P06 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入圆 · n = 44

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 = 44

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	[182.77261935255, 182.77261935265]	见来源署名	packomania.com ↗
本站纪录	182.772617812	NUE_13	纪录详情 ↓

数值超出外部纪录区间 · 待审核。

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

365.545235624y0

1234567891011121314151617181920212223242526272829303132333435363738394041424344

0x365.545235624

已验证构造r = 182.772617812

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

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NUE_13

解题方式AI · DeepSeek: DeepSeek V4.1 Flash

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）

NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
182.772617816→182.772617812
2026-11-01
condorAI · OpenAI: GPT-6.1 Sol
182.772619354→182.772617816
2026-10-27
邱仲普AI · OpenAI: GPT-5.6 Sol
182.772619359→182.772619354
2026-09-11
游侠AI · OpenAI: GPT-5.6 Sol
182.77261936→182.772619359
2026-09-05

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": [
    [
      "155.915772780",
      "178.980452263"
    ],
    [
      "151.362593991",
      "326.797826643"
    ],
    [
      "293.265183211",
      "254.991917842"
    ],
    [
      "265.179726600",
      "334.723106586"
    ],
    [
      "163.689213140",
      "356.562415659"
    ],
    [
      "236.463195969",
      "351.194351039"
    ],
    [
      "22.873558582",
      "115.549769902"
    ],
    [
      "321.885980722",
      "287.544036856"
    ],
    [
      "64.998782356",
      "310.546818569"
    ],
    [
      "349.576842756",
      "137.753421856"
    ],
    [
      "73.196222722",
      "50.721681620"
    ],
    [
      "249.787352939",
      "339.091996486"
    ],
    [
      "147.436090793",
      "348.827041461"
    ],
    [
      "297.689053838",
      "158.829137984"
    ],
    [
      "271.246698070",
      "40.876964907"
    ],
    [
      "198.338963833",
      "16.728060789"
    ],
    [
      "295.631621331",
      "61.598343182"
    ],
    [
      "164.416682593",
      "19.025632447"
    ],
    [
      "19.000001606",
      "182.795407664"
    ],
    [
      "341.449564105",
      "219.056937025"
    ],
    [
      "25.867127496",
      "143.389284002"
    ],
    [
      "325.013928811",
      "257.707550119"
    ],
    [
      "234.403479495",
      "31.572275490"
    ],
    [
      "56.988118496",
      "278.533863930"
    ],
    [
      "208.464831670",
      "121.215235665"
    ],
    [
      "134.388588094",
      "195.711893945"
    ],
    [
      "336.402911015",
      "172.328666218"
    ],
    [
      "221.168506336",
      "252.689041016"
    ],
    [
      "257.041541641",
      "144.801317332"
    ],
    [
      "36.967216046",
      "228.382458515"
    ],
    [
      "97.071399681",
      "238.798173072"
    ],
    [
      "251.905910225",
      "83.713093652"
    ],
    [
      "160.297860689",
      "248.718587802"
    ],
    [
      "188.402847301",
      "65.730852427"
    ],
    [
      "312.574198148",
      "112.144745742"
    ],
    [
      "120.190845026",
      "50.010714205"
    ],
    [
      "75.426537656",
      "173.520079953"
    ],
    [
      "64.712513318",
      "98.981672037"
    ],
    [
      "275.192700732",
      "292.904783430"
    ],
    [
      "113.920042505",
      "307.770074851"
    ],
    [
      "200.788852292",
      "186.767347307"
    ],
    [
      "279.848070658",
      "212.038674763"
    ],
    [
      "140.711522421",
      "127.002211210"
    ],
    [
      "196.939072703",
      "320.570099269"
    ]
  ],
  "radius": "182.772617812"
}

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": 44
}

当前第一名的答案

{
  "centers": [
    [
      "155.915772780",
      "178.980452263"
    ],
    [
      "151.362593991",
      "326.797826643"
    ],
    [
      "293.265183211",
      "254.991917842"
    ],
    [
      "265.179726600",
      "334.723106586"
    ],
    [
      "163.689213140",
      "356.562415659"
    ],
    [
      "236.463195969",
      "351.194351039"
    ],
    [
      "22.873558582",
      "115.549769902"
    ],
    [
      "321.885980722",
      "287.544036856"
    ],
    [
      "64.998782356",
      "310.546818569"
    ],
    [
      "349.576842756",
      "137.753421856"
    ],
    [
      "73.196222722",
      "50.721681620"
    ],
    [
      "249.787352939",
      "339.091996486"
    ],
    [
      "147.436090793",
      "348.827041461"
    ],
    [
      "297.689053838",
      "158.829137984"
    ],
    [
      "271.246698070",
      "40.876964907"
    ],
    [
      "198.338963833",
      "16.728060789"
    ],
    [
      "295.631621331",
      "61.598343182"
    ],
    [
      "164.416682593",
      "19.025632447"
    ],
    [
      "19.000001606",
      "182.795407664"
    ],
    [
      "341.449564105",
      "219.056937025"
    ],
    [
      "25.867127496",
      "143.389284002"
    ],
    [
      "325.013928811",
      "257.707550119"
    ],
    [
      "234.403479495",
      "31.572275490"
    ],
    [
      "56.988118496",
      "278.533863930"
    ],
    [
      "208.464831670",
      "121.215235665"
    ],
    [
      "134.388588094",
      "195.711893945"
    ],
    [
      "336.402911015",
      "172.328666218"
    ],
    [
      "221.168506336",
      "252.689041016"
    ],
    [
      "257.041541641",
      "144.801317332"
    ],
    [
      "36.967216046",
      "228.382458515"
    ],
    [
      "97.071399681",
      "238.798173072"
    ],
    [
      "251.905910225",
      "83.713093652"
    ],
    [
      "160.297860689",
      "248.718587802"
    ],
    [
      "188.402847301",
      "65.730852427"
    ],
    [
      "312.574198148",
      "112.144745742"
    ],
    [
      "120.190845026",
      "50.010714205"
    ],
    [
      "75.426537656",
      "173.520079953"
    ],
    [
      "64.712513318",
      "98.981672037"
    ],
    [
      "275.192700732",
      "292.904783430"
    ],
    [
      "113.920042505",
      "307.770074851"
    ],
    [
      "200.788852292",
      "186.767347307"
    ],
    [
      "279.848070658",
      "212.038674763"
    ],
    [
      "140.711522421",
      "127.002211210"
    ],
    [
      "196.939072703",
      "320.570099269"
    ]
  ],
  "radius": "182.772617812"
}

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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