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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 = 62

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 = 62
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[303.64963780365, 303.64963780375]见来源署名packomania.com ↗
本站纪录303.649637807zzzcy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

607.299275614y0
0x607.299275614
已验证构造r = 303.649637807

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 = 62 的当前纪录构型，303.649637807
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
    303.649637811→303.649637807
  2. Wilfred Heap · Packomania人工
    303.668337703→303.649637811
  3. 邱仲普AI · OpenAI: GPT-5.6 Sol
    303.668337708→303.668337703
  4. 游侠AI · OpenAI: GPT-5.6 Sol
    303.649637811→303.668337708
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": [
    [
      "562.226668223",
      "460.589843001"
    ],
    [
      "371.714790163",
      "85.045747201"
    ],
    [
      "31.667900361",
      "182.367630105"
    ],
    [
      "271.921647560",
      "455.728348147"
    ],
    [
      "102.923325900",
      "294.238421049"
    ],
    [
      "207.576348428",
      "185.023382261"
    ],
    [
      "122.884468903",
      "187.935134247"
    ],
    [
      "38.611305030",
      "411.432201644"
    ],
    [
      "218.991096434",
      "468.748233234"
    ],
    [
      "196.530151633",
      "30.297312053"
    ],
    [
      "126.030492680",
      "71.120715719"
    ],
    [
      "393.618344579",
      "26.242725193"
    ],
    [
      "292.794441669",
      "13.202780925"
    ],
    [
      "82.381587620",
      "490.553528159"
    ],
    [
      "474.643150486",
      "536.200346315"
    ],
    [
      "589.988802810",
      "276.223432895"
    ],
    [
      "57.052829326",
      "157.505774092"
    ],
    [
      "18.505407889",
      "286.664829204"
    ],
    [
      "516.333654800",
      "114.464740899"
    ],
    [
      "569.755831099",
      "401.750264204"
    ],
    [
      "400.884003447",
      "230.037371967"
    ],
    [
      "550.517903017",
      "439.232004318"
    ],
    [
      "483.534961308",
      "88.230303508"
    ],
    [
      "461.341019480",
      "411.620213419"
    ],
    [
      "537.934294205",
      "152.797655963"
    ],
    [
      "341.285566405",
      "578.736639050"
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    [
      "476.906676004",
      "361.712541250"
    ],
    [
      "323.035991927",
      "527.913878092"
    ],
    [
      "513.770907921",
      "403.867355989"
    ],
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      "421.357440563",
      "447.915025958"
    ],
    [
      "219.341169513",
      "562.936944032"
    ],
    [
      "372.428543053",
      "119.523868856"
    ],
    [
      "282.468594847",
      "573.469188366"
    ],
    [
      "171.920294312",
      "68.366176358"
    ],
    [
      "319.774807419",
      "362.361041643"
    ],
    [
      "267.466910756",
      "314.351834747"
    ],
    [
      "74.600967737",
      "440.096335442"
    ],
    [
      "165.467239403",
      "203.717376354"
    ],
    [
      "139.510806313",
      "399.920984551"
    ],
    [
      "423.586753884",
      "68.859902719"
    ],
    [
      "353.190255184",
      "189.943243178"
    ],
    [
      "258.326205166",
      "501.218494640"
    ],
    [
      "209.405714415",
      "135.625698212"
    ],
    [
      "295.984435394",
      "127.074378211"
    ],
    [
      "249.052759586",
      "51.290380598"
    ],
    [
      "340.012353880",
      "48.578885386"
    ],
    [
      "492.698776959",
      "476.887663416"
    ],
    [
      "402.396833335",
      "370.273110497"
    ],
    [
      "117.519458879",
      "131.052292081"
    ],
    [
      "545.569764093",
      "227.407974282"
    ],
    [
      "350.118234052",
      "281.891939124"
    ],
    [
      "550.967607112",
      "331.913165219"
    ],
    [
      "338.358722776",
      "448.376378009"
    ],
    [
      "405.807516218",
      "531.440571638"
    ],
    [
      "456.117055806",
      "282.391726247"
    ],
    [
      "59.992902180",
      "347.941152670"
    ],
    [
      "260.449880331",
      "221.616936871"
    ],
    [
      "236.439368622",
      "403.643055175"
    ],
    [
      "72.626158076",
      "231.893081018"
    ],
    [
      "156.077631774",
      "497.524981878"
    ],
    [
      "452.764831457",
      "166.403594431"
    ],
    [
      "170.053036072",
      "303.649637820"
    ]
  ],
  "radius": "303.649637807"
}
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": 62
}

当前第一名的答案

{
  "centers": [
    [
      "562.226668223",
      "460.589843001"
    ],
    [
      "371.714790163",
      "85.045747201"
    ],
    [
      "31.667900361",
      "182.367630105"
    ],
    [
      "271.921647560",
      "455.728348147"
    ],
    [
      "102.923325900",
      "294.238421049"
    ],
    [
      "207.576348428",
      "185.023382261"
    ],
    [
      "122.884468903",
      "187.935134247"
    ],
    [
      "38.611305030",
      "411.432201644"
    ],
    [
      "218.991096434",
      "468.748233234"
    ],
    [
      "196.530151633",
      "30.297312053"
    ],
    [
      "126.030492680",
      "71.120715719"
    ],
    [
      "393.618344579",
      "26.242725193"
    ],
    [
      "292.794441669",
      "13.202780925"
    ],
    [
      "82.381587620",
      "490.553528159"
    ],
    [
      "474.643150486",
      "536.200346315"
    ],
    [
      "589.988802810",
      "276.223432895"
    ],
    [
      "57.052829326",
      "157.505774092"
    ],
    [
      "18.505407889",
      "286.664829204"
    ],
    [
      "516.333654800",
      "114.464740899"
    ],
    [
      "569.755831099",
      "401.750264204"
    ],
    [
      "400.884003447",
      "230.037371967"
    ],
    [
      "550.517903017",
      "439.232004318"
    ],
    [
      "483.534961308",
      "88.230303508"
    ],
    [
      "461.341019480",
      "411.620213419"
    ],
    [
      "537.934294205",
      "152.797655963"
    ],
    [
      "341.285566405",
      "578.736639050"
    ],
    [
      "476.906676004",
      "361.712541250"
    ],
    [
      "323.035991927",
      "527.913878092"
    ],
    [
      "513.770907921",
      "403.867355989"
    ],
    [
      "421.357440563",
      "447.915025958"
    ],
    [
      "219.341169513",
      "562.936944032"
    ],
    [
      "372.428543053",
      "119.523868856"
    ],
    [
      "282.468594847",
      "573.469188366"
    ],
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      "171.920294312",
      "68.366176358"
    ],
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      "362.361041643"
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      "314.351834747"
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      "51.290380598"
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    ],
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      "448.376378009"
    ],
    [
      "405.807516218",
      "531.440571638"
    ],
    [
      "456.117055806",
      "282.391726247"
    ],
    [
      "59.992902180",
      "347.941152670"
    ],
    [
      "260.449880331",
      "221.616936871"
    ],
    [
      "236.439368622",
      "403.643055175"
    ],
    [
      "72.626158076",
      "231.893081018"
    ],
    [
      "156.077631774",
      "497.524981878"
    ],
    [
      "452.764831457",
      "166.403594431"
    ],
    [
      "170.053036072",
      "303.649637820"
    ]
  ],
  "radius": "303.649637807"
}

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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