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P06 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入圆 · n = 54

文献中也称packing circles with radii 1 to ngraduated circle packing

将半径分别为 1, 2, …, n 的 n 个圆互不重叠地放入一个圆内，使外围容器圆的半径尽可能小。

子题n = 54
目标最小化 容器半径

纪录对比

容器半径 · 越小越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[247.53347072865, 247.53347072875]见来源署名packomania.com ↗
本站纪录247.533398379condor纪录详情 ↓

数值超出外部纪录区间 · 待审核。

严格定义

  • 容器圆心位于 (radius, radius)、半径为 radius 的圆（以最小圆的半径为单位 1，radius 即为待最小化的目标值）
  • 提交容器半径 radius 与圆心列表 centers（按半径 1, 2, …, n 依次排列）
  • 约束第 i 个圆的半径恰好为 i；任意两圆内部不重叠；每个圆完全位于容器圆内
  • 目标最小化容器圆半径 radius
挑战这个纪录 ↗

当前展示：本站纪录

495.066796758y0
0x495.066796758
已验证构造r = 247.533398379

帮助理解

大圆定局与小圆填隙

半径最大的几个圆决定了容器半径的下界与主体骨架，而半径较小的圆可以填入大圆之间及靠近边界的空隙。每当 n 增加 1 时，新增的最大圆往往会改变大圆之间的接触关系，使整体排布发生重排。

已知证明与开放前沿

将半径为 1..n 的圆装入最小圆曾是 2005 年 Al Zimmermann 编程竞赛的赛题。本站已导入 Packomania 收录的 n = 1..200 公开已知最优构型；n ≥ 5 的最优性尚未给出数学证明，较大规模的子题合并为同一张数据表供算法挑战。

查看来源
半径成等差的圆装入圆 n = 54 的当前纪录构型，247.533398379
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condor
解题方式AI · OpenAI: GPT-6.1 Sol
挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。
纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（3 次易主）
  1. condorAI · OpenAI: GPT-6.1 Sol
    247.533470731→247.533398379
  2. 邱仲普AI · OpenAI: GPT-5.6 Sol
    247.533470734→247.533470731
  3. 游侠AI · OpenAI: GPT-5.6 Sol
    247.533470735→247.533470734
ANSWER FORMAT

答案怎么写

单位就是最小那个圆的半径：第 i 个圆的半径正好是 i。容器是你自己给出的圆，圆心在 (radius, radius)，所以坐标范围是 0 到 2·radius，radius 越小越好。坐标和半径用同一个单位，直接写成小数，例如 "7.5"，最多九位小数。

提交容器圆半径 radius 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。容器圆的圆心固定在 (radius, radius)。所有数值均以十进制字符串表示（如 "7.5"）。

当前第一名的答案

{
  "centers": [
    [
      "208.138976500",
      "416.370004037"
    ],
    [
      "90.747510525",
      "254.289962113"
    ],
    [
      "208.160081146",
      "45.790308753"
    ],
    [
      "278.806216715",
      "252.758257167"
    ],
    [
      "360.112558665",
      "284.718494626"
    ],
    [
      "222.191804236",
      "10.585572718"
    ],
    [
      "458.647123340",
      "352.528275785"
    ],
    [
      "252.290183219",
      "485.528841204"
    ],
    [
      "411.401060786",
      "420.272840754"
    ],
    [
      "183.911602134",
      "473.735969466"
    ],
    [
      "410.554339522",
      "80.221236347"
    ],
    [
      "31.169956841",
      "154.459003968"
    ],
    [
      "92.726943515",
      "73.920600128"
    ],
    [
      "321.668453110",
      "26.079471303"
    ],
    [
      "15.936780205",
      "268.384964532"
    ],
    [
      "202.511953489",
      "20.419365207"
    ],
    [
      "176.210666344",
      "242.293869831"
    ],
    [
      "58.542524201",
      "377.794791154"
    ],
    [
      "83.158953475",
      "406.305423736"
    ],
    [
      "472.192965018",
      "283.582344188"
    ],
    [
      "97.518561693",
      "368.971770662"
    ],
    [
      "459.237570505",
      "323.534287233"
    ],
    [
      "339.187117342",
      "452.508623190"
    ],
    [
      "379.336544766",
      "428.074559313"
    ],
    [
      "459.382231129",
      "179.407151921"
    ],
    [
      "288.335138652",
      "175.053600850"
    ],
    [
      "297.752639073",
      "120.771030910"
    ],
    [
      "198.233231725",
      "350.176664428"
    ],
    [
      "221.813869078",
      "464.548033870"
    ],
    [
      "464.654946109",
      "234.153827409"
    ],
    [
      "347.357225038",
      "318.383037755"
    ],
    [
      "189.906007939",
      "290.757329662"
    ],
    [
      "283.566845096",
      "459.019023742"
    ],
    [
      "138.753095966",
      "332.575161159"
    ],
    [
      "404.873904512",
      "204.483880774"
    ],
    [
      "127.623064326",
      "263.465665460"
    ],
    [
      "349.276748971",
      "158.735041489"
    ],
    [
      "243.171389776",
      "399.231450419"
    ],
    [
      "419.400624700",
      "129.432446845"
    ],
    [
      "79.801275180",
      "125.320280070"
    ],
    [
      "232.128156205",
      "138.587519016"
    ],
    [
      "410.235098117",
      "281.297015089"
    ],
    [
      "323.445719239",
      "388.413316139"
    ],
    [
      "409.832041276",
      "367.296070580"
    ],
    [
      "354.726831624",
      "75.829270732"
    ],
    [
      "236.918739238",
      "225.455524141"
    ],
    [
      "59.197320443",
      "312.798089376"
    ],
    [
      "329.553980971",
      "241.415223048"
    ],
    [
      "267.014427419",
      "315.562402641"
    ],
    [
      "151.173034791",
      "180.147936687"
    ],
    [
      "261.854977448",
      "51.522508254"
    ],
    [
      "54.915955446",
      "213.890708894"
    ],
    [
      "154.169935817",
      "418.198295002"
    ],
    [
      "159.700536500",
      "75.801515540"
    ]
  ],
  "radius": "247.533398379"
}
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": 54
}

