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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 = 55

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 = 55
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[254.48889026535, 254.48889026545]见来源署名packomania.com ↗
本站纪录254.488890268邱仲普纪录详情 ↓

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

508.977780536y0
0x508.977780536
已验证构造r = 254.488890268

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 = 55 的当前纪录构型，254.488890268
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者邱仲普
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公开方法与可复现线索（点击展开）＋

从 Packomania ccin55 的公开坐标（连续比值 254.4888902654）出发，按候选半径 254.488890268 缩放和平移。采用黄金角旋转 k=210（2π·frac(k·0.6180339887498948482)），将 55 个中心四舍五入到 1e-9 网格，并用整数平方逐项检查 55 个包含约束和 1485 个两两不重叠约束。为探索更低目标，254.488890267 在 k=48746 附近的 ±1、±2、±4、±8、±16、±32 整数修整模型均未得到可行证书；后续可扩大旋转和离散修整搜索。

在讨论区查看与回复 →
历史纪录（2 次易主）
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    254.488890271→254.488890268
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    254.488890272→254.488890271
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": [
    [
      "455.299675021",
      "321.554024516"
    ],
    [
      "15.087393084",
      "202.857931340"
    ],
    [
      "195.630770331",
      "28.867650214"
    ],
    [
      "194.395965240",
      "451.918623239"
    ],
    [
      "70.272332799",
      "422.546742931"
    ],
    [
      "284.518907884",
      "7.884761627"
    ],
    [
      "500.156445892",
      "269.763051952"
    ],
    [
      "10.707350234",
      "290.174351542"
    ],
    [
      "193.697895214",
      "16.809873724"
    ],
    [
      "64.254212065",
      "101.193926518"
    ],
    [
      "126.876786475",
      "47.123427093"
    ],
    [
      "181.058371624",
      "485.592375389"
    ],
    [
      "22.046237938",
      "189.214226961"
    ],
    [
      "480.652696558",
      "172.754049893"
    ],
    [
      "302.601078361",
      "19.936486851"
    ],
    [
      "414.339367309",
      "77.500823521"
    ],
    [
      "125.834518585",
      "141.956904609"
    ],
    [
      "38.721170992",
      "351.139232416"
    ],
    [
      "111.382977107",
      "272.825611627"
    ],
    [
      "327.320286797",
      "303.486903525"
    ],
    [
      "28.640310794",
      "313.243247932"
    ],
    [
      "262.520656721",
      "234.795968387"
    ],
    [
      "100.076531250",
      "426.953175556"
    ],
    [
      "283.456386686",
      "193.836290520"
    ],
    [
      "464.189962722",
      "347.339622014"
    ],
    [
      "479.108532473",
      "296.359993804"
    ],
    [
      "215.268574951",
      "247.766734302"
    ],
    [
      "69.951075964",
      "385.801542006"
    ],
    [
      "221.254881877",
      "477.515212303"
    ],
    [
      "168.046924737",
      "47.310106488"
    ],
    [
      "389.317489140",
      "216.001429866"
    ],
    [
      "365.150444464",
      "447.505280026"
    ],
    [
      "226.673961472",
      "188.860731565"
    ],
    [
      "152.369357087",
      "449.634894319"
    ],
    [
      "132.632296485",
      "378.951903177"
    ],
    [
      "109.728633219",
      "90.900966215"
    ],
    [
      "86.288316315",
      "322.888152511"
    ],
    [
      "339.416800468",
      "167.144867003"
    ],
    [
      "270.432576181",
      "131.683956220"
    ],
    [
      "323.894566455",
      "243.584779688"
    ],
    [
      "415.514313880",
      "394.661352678"
    ],
    [
      "66.350440673",
      "155.757492313"
    ],
    [
      "291.573741065",
      "462.047244026"
    ],
    [
      "463.324954275",
      "228.162642981"
    ],
    [
      "262.331949822",
      "302.255620745"
    ],
    [
      "185.935544016",
      "121.174829464"
    ],
    [
      "168.805830784",
      "305.361989355"
    ],
    [
      "246.042153429",
      "48.172835535"
    ],
    [
      "49.205832903",
      "245.293762095"
    ],
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      "423.980272737",
      "142.792882323"
    ],
    [
      "325.720945032",
      "374.468887901"
    ],
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      "143.688837115",
      "209.601159005"
    ],
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      "400.303042400",
      "301.900276681"
    ],
    [
      "344.374651115",
      "75.278552626"
    ],
    [
      "221.446167495",
      "393.515430101"
    ]
  ],
  "radius": "254.488890268"
}
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": 55
}

