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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 = 64

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 = 64
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[318.19092210195, 318.19092210205]见来源署名packomania.com ↗
本站纪录318.190922106zzzcy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

636.381844212y0
0x636.381844212
已验证构造r = 318.190922106

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 = 64 的当前纪录构型，318.190922106
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
    318.19092211→318.190922106
  2. Wilfred Heap · Packomania人工
    318.223159147→318.19092211
  3. 邱仲普AI · OpenAI: GPT-5.6 Sol
    318.223159149→318.223159147
  4. 游侠AI · OpenAI: GPT-5.6 Sol
    318.19092211→318.223159149
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": [
    [
      "462.267594545",
      "37.811671633"
    ],
    [
      "623.350595391",
      "400.936104728"
    ],
    [
      "478.607589532",
      "484.801774774"
    ],
    [
      "581.745449155",
      "422.540841820"
    ],
    [
      "195.830335027",
      "345.916997607"
    ],
    [
      "559.284908199",
      "121.369388922"
    ],
    [
      "246.565012441",
      "15.397826628"
    ],
    [
      "442.950776822",
      "34.247393704"
    ],
    [
      "28.664807296",
      "209.767243540"
    ],
    [
      "604.754763899",
      "204.938006787"
    ],
    [
      "150.112101490",
      "62.072794641"
    ],
    [
      "19.021569939",
      "379.550356588"
    ],
    [
      "90.301769332",
      "115.193303935"
    ],
    [
      "533.648495423",
      "532.805919973"
    ],
    [
      "599.688142372",
      "430.787176799"
    ],
    [
      "286.695697941",
      "294.269654677"
    ],
    [
      "537.297164969",
      "111.549606209"
    ],
    [
      "272.546555461",
      "21.490430856"
    ],
    [
      "415.198683919",
      "35.163181384"
    ],
    [
      "377.029607389",
      "442.961125058"
    ],
    [
      "31.474082571",
      "239.986714361"
    ],
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      "165.620955570",
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    ],
    [
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    ],
    [
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      "486.810246100",
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    [
      "470.168124519",
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      "537.565683174",
      "463.601292043"
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      "250.629680422",
      "508.854655583"
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      "355.608935734",
      "270.187178540"
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  ],
  "radius": "318.190922106"
}
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": 64
}

当前第一名的答案

{
  "centers": [
    [
      "462.267594545",
      "37.811671633"
    ],
    [
      "623.350595391",
      "400.936104728"
    ],
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      "478.607589532",
      "484.801774774"
    ],
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      "581.745449155",
      "422.540841820"
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      "195.830335027",
      "345.916997607"
    ],
    [
      "559.284908199",
      "121.369388922"
    ],
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      "246.565012441",
      "15.397826628"
    ],
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      "442.950776822",
      "34.247393704"
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      "209.767243540"
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      "204.938006787"
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      "62.072794641"
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  ],
  "radius": "318.190922106"
}

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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