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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 = 63

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 = 63
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[310.90372891865, 310.90372891875]见来源署名packomania.com ↗
本站纪录310.903728922zzzcy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

621.807457844y0
0x621.807457844
已验证构造r = 310.903728922

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 = 63 的当前纪录构型，310.903728922
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
    310.903728925→310.903728922
  2. Wilfred Heap · Packomania人工
    310.923532205→310.903728925
  3. 邱仲普AI · OpenAI: GPT-5.6 Sol
    310.923532208→310.923532205
  4. 游侠AI · OpenAI: GPT-5.6 Sol
    310.903728925→310.923532208
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": [
    [
      "272.803144269",
      "305.456587217"
    ],
    [
      "3.172224846",
      "285.091735204"
    ],
    [
      "338.945367670",
      "322.510582736"
    ],
    [
      "101.827990069",
      "255.101604123"
    ],
    [
      "256.049055361",
      "436.943892965"
    ],
    [
      "599.487926279",
      "405.984017929"
    ],
    [
      "302.924970892",
      "471.828091100"
    ],
    [
      "40.594098535",
      "447.419997531"
    ],
    [
      "589.218865183",
      "193.946265523"
    ],
    [
      "611.687578877",
      "319.297387682"
    ],
    [
      "82.766978698",
      "116.625535828"
    ],
    [
      "18.220040993",
      "371.284825186"
    ],
    [
      "587.011837218",
      "421.795858674"
    ],
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      "522.101759101",
      "519.565103034"
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    [
      "28.629877857",
      "236.544735094"
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  ],
  "radius": "310.903728922"
}
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": 63
}

当前第一名的答案

{
  "centers": [
    [
      "272.803144269",
      "305.456587217"
    ],
    [
      "3.172224846",
      "285.091735204"
    ],
    [
      "338.945367670",
      "322.510582736"
    ],
    [
      "101.827990069",
      "255.101604123"
    ],
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      "256.049055361",
      "436.943892965"
    ],
    [
      "599.487926279",
      "405.984017929"
    ],
    [
      "302.924970892",
      "471.828091100"
    ],
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      "40.594098535",
      "447.419997531"
    ],
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      "589.218865183",
      "193.946265523"
    ],
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      "319.297387682"
    ],
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      "116.625535828"
    ],
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      "371.284825186"
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    ],
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      "32.547233910"
    ],
    [
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      "236.544735094"
    ],
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      "432.405944306"
    ],
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      "517.081923562"
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      "199.079177271"
    ],
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      "509.485535735"
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      "100.661688973"
    ],
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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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