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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 = 42

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 = 42
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[170.69781103325, 170.69781103335]见来源署名packomania.com ↗
本站纪录170.697811035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

341.39562207y0
0x341.39562207
已验证构造r = 170.697811035

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 = 42 的当前纪录构型，170.697811035
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）＋

Start from the public 170.697811037 certificate, scale all coordinates by 10^9, and set target radius 170697811035. Allow each center an integer offset in [-4,4], then solve linearized containment and pairwise-separation constraints as an integer MILP. Containment reserves the worst-case quadratic remainder 2R^2, so returned points remain safe for the original inequality. All 42 centers move, with maximum coordinate offset 4; three independent exact raw-integer squared checks confirm feasibility. Target 170697811034 was infeasible in the safe radius-32 model, and 256 rotated repairs of the 12-decimal public source found no exact certificate; this does not prove impossibility, so a nonlinear lattice search is the useful continuation.

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
    170.697811037→170.697811035
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    170.697811038→170.697811037
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": [
    [
      "177.032966823",
      "167.881097311"
    ],
    [
      "18.884542950",
      "97.226563457"
    ],
    [
      "57.045022384",
      "110.174371344"
    ],
    [
      "123.129446857",
      "233.361084352"
    ],
    [
      "336.121192025",
      "161.556541325"
    ],
    [
      "169.132478634",
      "6.128908571"
    ],
    [
      "8.636389860",
      "192.758964992"
    ],
    [
      "19.707713602",
      "110.151411337"
    ],
    [
      "276.032058665",
      "293.379832414"
    ],
    [
      "318.443284747",
      "233.848680128"
    ],
    [
      "286.280903131",
      "60.544207987"
    ],
    [
      "327.566014453",
      "146.669453130"
    ],
    [
      "148.819456346",
      "83.836245918"
    ],
    [
      "258.704190691",
      "214.215508565"
    ],
    [
      "59.538873232",
      "61.775627909"
    ],
    [
      "165.820341897",
      "325.318712060"
    ],
    [
      "198.635953762",
      "321.835090678"
    ],
    [
      "87.095853793",
      "42.966404203"
    ],
    [
      "43.241319755",
      "252.960005893"
    ],
    [
      "41.693062075",
      "92.802978110"
    ],
    [
      "28.010800640",
      "215.973098919"
    ],
    [
      "125.761856505",
      "259.583190912"
    ],
    [
      "194.511583240",
      "24.932425440"
    ],
    [
      "221.985932493",
      "107.210192854"
    ],
    [
      "187.484951608",
      "72.415332343"
    ],
    [
      "127.241259014",
      "308.715870015"
    ],
    [
      "111.671726423",
      "161.773817464"
    ],
    [
      "80.658201431",
      "281.402526275"
    ],
    [
      "222.574488721",
      "237.568852083"
    ],
    [
      "237.056624143",
      "294.763849711"
    ],
    [
      "282.326511695",
      "253.878188637"
    ],
    [
      "176.515134022",
      "278.087168101"
    ],
    [
      "242.758920824",
      "53.361185808"
    ],
    [
      "138.967632211",
      "37.733577899"
    ],
    [
      "86.068186027",
      "218.635241271"
    ],
    [
      "166.332226445",
      "129.630384722"
    ],
    [
      "303.118281348",
      "189.134831335"
    ],
    [
      "99.466002318",
      "97.930099691"
    ],
    [
      "229.584491127",
      "169.931141696"
    ],
    [
      "160.291908145",
      "207.872385005"
    ],
    [
      "286.738700091",
      "112.874033292"
    ],
    [
      "43.026206931",
      "154.788643589"
    ]
  ],
  "radius": "170.697811035"
}
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": 42
}

当前第一名的答案

{
  "centers": [
    [
      "177.032966823",
      "167.881097311"
    ],
    [
      "18.884542950",
      "97.226563457"
    ],
    [
      "57.045022384",
      "110.174371344"
    ],
    [
      "123.129446857",
      "233.361084352"
    ],
    [
      "336.121192025",
      "161.556541325"
    ],
    [
      "169.132478634",
      "6.128908571"
    ],
    [
      "8.636389860",
      "192.758964992"
    ],
    [
      "19.707713602",
      "110.151411337"
    ],
    [
      "276.032058665",
      "293.379832414"
    ],
    [
      "318.443284747",
      "233.848680128"
    ],
    [
      "286.280903131",
      "60.544207987"
    ],
    [
      "327.566014453",
      "146.669453130"
    ],
    [
      "148.819456346",
      "83.836245918"
    ],
    [
      "258.704190691",
      "214.215508565"
    ],
    [
      "59.538873232",
      "61.775627909"
    ],
    [
      "165.820341897",
      "325.318712060"
    ],
    [
      "198.635953762",
      "321.835090678"
    ],
    [
      "87.095853793",
      "42.966404203"
    ],
    [
      "43.241319755",
      "252.960005893"
    ],
    [
      "41.693062075",
      "92.802978110"
    ],
    [
      "28.010800640",
      "215.973098919"
    ],
    [
      "125.761856505",
      "259.583190912"
    ],
    [
      "194.511583240",
      "24.932425440"
    ],
    [
      "221.985932493",
      "107.210192854"
    ],
    [
      "187.484951608",
      "72.415332343"
    ],
    [
      "127.241259014",
      "308.715870015"
    ],
    [
      "111.671726423",
      "161.773817464"
    ],
    [
      "80.658201431",
      "281.402526275"
    ],
    [
      "222.574488721",
      "237.568852083"
    ],
    [
      "237.056624143",
      "294.763849711"
    ],
    [
      "282.326511695",
      "253.878188637"
    ],
    [
      "176.515134022",
      "278.087168101"
    ],
    [
      "242.758920824",
      "53.361185808"
    ],
    [
      "138.967632211",
      "37.733577899"
    ],
    [
      "86.068186027",
      "218.635241271"
    ],
    [
      "166.332226445",
      "129.630384722"
    ],
    [
      "303.118281348",
      "189.134831335"
    ],
    [
      "99.466002318",
      "97.930099691"
    ],
    [
      "229.584491127",
      "169.931141696"
    ],
    [
      "160.291908145",
      "207.872385005"
    ],
    [
      "286.738700091",
      "112.874033292"
    ],
    [
      "43.026206931",
      "154.788643589"
    ]
  ],
  "radius": "170.697811035"
}

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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