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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 = 58

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 = 58
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[275.15543883945, 275.15543883955]见来源署名packomania.com ↗
本站纪录275.155438841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

550.310877682y0
0x550.310877682
已验证构造r = 275.155438841

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 = 58 的当前纪录构型，275.155438841
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 ccin58 的公开原始坐标出发，以高精度数值在黄金角旋转 k=52640 附近搜索九位网格；在目标半径 275.155438841 下，先对直接四舍五入构型建立 ±4 个整数单位的局部线性化修整模型，得到可行修整。最终用整数平方逐项复核 58 个包含约束和 1653 个两两不重叠约束；更低目标尚未形成证书，可继续扩大旋转和整数修整范围。

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
    275.155438848→275.155438841
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    275.155438849→275.155438848
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": [
    [
      "26.885547582",
      "163.664543538"
    ],
    [
      "145.452867412",
      "450.544167255"
    ],
    [
      "246.440569754",
      "507.419936187"
    ],
    [
      "238.542784537",
      "541.378639260"
    ],
    [
      "152.216478063",
      "173.971469374"
    ],
    [
      "234.287823763",
      "255.401559609"
    ],
    [
      "223.106624320",
      "466.308784633"
    ],
    [
      "227.894616890",
      "374.886018380"
    ],
    [
      "16.800308541",
      "339.016214752"
    ],
    [
      "38.296222368",
      "158.322084727"
    ],
    [
      "402.860961791",
      "506.355459095"
    ],
    [
      "13.285148691",
      "249.213135600"
    ],
    [
      "534.833377318",
      "310.850537961"
    ],
    [
      "209.846892557",
      "22.300545012"
    ],
    [
      "524.218536637",
      "200.123706040"
    ],
    [
      "488.974206601",
      "421.589984063"
    ],
    [
      "456.519896002",
      "91.440539386"
    ],
    [
      "336.106526770",
      "25.327728998"
    ],
    [
      "447.328873015",
      "298.150593204"
    ],
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      "456.405076047"
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    ]
  ],
  "radius": "275.155438841"
}
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": 58
}

当前第一名的答案

{
  "centers": [
    [
      "26.885547582",
      "163.664543538"
    ],
    [
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    ],
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      "507.419936187"
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      "238.542784537",
      "541.378639260"
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      "13.285148691",
      "249.213135600"
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  ],
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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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