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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 = 92

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 = 92
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[0.0935922457546945875274840950585, 0.0935922457546945875274840950595]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

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：起步布局

2y0
0x2
已验证构造r = 0.093592245

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 = 92 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

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。
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"。

起步布局答案

{
  "centers": [
    [
      "0.919048878",
      "0.097214348"
    ],
    [
      "1.106214995",
      "0.099837019"
    ],
    [
      "1.28885132",
      "0.140849296"
    ],
    [
      "0.669476616",
      "0.156003964"
    ],
    [
      "1.459168888",
      "0.218502112"
    ],
    [
      "0.503160549",
      "0.241893426"
    ],
    [
      "0.825535196",
      "0.259366207"
    ],
    [
      "1.162015484",
      "0.278510854"
    ],
    [
      "1.6099041",
      "0.329483776"
    ],
    [
      "0.65921913",
      "0.34525567"
    ],
    [
      "0.989145946",
      "0.350303309"
    ],
    [
      "1.332333052",
      "0.35616367"
    ],
    [
      "0.358033391",
      "0.360114159"
    ],
    [
      "0.81675962",
      "0.446344877"
    ],
    [
      "1.137754805",
      "0.464116494"
    ],
    [
      "1.73462849",
      "0.469061211"
    ],
    [
      "1.485695363",
      "0.471765081"
    ],
    [
      "0.240284435",
      "0.505624356"
    ],
    [
      "0.459571451",
      "0.517365729"
    ],
    [
      "0.646184121",
      "0.531985749"
    ],
    [
      "1.308072373",
      "0.541769309"
    ],
    [
      "0.971955108",
      "0.550998565"
    ],
    [
      "1.640838538",
      "0.631053429"
    ],
    [
      "1.82802289",
      "0.631281801"
    ],
    [
      "0.803724611",
      "0.633074956"
    ],
    [
      "1.130097037",
      "0.651144279"
    ],
    [
      "1.455479893",
      "0.657134196"
    ],
    [
      "0.341822495",
      "0.662875926"
    ],
    [
      "0.152740428",
      "0.677944983"
    ],
    [
      "0.624701101",
      "0.718070715"
    ],
    [
      "0.958920099",
      "0.737728644"
    ],
    [
      "1.291085508",
      "0.746646877"
    ],
    [
      "1.737913256",
      "0.795349737"
    ],
    [
      "0.460382575",
      "0.80772599"
    ],
    [
      "1.55204264",
      "0.81748902"
    ],
    [
      "0.784913667",
      "0.819311854"
    ],
    [
      "0.260148082",
      "0.831301946"
    ],
    [
      "1.117062029",
      "0.837874358"
    ],
    [
      "1.89965561",
      "0.88956993"
    ],
    [
      "0.619325243",
      "0.906595917"
    ],
    [
      "1.387648255",
