
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
    0.132049593→0.132049594
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": [
    [
      "1.843495211",
      "0.810554564"
    ],
    [
      "1.750114640",
      "0.563346896"
    ],
    [
      "1.859751690",
      "1.119016545"
    ],
    [
      "1.584071912",
      "0.357973592"
    ],
    [
      "1.637239519",
      "0.976760737"
    ],
    [
      "1.717796643",
      "1.487960947"
    ],
    [
      "1.544741337",
      "0.729389624"
    ],
    [
      "1.625298461",
      "1.240589834"
    ],
    [
      "1.358259079",
      "0.209544989"
    ],
    [
      "1.378698609",
      "0.524016320"
    ],
    [
      "1.402786289",
      "1.098334026"
    ],
    [
      "1.457317884",
      "1.444381166"
    ],
    [
      "1.505421829",
      "1.705490940"
    ],
    [
      "1.310288107",
      "0.850962913"
    ],
    [
      "1.156186437",
      "0.381760512"
    ],
    [
      "1.063688255",
      "0.134389399"
    ],
    [
      "1.144245379",
      "0.645589609"
    ],
    [
      "1.234805712",
      "1.302125358"
    ],
    [
      "1.142307530",
      "1.054754245"
    ],
    [
      "1.222864654",
      "1.565954455"
    ],
    [
      "1.263975223",
      "1.826834317"
    ],
    [
      "0.895707678",
      "0.338180731"
    ],
    [
      "0.976264802",
      "0.849380941"
    ],
    [
      "0.883766620",
      "0.602009828"
    ],
    [
      "0.973925850",
      "1.343235927"
    ],
    [
      "0.881427668",
      "1.095864814"
    ],
    [
      "0.961984792",
      "1.607065024"
    ],
    [
      "0.673195507",
      "0.195924923"
    ],
    [
      "1.003095361",
      "1.867944886"
    ],
    [
      "0.661254448",
      "0.459754020"
    ],
    [
      "0.715786043",
      "0.805801160"
    ],
    [
      "0.713447091",
      "1.299656146"
    ],
    [
      "0.620948909",
      "1.052285033"
    ],
    [
      "0.701506033",
      "1.563485243"
    ],
    [
      "0.742157724",
      "1.825965139"
    ],
    [
      "0.493273871",
      "0.663545352"
    ],
    [
      "0.417063455",
      "0.357074128"
    ],
    [
      "0.398436737",
      "0.910029225"
    ],
    [
      "0.478993861",
      "1.421229435"
    ],
    [
      "0.386495679",
      "1.173858322"
    ],
    [
      "0.246789999",
      "0.568708218"
    ],
    [
      "0.467052803",
      "1.685058532"
    ],
    [
      "0.151952865",
      "0.815192090"
    ],
    [
      "0.218515102",
      "1.377649654"
    ],
    [
      "0.136316874",
      "1.084971469"
    ]
  ],
  "radius": "0.132049594"
}
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": 45
}

当前第一名的答案

{
  "centers": [
    [
      "1.843495211",
      "0.810554564"
    ],
    [
      "1.750114640",
      "0.563346896"
    ],
    [
      "1.859751690",
      "1.119016545"
    ],
    [
      "1.584071912",
      "0.357973592"
    ],
    [
      "1.637239519",
      "0.976760737"
    ],
    [
      "1.717796643",
      "1.487960947"
    ],
    [
      "1.544741337",
      "0.729389624"
    ],
    [
      "1.625298461",
      "1.240589834"
    ],
    [
      "1.358259079",
      "0.209544989"
    ],
    [
      "1.378698609",
      "0.524016320"
    ],
    [
      "1.402786289",
      "1.098334026"
    ],
    [
      "1.457317884",
      "1.444381166"
    ],
    [
      "1.505421829",
      "1.705490940"
    ],
    [
      "1.310288107",
      "0.850962913"
    ],
    [
      "1.156186437",
      "0.381760512"
    ],
    [
      "1.063688255",
      "0.134389399"
    ],
    [
      "1.144245379",
      "0.645589609"
    ],
    [
      "1.234805712",
      "1.302125358"
    ],
    [
      "1.142307530",
      "1.054754245"
    ],
    [
      "1.222864654",
      "1.565954455"
    ],
    [
      "1.263975223",
      "1.826834317"
    ],
    [
      "0.895707678",
      "0.338180731"
    ],
    [
      "0.976264802",
      "0.849380941"
    ],
    [
      "0.883766620",
      "0.602009828"
    ],
    [
      "0.973925850",
      "1.343235927"
    ],
    [
      "0.881427668",
      "1.095864814"
    ],
    [
      "0.961984792",
      "1.607065024"
    ],
    [
      "0.673195507",
      "0.195924923"
    ],
    [
      "1.003095361",
      "1.867944886"
    ],
    [
      "0.661254448",
      "0.459754020"
    ],
    [
      "0.715786043",
      "0.805801160"
    ],
    [
      "0.713447091",
      "1.299656146"
    ],
    [
      "0.620948909",
      "1.052285033"
    ],
    [
      "0.701506033",
      "1.563485243"
    ],
    [
      "0.742157724",
      "1.825965139"
    ],
    [
      "0.493273871",
      "0.663545352"
    ],
    [
      "0.417063455",
      "0.357074128"
    ],
    [
      "0.398436737",
      "0.910029225"
    ],
    [
      "0.478993861",
      "1.421229435"
    ],
    [
      "0.386495679",
      "1.173858322"
    ],
    [
      "0.246789999",
      "0.568708218"
    ],
    [
      "0.467052803",
      "1.685058532"
    ],
    [
      "0.151952865",
      "0.815192090"
    ],
    [
      "0.218515102",
      "1.377649654"
    ],
    [
      "0.136316874",
      "1.084971469"
    ]
  ],
  "radius": "0.132049594"
}

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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