当前第一名的答案

{
  "centers": [
    [
      "455.299675021",
      "321.554024516"
    ],
    [
      "15.087393084",
      "202.857931340"
    ],
    [
      "195.630770331",
      "28.867650214"
    ],
    [
      "194.395965240",
      "451.918623239"
    ],
    [
      "70.272332799",
      "422.546742931"
    ],
    [
      "284.518907884",
      "7.884761627"
    ],
    [
      "500.156445892",
      "269.763051952"
    ],
    [
      "10.707350234",
      "290.174351542"
    ],
    [
      "193.697895214",
      "16.809873724"
    ],
    [
      "64.254212065",
      "101.193926518"
    ],
    [
      "126.876786475",
      "47.123427093"
    ],
    [
      "181.058371624",
      "485.592375389"
    ],
    [
      "22.046237938",
      "189.214226961"
    ],
    [
      "480.652696558",
      "172.754049893"
    ],
    [
      "302.601078361",
      "19.936486851"
    ],
    [
      "414.339367309",
      "77.500823521"
    ],
    [
      "125.834518585",
      "141.956904609"
    ],
    [
      "38.721170992",
      "351.139232416"
    ],
    [
      "111.382977107",
      "272.825611627"
    ],
    [
      "327.320286797",
      "303.486903525"
    ],
    [
      "28.640310794",
      "313.243247932"
    ],
    [
      "262.520656721",
      "234.795968387"
    ],
    [
      "100.076531250",
      "426.953175556"
    ],
    [
      "283.456386686",
      "193.836290520"
    ],
    [
      "464.189962722",
      "347.339622014"
    ],
    [
      "479.108532473",
      "296.359993804"
    ],
    [
      "215.268574951",
      "247.766734302"
    ],
    [
      "69.951075964",
      "385.801542006"
    ],
    [
      "221.254881877",
      "477.515212303"
    ],
    [
      "168.046924737",
      "47.310106488"
    ],
    [
      "389.317489140",
      "216.001429866"
    ],
    [
      "365.150444464",
      "447.505280026"
    ],
    [
      "226.673961472",
      "188.860731565"
    ],
    [
      "152.369357087",
      "449.634894319"
    ],
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      "132.632296485",
      "378.951903177"
    ],
    [
      "109.728633219",
      "90.900966215"
    ],
    [
      "86.288316315",
      "322.888152511"
    ],
    [
      "339.416800468",
      "167.144867003"
    ],
    [
      "270.432576181",
      "131.683956220"
    ],
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      "323.894566455",
      "243.584779688"
    ],
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      "415.514313880",
      "394.661352678"
    ],
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      "155.757492313"
    ],
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      "462.047244026"
    ],
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      "463.324954275",
      "228.162642981"
    ],
    [
      "262.331949822",
      "302.255620745"
    ],
    [
      "185.935544016",
      "121.174829464"
    ],
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      "168.805830784",
      "305.361989355"
    ],
    [
      "246.042153429",
      "48.172835535"
    ],
    [
      "49.205832903",
      "245.293762095"
    ],
    [
      "423.980272737",
      "142.792882323"
    ],
    [
      "325.720945032",
      "374.468887901"
    ],
    [
      "143.688837115",
      "209.601159005"
    ],
    [
      "400.303042400",
      "301.900276681"
    ],
    [
      "344.374651115",
      "75.278552626"
    ],
    [
      "221.446167495",
      "393.515430101"
    ]
  ],
  "radius": "254.488890268"
}

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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