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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 = 39

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 = 39
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[153.19974335205, 153.19974335215]见来源署名packomania.com ↗
本站纪录153.199743354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

306.399486708y0
0x306.399486708
已验证构造r = 153.199743354

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 = 39 的当前纪录构型，153.199743354
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）＋

Starting from the public 153.199743355 certificate, scale every coordinate by 10^9. Allow each center an integer offset in [-2,2], set the target radius to 153199743354, and solve the linearized containment and pairwise-separation constraints as an integer MILP; for containment reserve the worst-case quadratic term 2R^2. The resulting certificate moves all 39 centers by at most 2 integer units. Rechecking every raw squared inequality with exact integers independently confirms feasibility. A conservative [-4,4] linearized model found no solution at 153199743353; a useful continuation is a larger-neighborhood nonlinear integer search.

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
    153.199743355→153.199743354
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    153.199743356→153.199743355
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": [
    [
      "142.129111348",
      "157.435619655"
    ],
    [
      "259.389943914",
      "45.621946229"
    ],
    [
      "250.313596035",
      "117.451522765"
    ],
    [
      "61.241509149",
      "35.809261272"
    ],
    [
      "19.963936090",
      "88.411845886"
    ],
    [
      "276.263138821",
      "233.794872072"
    ],
    [
      "166.044979578",
      "298.766703123"
    ],
    [
      "237.828470156",
      "271.130161152"
    ],
    [
      "15.164292328",
      "194.904233145"
    ],
    [
      "156.167574903",
      "10.036530800"
    ],
    [
      "77.380229111",
      "32.934742659"
    ],
    [
      "20.335205541",
      "105.497602541"
    ],
    [
      "292.617688942",
      "138.414464037"
    ],
    [
      "115.498556906",
      "87.225547757"
    ],
    [
      "290.772072983",
      "166.353571069"
    ],
    [
      "262.388049539",
      "70.123855187"
    ],
    [
      "106.894909779",
      "281.263074378"
    ],
    [
      "237.207664346",
      "47.267501739"
    ],
    [
      "142.490495053",
      "286.946901533"
    ],
    [
      "170.718615676",
      "207.719440251"
    ],
    [
      "135.298711316",
      "186.567123487"
    ],
    [
      "248.758614528",
      "243.057179895"
    ],
    [
      "274.946169618",
      "107.050194479"
    ],
    [
      "40.392494181",
      "216.184648026"
    ],
    [
      "271.902530832",
      "201.623110037"
    ],
    [
      "74.275662865",
      "252.953260727"
    ],
    [
      "88.365820820",
      "197.160337625"
    ],
    [
      "218.916641714",
      "202.846038676"
    ],
    [
      "158.383399615",
      "90.370427063"
    ],
    [
      "247.244470984",
      "152.234438337"
    ],
    [
      "129.339859550",
      "238.224572956"
    ],
    [
      "188.236874450",
      "37.174844158"
    ],
    [
      "220.183727661",
      "95.342293260"
    ],
    [
      "122.243218363",
      "38.089915902"
    ],
    [
      "111.166378414",
      "136.033664188"
    ],
    [
      "180.410569325",
      "151.725764917"
    ],
    [
      "192.893689427",
      "262.409500247"
    ],
    [
      "63.954335527",
      "80.355813956"
    ],
    [
      "39.000000001",
      "153.200039112"
    ]
  ],
  "radius": "153.199743354"
}
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": 39
}

当前第一名的答案

{
  "centers": [
    [
      "142.129111348",
      "157.435619655"
    ],
    [
      "259.389943914",
      "45.621946229"
    ],
    [
      "250.313596035",
      "117.451522765"
    ],
    [
      "61.241509149",
      "35.809261272"
    ],
    [
      "19.963936090",
      "88.411845886"
    ],
    [
      "276.263138821",
      "233.794872072"
    ],
    [
      "166.044979578",
      "298.766703123"
    ],
    [
      "237.828470156",
      "271.130161152"
    ],
    [
      "15.164292328",
      "194.904233145"
    ],
    [
      "156.167574903",
      "10.036530800"
    ],
    [
      "77.380229111",
      "32.934742659"
    ],
    [
      "20.335205541",
      "105.497602541"
    ],
    [
      "292.617688942",
      "138.414464037"
    ],
    [
      "115.498556906",
      "87.225547757"
    ],
    [
      "290.772072983",
      "166.353571069"
    ],
    [
      "262.388049539",
      "70.123855187"
    ],
    [
      "106.894909779",
      "281.263074378"
    ],
    [
      "237.207664346",
      "47.267501739"
    ],
    [
      "142.490495053",
      "286.946901533"
    ],
    [
      "170.718615676",
      "207.719440251"
    ],
    [
      "135.298711316",
      "186.567123487"
    ],
    [
      "248.758614528",
      "243.057179895"
    ],
    [
      "274.946169618",
      "107.050194479"
    ],
    [
      "40.392494181",
      "216.184648026"
    ],
    [
      "271.902530832",
      "201.623110037"
    ],
    [
      "74.275662865",
      "252.953260727"
    ],
    [
      "88.365820820",
      "197.160337625"
    ],
    [
      "218.916641714",
      "202.846038676"
    ],
    [
      "158.383399615",
      "90.370427063"
    ],
    [
      "247.244470984",
      "152.234438337"
    ],
    [
      "129.339859550",
      "238.224572956"
    ],
    [
      "188.236874450",
      "37.174844158"
    ],
    [
      "220.183727661",
      "95.342293260"
    ],
    [
      "122.243218363",
      "38.089915902"
    ],
    [
      "111.166378414",
      "136.033664188"
    ],
    [
      "180.410569325",
      "151.725764917"
    ],
    [
      "192.893689427",
      "262.409500247"
    ],
    [
      "63.954335527",
      "80.355813956"
    ],
    [
      "39.000000001",
      "153.200039112"
    ]
  ],
  "radius": "153.199743354"
}

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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