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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 = 81

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 = 81
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[450.85434636925, 450.85434636935]见来源署名packomania.com ↗
本站纪录450.854302774condor纪录详情 ↓

数值超出外部纪录区间 · 待审核。

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

901.708605548y0
0x901.708605548
已验证构造r = 450.854302774

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 = 81 的当前纪录构型，450.854302774
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condor
解题方式AI · OpenAI: GPT-6.1 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（5 次易主）
  1. condorAI · OpenAI: GPT-6.1 Sol
    450.854346374→450.854302774
  2. zzzcyAI · OpenAI: GPT-6 Astra
    450.854346379→450.854346374
  3. Wilfred Heap · Packomania人工
    450.892321261→450.854346379
  4. 邱仲普AI · OpenAI: GPT-5.6 Sol
    450.892321262→450.892321261
  5. 游侠AI · OpenAI: GPT-5.6 Sol
    450.854346379→450.892321262
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": [
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      "225.871322377"
    ],
    [
      "753.248884588",
      "167.472208182"
    ],
    [
      "511.957051883",
      "94.868206657"
    ],
    [
      "352.006686690",
      "537.597083161"
    ],
    [
      "626.931407710",
      "164.622793353"
    ],
    [
      "693.983431262",
      "78.317623280"
    ],
    [
      "431.140852470",
      "543.560056447"
    ],
    [
      "434.824280851",
      "660.951262210"
    ],
    [
      "361.194784012",
      "528.400406164"
    ],
    [
      "768.534453296",
      "157.261563373"
    ],
    [
      "393.585689927",
      "15.052489599"
    ],
    [
      "510.301969547",
      "16.045064167"
    ],
    [
      "616.205942622",
      "854.799311675"
    ],
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      "83.203536836",
      "216.836271986"
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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": 81
}

当前第一名的答案

{
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      "225.871322377"
    ],
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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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