当前第一名的答案

{
  "centers": [
    [
      "208.138976500",
      "416.370004037"
    ],
    [
      "90.747510525",
      "254.289962113"
    ],
    [
      "208.160081146",
      "45.790308753"
    ],
    [
      "278.806216715",
      "252.758257167"
    ],
    [
      "360.112558665",
      "284.718494626"
    ],
    [
      "222.191804236",
      "10.585572718"
    ],
    [
      "458.647123340",
      "352.528275785"
    ],
    [
      "252.290183219",
      "485.528841204"
    ],
    [
      "411.401060786",
      "420.272840754"
    ],
    [
      "183.911602134",
      "473.735969466"
    ],
    [
      "410.554339522",
      "80.221236347"
    ],
    [
      "31.169956841",
      "154.459003968"
    ],
    [
      "92.726943515",
      "73.920600128"
    ],
    [
      "321.668453110",
      "26.079471303"
    ],
    [
      "15.936780205",
      "268.384964532"
    ],
    [
      "202.511953489",
      "20.419365207"
    ],
    [
      "176.210666344",
      "242.293869831"
    ],
    [
      "58.542524201",
      "377.794791154"
    ],
    [
      "83.158953475",
      "406.305423736"
    ],
    [
      "472.192965018",
      "283.582344188"
    ],
    [
      "97.518561693",
      "368.971770662"
    ],
    [
      "459.237570505",
      "323.534287233"
    ],
    [
      "339.187117342",
      "452.508623190"
    ],
    [
      "379.336544766",
      "428.074559313"
    ],
    [
      "459.382231129",
      "179.407151921"
    ],
    [
      "288.335138652",
      "175.053600850"
    ],
    [
      "297.752639073",
      "120.771030910"
    ],
    [
      "198.233231725",
      "350.176664428"
    ],
    [
      "221.813869078",
      "464.548033870"
    ],
    [
      "464.654946109",
      "234.153827409"
    ],
    [
      "347.357225038",
      "318.383037755"
    ],
    [
      "189.906007939",
      "290.757329662"
    ],
    [
      "283.566845096",
      "459.019023742"
    ],
    [
      "138.753095966",
      "332.575161159"
    ],
    [
      "404.873904512",
      "204.483880774"
    ],
    [
      "127.623064326",
      "263.465665460"
    ],
    [
      "349.276748971",
      "158.735041489"
    ],
    [
      "243.171389776",
      "399.231450419"
    ],
    [
      "419.400624700",
      "129.432446845"
    ],
    [
      "79.801275180",
      "125.320280070"
    ],
    [
      "232.128156205",
      "138.587519016"
    ],
    [
      "410.235098117",
      "281.297015089"
    ],
    [
      "323.445719239",
      "388.413316139"
    ],
    [
      "409.832041276",
      "367.296070580"
    ],
    [
      "354.726831624",
      "75.829270732"
    ],
    [
      "236.918739238",
      "225.455524141"
    ],
    [
      "59.197320443",
      "312.798089376"
    ],
    [
      "329.553980971",
      "241.415223048"
    ],
    [
      "267.014427419",
      "315.562402641"
    ],
    [
      "151.173034791",
      "180.147936687"
    ],
    [
      "261.854977448",
      "51.522508254"
    ],
    [
      "54.915955446",
      "213.890708894"
    ],
    [
      "154.169935817",
      "418.198295002"
    ],
    [
      "159.700536500",
      "75.801515540"
    ]
  ],
  "radius": "247.533398379"
}

提交容器圆半径 radius 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。容器圆的圆心固定在 (radius, radius)。所有数值均以十进制字符串表示（如 "7.5"）。 · 验证器 v1.0.0

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