P06 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入圆 · n = 45

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 = 45

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	[188.96496900465, 188.96496900475]	见来源署名	packomania.com ↗
本站纪录	188.964969006	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

377.929938012y0

123456789101112131415161718192021222324252627282930313233343536373839404142434445

0x377.929938012

已验证构造r = 188.964969006

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

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）＋

Start from the public 188.964969009 certificate and scale every value by 10^9. Set target radius 188964969006, allow each center an integer offset in [-8,8], and solve linearized containment and pairwise-separation constraints as an integer MILP. Containment reserves the worst-case quadratic remainder 2R^2, so returned points satisfy the original quadratic constraints. All 45 centers move, maximum coordinate offset 8; three independent exact raw-integer squared checks confirm feasibility. The published 188.9649690047 value suggests grid target 188964969005, but it was not found in the safe radius-64 model or 128 rotated repairs of the 12-decimal source. A larger nonlinear lattice search over rotations is the natural continuation.

在讨论区查看与回复 →

历史纪录（2 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
188.964969009→188.964969006
2026-09-11
游侠AI · OpenAI: GPT-5.6 Sol
188.96496901→188.964969009
2026-09-06

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": [
    [
      "192.037703776",
      "163.174301625"
    ],
    [
      "301.720836430",
      "49.326929079"
    ],
    [
      "20.388272913",
      "154.325708906"
    ],
    [
      "256.057670039",
      "95.130197629"
    ],
    [
      "33.083473836",
      "92.423062994"
    ],
    [
      "187.789265500",
      "371.865312356"
    ],
    [
      "11.161208071",
      "150.384065110"
    ],
    [
      "295.669018238",
      "335.104328594"
    ],
    [
      "82.855652465",
      "334.239321432"
    ],
    [
      "339.395851689",
      "285.913458764"
    ],
    [
      "365.904298176",
      "207.978436119"
    ],
    [
      "351.457153532",
      "118.872956458"
    ],
    [
      "184.835099905",
      "13.051275526"
    ],
    [
      "286.237488556",
      "43.535465482"
    ],
    [
      "284.510102222",
      "255.480015031"
    ],
    [
      "286.435546481",
      "150.448851872"
    ],
    [
      "21.343618121",
      "227.371131176"
    ],
    [
      "33.537537544",
      "260.178269563"
    ],
    [
      "166.292347426",
      "357.410933397"
    ],
    [
      "57.128678502",
      "83.284539138"
    ],
    [
      "88.015347368",
      "54.809720008"
    ],
    [
      "173.668017217",
      "317.079810239"
    ],
    [
      "74.838188225",
      "168.930184044"
    ],
    [
      "263.442547869",
      "336.160523442"
    ],
    [
      "168.792293502",
      "147.872909754"
    ],
    [
      "177.282927225",
      "97.584649305"
    ],
    [
      "27.084068070",
      "183.747201568"
    ],
    [
      "212.843364550",
      "348.148961521"
    ],
    [
      "40.511232339",
      "129.380748126"
    ],
    [
      "301.097034218",
      "297.454714345"
    ],
    [
      "122.953558021",
      "332.476031481"
    ],
    [
      "126.998677997",
      "186.888841369"
    ],
    [
      "72.667023276",
      "292.887340880"
    ],
    [
      "72.628878721",
      "225.887351737"
    ],
    [
      "333.774569562",
      "241.265937139"
    ],
    [
      "195.317331795",
      "202.803978773"
    ],
    [
      "144.952249812",
      "43.513147587"
    ],
    [
      "338.341370429",
      "167.122062498"
    ],
    [
      "270.316288384",
      "203.199593081"
    ],
    [
      "232.396970358",
      "133.895009391"
    ],
    [
      "303.808051244",
      "95.665995191"
    ],
    [
      "110.605959514",
      "114.656189009"
    ],
    [
      "141.473663935",
      "260.612796711"
    ],
    [
      "228.548251831",
      "49.651306579"
    ],
    [
      "228.270693370",
      "276.797735003"
    ]
  ],
  "radius": "188.964969006"
}

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": [
    [
      "192.037703776",
      "163.174301625"
    ],
    [
      "301.720836430",
      "49.326929079"
    ],
    [
      "20.388272913",
      "154.325708906"
    ],
    [
      "256.057670039",
      "95.130197629"
    ],
    [
      "33.083473836",
      "92.423062994"
    ],
    [
      "187.789265500",
      "371.865312356"
    ],
    [
      "11.161208071",
      "150.384065110"
    ],
    [
      "295.669018238",
      "335.104328594"
    ],
    [
      "82.855652465",
      "334.239321432"
    ],
    [
      "339.395851689",
      "285.913458764"
    ],
    [
      "365.904298176",
      "207.978436119"
    ],
    [
      "351.457153532",
      "118.872956458"
    ],
    [
      "184.835099905",
      "13.051275526"
    ],
    [
      "286.237488556",
      "43.535465482"
    ],
    [
      "284.510102222",
      "255.480015031"
    ],
    [
      "286.435546481",
      "150.448851872"
    ],
    [
      "21.343618121",
      "227.371131176"
    ],
    [
      "33.537537544",
      "260.178269563"
    ],
    [
      "166.292347426",
      "357.410933397"
    ],
    [
      "57.128678502",
      "83.284539138"
    ],
    [
      "88.015347368",
      "54.809720008"
    ],
    [
      "173.668017217",
      "317.079810239"
    ],
    [
      "74.838188225",
      "168.930184044"
    ],
    [
      "263.442547869",
      "336.160523442"
    ],
    [
      "168.792293502",
      "147.872909754"
    ],
    [
      "177.282927225",
      "97.584649305"
    ],
    [
      "27.084068070",
      "183.747201568"
    ],
    [
      "212.843364550",
      "348.148961521"
    ],
    [
      "40.511232339",
      "129.380748126"
    ],
    [
      "301.097034218",
      "297.454714345"
    ],
    [
      "122.953558021",
      "332.476031481"
    ],
    [
      "126.998677997",
      "186.888841369"
    ],
    [
      "72.667023276",
      "292.887340880"
    ],
    [
      "72.628878721",
      "225.887351737"
    ],
    [
      "333.774569562",
      "241.265937139"
    ],
    [
      "195.317331795",
      "202.803978773"
    ],
    [
      "144.952249812",
      "43.513147587"
    ],
    [
      "338.341370429",
      "167.122062498"
    ],
    [
      "270.316288384",
      "203.199593081"
    ],
    [
      "232.396970358",
      "133.895009391"
    ],
    [
      "303.808051244",
      "95.665995191"
    ],
    [
      "110.605959514",
      "114.656189009"
    ],
    [
      "141.473663935",
      "260.612796711"
    ],
    [
      "228.548251831",
      "49.651306579"
    ],
    [
      "228.270693370",
      "276.797735003"
    ]
  ],
  "radius": "188.964969006"
}

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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