      "0.907001701"
    ],
    [
      "0.096881154",
      "0.922854902"
    ],
    [
      "0.946031571",
      "0.924468891"
    ],
    [
      "1.733236319",
      "0.982475791"
    ],
    [
      "1.221499905",
      "0.993215157"
    ],
    [
      "0.453178579",
      "0.994771804"
    ],
    [
      "1.547365703",
      "1.004615074"
    ],
    [
      "0.77770967",
      "1.006357668"
    ],
    [
      "0.267484071",
      "1.018342629"
    ],
    [
      "1.903154913",
      "1.07672171"
    ],
    [
      "1.054223449",
      "1.077218854"
    ],
    [
      "0.612121246",
      "1.09364173"
    ],
    [
      "1.379237123",
      "1.09399712"
    ],
    [
      "0.10029272",
      "1.110008302"
    ],
    [
      "0.881570623",
      "1.162084789"
    ],
    [
      "1.740590169",
      "1.169515773"
    ],
    [
      "1.213088773",
      "1.180210576"
    ],
    [
      "0.447026044",
      "1.181855155"
    ],
    [
      "1.538954571",
      "1.191610493"
    ],
    [
      "0.261331537",
      "1.205425979"
    ],
    [
      "0.709926609",
      "1.253241683"
    ],
    [
      "1.03873273",
      "1.263761266"
    ],
    [
      "1.373376443",
      "1.281149865"
    ],
    [
      "1.84557084",
      "1.324511973"
    ],
    [
      "1.656909281",
      "1.336953949"
    ],
    [
      "0.544831407",
      "1.341455107"
    ],
    [
      "0.870787835",
      "1.348958451"
    ],
    [
      "1.195178675",
      "1.366536264"
    ],
    [
      "0.35986074",
      "1.370158523"
    ],
    [
      "0.172676291",
      "1.370284347"
    ],
    [
      "1.027949943",
      "1.450634927"
    ],
    [
      "0.690850278",
      "1.45857267"
    ],
    [
      "1.352951151",
      "1.467263021"
    ],
    [
      "1.539495163",
      "1.482734465"
    ],
    [
      "1.76085012",
      "1.492627761"
    ],
    [
      "0.266377482",
      "1.532327923"
    ],
    [
      "0.51843619",
      "1.532605475"
    ],
    [
      "0.861314478",
      "1.535903067"
    ],
    [
      "1.184395887",
      "1.553409926"
    ],
    [
      "1.643436002",
      "1.638408277"
    ],
    [
      "1.017504455",
      "1.639403405"
    ],
    [
      "0.668885828",
      "1.644464027"
    ],
    [
      "1.342168364",
      "1.654136682"
    ],
    [
      "0.391365762",
      "1.671669101"
    ],
    [
      "0.839350028",
      "1.721794424"
    ],
    [
      "1.176050196",
      "1.740408277"
    ],
    [
      "1.498581015",
      "1.756962343"
    ],
    [
      "0.542310706",
      "1.782365341"
    ],
    [
      "1.332462847",
      "1.843233937"
    ],
    [
      "0.712774906",
      "1.859695738"
    ],
    [
      "0.895488508",
      "1.900362352"
    ],
    [
      "1.082659253",
      "1.902630857"
    ]
  ],
  "radius": "0.093592245"
}
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": 92
}

