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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 = 43

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 = 43
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[176.73265624195, 176.73265624205]见来源署名packomania.com ↗
本站纪录176.732656243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

353.465312486y0
0x353.465312486
已验证构造r = 176.732656243

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 = 43 的当前纪录构型，176.732656243
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 176.732656246 certificate and scale every value by 10^9. Set target radius 176732656243, the safe nine-place ceiling of the published best-known interval. Allow each center an integer offset in [-16,16] and solve linearized containment and pairwise-separation constraints as an integer MILP. Containment reserves the worst-case quadratic remainder 2R^2, so returned points satisfy the original quadratic inequality. All 43 centers move, maximum coordinate offset 16; three independent exact raw-integer squared checks confirm feasibility. Target 176732656242 was infeasible in safe neighborhoods through radius 64; this is local, not a global impossibility proof, and rotation-aware nonlinear lattice search is the next step.

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
    176.732656246→176.732656243
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    176.732656247→176.732656246
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": [
    [
      "162.075966799",
      "160.181949175"
    ],
    [
      "151.594667609",
      "345.073407002"
    ],
    [
      "143.270843769",
      "322.566051957"
    ],
    [
      "261.066739817",
      "98.377121570"
    ],
    [
      "300.404004120",
      "57.728586105"
    ],
    [
      "24.652011188",
      "254.325623446"
    ],
    [
      "13.436063415",
      "130.447579659"
    ],
    [
      "10.569560170",
      "205.968969748"
    ],
    [
      "221.693565233",
      "338.264526198"
    ],
    [
      "284.325825484",
      "304.045580894"
    ],
    [
      "333.240859063",
      "122.222605200"
    ],
    [
      "139.091628554",
      "337.107221144"
    ],
    [
      "333.899202180",
      "222.536117918"
    ],
    [
      "42.922318092",
      "84.253702626"
    ],
    [
      "161.642653650",
      "272.777619539"
    ],
    [
      "279.647813251",
      "53.268259590"
    ],
    [
      "30.168240671",
      "113.232499247"
    ],
    [
      "251.169442305",
      "78.632720167"
    ],
    [
      "203.165301463",
      "21.271758892"
    ],
    [
      "108.902160912",
      "35.438107742"
    ],
    [
      "29.966045295",
      "227.849818462"
    ],
    [
      "50.217157581",
      "265.817017981"
    ],
    [
      "243.627430761",
      "38.317271485"
    ],
    [
      "74.665872115",
      "63.117716329"
    ],
    [
      "175.246708881",
      "128.262255048"
    ],
    [
      "73.177563470",
      "113.105896337"
    ],
    [
      "246.097156253",
      "199.345730864"
    ],
    [
      "156.518336666",
      "29.380071206"
    ],
    [
      "245.297204780",
      "307.577565124"
    ],
    [
      "143.828304869",
      "230.030659118"
    ],
    [
      "83.095257514",
      "224.247431204"
    ],
    [
      "297.945343760",
      "97.643932850"
    ],
    [
      "187.862342966",
      "184.899052303"
    ],
    [
      "199.347197038",
      "74.209623058"
    ],
    [
      "182.190911135",
      "318.360172281"
    ],
    [
      "129.633492270",
      "87.459371665"
    ],
    [
      "209.130350899",
      "251.589920033"
    ],
    [
      "238.199419757",
      "134.827316761"
    ],
    [
      "39.249107713",
      "168.560495256"
    ],
    [
      "118.346236732",
      "162.616526098"
    ],
    [
      "289.055585906",
      "252.902728851"
    ],
    [
      "108.264263103",
      "292.771304108"
    ],
    [
      "310.367091992",
      "171.608114526"
    ]
  ],
  "radius": "176.732656243"
}
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": 43
}

当前第一名的答案

{
  "centers": [
    [
      "162.075966799",
      "160.181949175"
    ],
    [
      "151.594667609",
      "345.073407002"
    ],
    [
      "143.270843769",
      "322.566051957"
    ],
    [
      "261.066739817",
      "98.377121570"
    ],
    [
      "300.404004120",
      "57.728586105"
    ],
    [
      "24.652011188",
      "254.325623446"
    ],
    [
      "13.436063415",
      "130.447579659"
    ],
    [
      "10.569560170",
      "205.968969748"
    ],
    [
      "221.693565233",
      "338.264526198"
    ],
    [
      "284.325825484",
      "304.045580894"
    ],
    [
      "333.240859063",
      "122.222605200"
    ],
    [
      "139.091628554",
      "337.107221144"
    ],
    [
      "333.899202180",
      "222.536117918"
    ],
    [
      "42.922318092",
      "84.253702626"
    ],
    [
      "161.642653650",
      "272.777619539"
    ],
    [
      "279.647813251",
      "53.268259590"
    ],
    [
      "30.168240671",
      "113.232499247"
    ],
    [
      "251.169442305",
      "78.632720167"
    ],
    [
      "203.165301463",
      "21.271758892"
    ],
    [
      "108.902160912",
      "35.438107742"
    ],
    [
      "29.966045295",
      "227.849818462"
    ],
    [
      "50.217157581",
      "265.817017981"
    ],
    [
      "243.627430761",
      "38.317271485"
    ],
    [
      "74.665872115",
      "63.117716329"
    ],
    [
      "175.246708881",
      "128.262255048"
    ],
    [
      "73.177563470",
      "113.105896337"
    ],
    [
      "246.097156253",
      "199.345730864"
    ],
    [
      "156.518336666",
      "29.380071206"
    ],
    [
      "245.297204780",
      "307.577565124"
    ],
    [
      "143.828304869",
      "230.030659118"
    ],
    [
      "83.095257514",
      "224.247431204"
    ],
    [
      "297.945343760",
      "97.643932850"
    ],
    [
      "187.862342966",
      "184.899052303"
    ],
    [
      "199.347197038",
      "74.209623058"
    ],
    [
      "182.190911135",
      "318.360172281"
    ],
    [
      "129.633492270",
      "87.459371665"
    ],
    [
      "209.130350899",
      "251.589920033"
    ],
    [
      "238.199419757",
      "134.827316761"
    ],
    [
      "39.249107713",
      "168.560495256"
    ],
    [
      "118.346236732",
      "162.616526098"
    ],
    [
      "289.055585906",
      "252.902728851"
    ],
    [
      "108.264263103",
      "292.771304108"
    ],
    [
      "310.367091992",
      "171.608114526"
    ]
  ],
  "radius": "176.732656243"
}

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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