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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 = 67

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 = 67
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[340.43598803865, 340.43598803875]见来源署名packomania.com ↗
本站纪录340.435988043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

680.871976086y0
0x680.871976086
已验证构造r = 340.435988043

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 = 67 的当前纪录构型，340.435988043
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 67 圆九位整数证书出发，将容器半径目标降至 340.435988043，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 67 个包含约束与 2211 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采用全局重启。

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
    340.435988046→340.435988043
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    340.435988047→340.435988046
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": [
    [
      "233.209050409",
      "25.444528975"
    ],
    [
      "329.692345277",
      "582.267486352"
    ],
    [
      "17.533724386",
      "434.219774782"
    ],
    [
      "329.850632637",
      "670.118493592"
    ],
    [
      "7.954383313",
      "360.340108934"
    ],
    [
      "175.616557538",
      "187.797606163"
    ],
    [
      "186.358110598",
      "373.516889269"
    ],
    [
      "235.516058679",
      "418.252504082"
    ],
    [
      "564.883960984",
      "584.007603237"
    ],
    [
      "658.365347393",
      "424.793284547"
    ],
    [
      "116.363694048",
      "98.987543614"
    ],
    [
      "155.790884899",
      "611.993465352"
    ],
    [
      "77.975168725",
      "536.213015777"
    ],
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      "32.725771638",
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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": 67
}

当前第一名的答案

{
  "centers": [
    [
      "233.209050409",
      "25.444528975"
    ],
    [
      "329.692345277",
      "582.267486352"
    ],
    [
      "17.533724386",
      "434.219774782"
    ],
    [
      "329.850632637",
      "670.118493592"
    ],
    [
      "7.954383313",
      "360.340108934"
    ],
    [
      "175.616557538",
      "187.797606163"
    ],
    [
      "186.358110598",
      "373.516889269"
    ],
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      "235.516058679",
      "418.252504082"
    ],
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      "564.883960984",
      "584.007603237"
    ],
    [
      "658.365347393",
      "424.793284547"
    ],
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      "116.363694048",
      "98.987543614"
    ],
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      "611.993465352"
    ],
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      "331.842223195"
    ],
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      "470.773715912"
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      "30.707398642"
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      "27.002841844"
    ],
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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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