起步布局答案

{
  "centers": [
    [
      "0.919048878",
      "0.097214348"
    ],
    [
      "1.106214995",
      "0.099837019"
    ],
    [
      "1.28885132",
      "0.140849296"
    ],
    [
      "0.669476616",
      "0.156003964"
    ],
    [
      "1.459168888",
      "0.218502112"
    ],
    [
      "0.503160549",
      "0.241893426"
    ],
    [
      "0.825535196",
      "0.259366207"
    ],
    [
      "1.162015484",
      "0.278510854"
    ],
    [
      "1.6099041",
      "0.329483776"
    ],
    [
      "0.65921913",
      "0.34525567"
    ],
    [
      "0.989145946",
      "0.350303309"
    ],
    [
      "1.332333052",
      "0.35616367"
    ],
    [
      "0.358033391",
      "0.360114159"
    ],
    [
      "0.81675962",
      "0.446344877"
    ],
    [
      "1.137754805",
      "0.464116494"
    ],
    [
      "1.73462849",
      "0.469061211"
    ],
    [
      "1.485695363",
      "0.471765081"
    ],
    [
      "0.240284435",
      "0.505624356"
    ],
    [
      "0.459571451",
      "0.517365729"
    ],
    [
      "0.646184121",
      "0.531985749"
    ],
    [
      "1.308072373",
      "0.541769309"
    ],
    [
      "0.971955108",
      "0.550998565"
    ],
    [
      "1.640838538",
      "0.631053429"
    ],
    [
      "1.82802289",
      "0.631281801"
    ],
    [
      "0.803724611",
      "0.633074956"
    ],
    [
      "1.130097037",
      "0.651144279"
    ],
    [
      "1.455479893",
      "0.657134196"
    ],
    [
      "0.341822495",
      "0.662875926"
    ],
    [
      "0.152740428",
      "0.677944983"
    ],
    [
      "0.624701101",
      "0.718070715"
    ],
    [
      "0.958920099",
      "0.737728644"
    ],
    [
      "1.291085508",
      "0.746646877"
    ],
    [
      "1.737913256",
      "0.795349737"
    ],
    [
      "0.460382575",
      "0.80772599"
    ],
    [
      "1.55204264",
      "0.81748902"
    ],
    [
      "0.784913667",
      "0.819311854"
    ],
    [
      "0.260148082",
      "0.831301946"
    ],
    [
      "1.117062029",
      "0.837874358"
    ],
    [
      "1.89965561",
      "0.88956993"
    ],
    [
      "0.619325243",
      "0.906595917"
    ],
    [
      "1.387648255",
      "0.907001701"
    ],
    [
      "0.096881154",
      "0.922854902"
    ],
    [
      "0.946031571",
      "0.924468891"
    ],
    [
      "1.733236319",
      "0.982475791"
    ],
    [
      "1.221499905",
      "0.993215157"
    ],
    [
      "0.453178579",
      "0.994771804"
    ],
    [
      "1.547365703",
      "1.004615074"
    ],
    [
      "0.77770967",
      "1.006357668"
    ],
    [
      "0.267484071",
      "1.018342629"
    ],
    [
      "1.903154913",
      "1.07672171"
    ],
    [
      "1.054223449",
      "1.077218854"
    ],
    [
      "0.612121246",
      "1.09364173"
    ],
    [
      "1.379237123",
      "1.09399712"
    ],
    [
      "0.10029272",
      "1.110008302"
    ],
    [
      "0.881570623",
      "1.162084789"
    ],
    [
      "1.740590169",
      "1.169515773"
    ],
    [
      "1.213088773",
      "1.180210576"
    ],
    [
      "0.447026044",
      "1.181855155"
    ],
    [
      "1.538954571",
      "1.191610493"
    ],
    [
      "0.261331537",
      "1.205425979"
    ],
    [
      "0.709926609",
      "1.253241683"
    ],
    [
      "1.03873273",
      "1.263761266"
    ],
    [
      "1.373376443",
      "1.281149865"
    ],
    [
      "1.84557084",
      "1.324511973"
    ],
    [
      "1.656909281",
      "1.336953949"
    ],
    [
      "0.544831407",
      "1.341455107"
    ],
    [
      "0.870787835",
      "1.348958451"
    ],
    [
      "1.195178675",
      "1.366536264"
    ],
    [
      "0.35986074",
      "1.370158523"
    ],
    [
      "0.172676291",
      "1.370284347"
    ],
    [
      "1.027949943",
      "1.450634927"
    ],
    [
      "0.690850278",
      "1.45857267"
    ],
    [
      "1.352951151",
      "1.467263021"
    ],
    [
      "1.539495163",
      "1.482734465"
    ],
    [
      "1.76085012",
      "1.492627761"
    ],
    [
      "0.266377482",
      "1.532327923"
    ],
    [
      "0.51843619",
      "1.532605475"
    ],
    [
      "0.861314478",
      "1.535903067"
    ],
    [
      "1.184395887",
      "1.553409926"
    ],
    [
      "1.643436002",
      "1.638408277"
    ],
    [
      "1.017504455",
      "1.639403405"
    ],
    [
      "0.668885828",
      "1.644464027"
    ],
    [
      "1.342168364",
      "1.654136682"
    ],
    [
      "0.391365762",
      "1.671669101"
    ],
    [
      "0.839350028",
      "1.721794424"
    ],
    [
      "1.176050196",
      "1.740408277"
    ],
    [
      "1.498581015",
      "1.756962343"
    ],
    [
      "0.542310706",
      "1.782365341"
    ],
    [
      "1.332462847",
      "1.843233937"
    ],
    [
      "0.712774906",
      "1.859695738"
    ],
    [
      "0.895488508",
      "1.900362352"
    ],
    [
      "1.082659253",
      "1.902630857"
    ]
  ],
  "radius": "0.093592245"
}

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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