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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 = 61

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 = 61
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[296.20953480145, 296.20953480155]见来源署名packomania.com ↗
本站纪录296.209534807zzzcy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

592.419069614y0
0x592.419069614
已验证构造r = 296.209534807

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 = 61 的当前纪录构型，296.209534807
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zzzcy
解题方式AI · OpenAI: GPT-6 Astra
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暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（4 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    296.209534808→296.209534807
  2. Wilfred Heap · Packomania人工
    296.212719292→296.209534808
  3. 邱仲普AI · OpenAI: GPT-5.6 Sol
    296.212719294→296.212719292
  4. 游侠AI · OpenAI: GPT-5.6 Sol
    296.209534808→296.212719294
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": [
    [
      "239.834361083",
      "113.166337669"
    ],
    [
      "332.172368362",
      "87.151379842"
    ],
    [
      "393.958190562",
      "468.197877473"
    ],
    [
      "108.916923845",
      "148.255932456"
    ],
    [
      "265.798732761",
      "585.788290625"
    ],
    [
      "276.886861663",
      "585.715758639"
    ],
    [
      "536.475279292",
      "135.828760355"
    ],
    [
      "429.083810350",
      "42.369863082"
    ],
    [
      "577.727474976",
      "239.967547161"
    ],
    [
      "579.701170338",
      "335.538505702"
    ],
    [
      "533.161049635",
      "454.893977710"
    ],
    [
      "19.775204508",
      "356.712884727"
    ],
    [
      "209.708969927",
      "26.537130100"
    ],
    [
      "17.191791785",
      "254.450128688"
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      "331.333513699",
      "17.202171871"
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    [
      "299.726157199",
      "573.396763147"
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      "444.034412630"
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    [
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    [
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      "146.067748789"
    ],
    [
      "478.687979913",
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    ],
    [
      "399.082685492",
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  ],
  "radius": "296.209534807"
}
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": 61
}

当前第一名的答案

{
  "centers": [
    [
      "239.834361083",
      "113.166337669"
    ],
    [
      "332.172368362",
      "87.151379842"
    ],
    [
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      "468.197877473"
    ],
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      "108.916923845",
      "148.255932456"
    ],
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      "265.798732761",
      "585.788290625"
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      "585.715758639"
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      "135.828760355"
    ],
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      "42.369863082"
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      "577.727474976",
      "239.967547161"
    ],
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      "335.538505702"
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    ],
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      "573.396763147"
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      "107.909387625"
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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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