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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 = 50

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 = 50
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[220.56540026235, 220.56540026245]见来源署名packomania.com ↗
本站纪录220.565400264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

441.130800528y0
0x441.130800528
已验证构造r = 220.565400264

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 = 50 的当前纪录构型，220.565400264
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 ccin50 的单位容器归一化坐标和公开半径比 220.5654002624。固定九位目标 R=220.565400264，以 golden-angle 序列的第 23739 个全局旋转映射圆心：(x,y) -> (R,R)+R*Rot(theta)*(x,y)，先四舍五入到 1e-9 网格。直接舍入仍有 2 个约束违例；随后在每个坐标 ±2 个整数单位内解整数可行性模型。容器约束在线性化时预留 2*2^2 的二次余量，圆间约束利用位移差平方项非负，因此模型解对原二次约束是安全的。最后以整数平方完整检查 50 个容器约束和 1225 个圆对约束，全部通过。尝试 R=220.565400263 时，测试的当前纪录邻域和公开坐标旋转邻域均未得到可行解；后续可从更高精度原始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
    220.565400268→220.565400264
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    220.565400269→220.565400268
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

{
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      "411.382382676"
    ],
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      "436.199128085",
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    ],
    [
      "24.048018266",
      "313.737640414"
    ],
    [
      "12.767540893",
      "281.379239733"
    ],
    [
      "6.945038754",
      "237.664532144"
    ],
    [
      "150.479358717",
      "17.851020024"
    ],
    [
      "286.359483385",
      "17.414791235"
    ],
    [
      "67.640953243",
      "73.071692911"
    ],
    [
      "424.744180390",
      "165.257234459"
    ],
    [
      "430.458978317",
      "236.529035495"
    ],
    [
      "102.581176363",
      "393.762917273"
    ],
    [
      "23.517156866",
      "152.243643591"
    ],
    [
      "192.245788132",
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  "radius": "220.565400264"
}
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": 50
}

当前第一名的答案

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    ],
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      "313.737640414"
    ],
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      "12.767540893",
      "281.379239733"
    ],
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      "237.664532144"
    ],
    [
      "150.479358717",
      "17.851020024"
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      "286.359483385",
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      "73.071692911"
    ],
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      "165.257234459"
    ],
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      "236.529035495"
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}

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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