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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 = 68

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 = 68
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[347.98679458815, 347.98679458825]见来源署名packomania.com ↗
本站纪录347.986794592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

695.973589184y0
0x695.973589184
已验证构造r = 347.986794592

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 = 68 的当前纪录构型，347.986794592
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）＋

从实时页面的 68 圆九位整数证书出发，将容器半径目标降至 347.986794592，在每个圆心 ±16 个整数单位窗口内建立线性化修复模型并由 HiGHS 求解；独立整数检查确认全部 68 个包含约束与 2278 个成对分离约束。该解严格领先 3e-9；更低目标可扩大窗口或进行全局重启。

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
    347.986794595→347.986794592
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    347.986794596→347.986794595
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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    ],
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      "297.317069556",
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    ],
    [
      "334.768501496",
      "571.959951918"
    ],
    [
      "602.410489715",
      "368.177898877"
    ],
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      "504.805651114",
      "74.769358736"
    ],
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      "73.946243694",
      "434.808089871"
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    [
      "370.019721377",
      "432.859335797"
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      "149.181815574",
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    ],
    [
      "80.891556669",
      "555.047506624"
    ],
    [
      "256.404388672",
      "23.948423393"
    ],
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  "radius": "347.986794592"
}
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": 68
}

当前第一名的答案

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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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