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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 = 51

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 = 51
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[227.61188890825, 227.61188890835]见来源署名packomania.com ↗
本站纪录227.61188891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

455.22377782y0
0x455.22377782
已验证构造r = 227.61188891

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 = 51 的当前纪录构型，227.61188891
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）＋

使用 Packomania ccin51 的单位圆归一化坐标与公开半径比 227.6118889083。固定九位目标 R=227.611888910，取 golden-angle 序列第 393 个全局旋转，按 (R,R)+R*Rot(theta)*(x,y) 映射并舍入到 1e-9 网格。直接最近舍入剩 2 个约束违例；随后允许每个坐标移动 ±2 个整数单位，解整数可行性模型。容器线性约束预留最坏二次余量，圆间约束利用位移差平方项非负，所以模型解对原二次不等式安全。最终用整数平方检查 51 个容器约束和 1275 个圆对约束，全部通过。目标 227.611888909 对当前十二位公开坐标已有轻微连续约束亏损，后续可从更高精度坐标或非线性重解继续。

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
    227.611888913→227.61188891
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    227.611888914→227.611888913
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": [
    [
      "175.340520944",
      "250.578030982"
    ],
    [
      "180.090410921",
      "12.081956580"
    ],
    [
      "153.612915620",
      "438.169005298"
    ],
    [
      "86.636505461",
      "203.742080655"
    ],
    [
      "330.996207253",
      "32.399662574"
    ],
    [
      "15.948564005",
      "293.166286810"
    ],
    [
      "391.950654232",
      "85.732742846"
    ],
    [
      "430.861393439",
      "308.485120694"
    ],
    [
      "96.974952358",
      "402.495047106"
    ],
    [
      "261.695736197",
      "13.214985362"
    ],
    [
      "276.274606002",
      "438.650107206"
    ],
    [
      "443.223288687",
      "228.071155145"
    ],
    [
      "160.412786322",
      "24.154391026"
    ],
    [
      "66.821535017",
      "87.087621561"
    ],
    [
      "17.835289315",
      "193.134647616"
    ],
    [
      "33.733869082",
      "312.411092653"
    ],
    [
      "205.867786577",
      "198.450090869"
    ],
    [
      "113.852457443",
      "276.079014574"
    ],
    [
      "133.811156134",
      "142.882507207"
    ],
    [
      "322.678574438",
      "292.202765948"
    ],
    [
      "364.288287553",
      "292.757793771"
    ],
    [
      "176.878464688",
      "426.754691230"
    ],
    [
      "370.553745583",
      "374.014247818"
    ],
    [
      "248.404132688",
      "362.072341477"
    ],
    [
      "133.389286478",
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    [
      "342.045196950",
      "334.161234696"
    ],
    [
      "397.098409836",
      "120.531769771"
    ],
    [
      "287.470627550",
      "325.900384990"
    ],
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      "397.928656630",
      "329.766218933"
    ],
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      "291.220760604",
      "40.517340405"
    ],
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      "230.758351474",
      "309.630236178"
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      "266.824221449"
    ],
    [
      "214.880487730",
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      "163.684480148",
      "290.990797833"
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      "420.112634189"
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      "415.133118038",
      "266.993415009"
    ],
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      "346.391168328",
      "78.533470034"
    ],
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      "340.684677035",
      "153.316061305"
    ],
    [
      "276.679212296",
      "108.802979218"
    ],
    [
      "410.857463095",
      "187.371605591"
    ],
    [
      "183.220474674",
      "363.705648977"
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      "233.452692568"
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  ],
  "radius": "227.611888910"
}
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": 51
}

当前第一名的答案

{
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      "175.340520944",
      "250.578030982"
    ],
    [
      "180.090410921",
      "12.081956580"
    ],
    [
      "153.612915620",
      "438.169005298"
    ],
    [
      "86.636505461",
      "203.742080655"
    ],
    [
      "330.996207253",
      "32.399662574"
    ],
    [
      "15.948564005",
      "293.166286810"
    ],
    [
      "391.950654232",
      "85.732742846"
    ],
    [
      "430.861393439",
      "308.485120694"
    ],
    [
      "96.974952358",
      "402.495047106"
    ],
    [
      "261.695736197",
      "13.214985362"
    ],
    [
      "276.274606002",
      "438.650107206"
    ],
    [
      "443.223288687",
      "228.071155145"
    ],
    [
      "160.412786322",
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    ],
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      "17.835289315",
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    ],
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    [
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  ],
  "radius": "227.611888910"
}

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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