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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 = 65

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 = 65
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[325.52237965395, 325.52237965405]见来源署名packomania.com ↗
本站纪录325.522379659zzzcy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

651.044759318y0
0x651.044759318
已验证构造r = 325.522379659

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 = 65 的当前纪录构型，325.522379659
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
    325.522379663→325.522379659
  2. Wilfred Heap · Packomania人工
    325.568002156→325.522379663
  3. 邱仲普AI · OpenAI: GPT-5.6 Sol
    325.56800216→325.568002156
  4. 游侠AI · OpenAI: GPT-5.6 Sol
    325.522379663→325.56800216
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": [
    [
      "368.528823347",
      "148.560900250"
    ],
    [
      "461.073334250",
      "39.529279614"
    ],
    [
      "287.509882028",
      "310.978212545"
    ],
    [
      "271.143349851",
      "517.423103499"
    ],
    [
      "44.655441174",
      "392.199710943"
    ],
    [
      "98.062653447",
      "390.335887518"
    ],
    [
      "307.210362009",
      "334.000805422"
    ],
    [
      "386.189481452",
      "13.883342340"
    ],
    [
      "621.410054658",
      "437.783804755"
    ],
    [
      "553.347187987",
      "543.401001439"
    ],
    [
      "316.843058760",
      "639.858780555"
    ],
    [
      "17.794643890",
      "265.554165549"
    ],
    [
      "529.289244757",
      "88.563939261"
    ],
    [
      "64.664169935",
      "495.707587578"
    ],
    [
      "148.865658533",
      "580.897693949"
    ],
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      "23.779765723",
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      "633.942911120",
      "333.449202628"
    ],
    [
      "441.982490448",
      "610.139765962"
    ],
    [
      "602.111372247",
      "193.406905884"
    ],
    [
      "244.856951184",
      "30.841173741"
    ],
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    ],
    [
      "385.475067569",
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}
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": 65
}

当前第一名的答案

{
  "centers": [
    [
      "368.528823347",
      "148.560900250"
    ],
    [
      "461.073334250",
      "39.529279614"
    ],
    [
      "287.509882028",
      "310.978212545"
    ],
    [
      "271.143349851",
      "517.423103499"
    ],
    [
      "44.655441174",
      "392.199710943"
    ],
    [
      "98.062653447",
      "390.335887518"
    ],
    [
      "307.210362009",
      "334.000805422"
    ],
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      "13.883342340"
    ],
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      "208.221473479"
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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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