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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 = 59

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 = 59
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[282.23016971335, 282.23016971345]见来源署名packomania.com ↗
本站纪录282.230169715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

564.46033943y0
0x564.46033943
已验证构造r = 282.230169715

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 = 59 的当前纪录构型，282.230169715
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）＋

从 Packomania ccin59 公开原始坐标出发，使用黄金角旋转 k=1548 和高精度缩放，在目标半径 282.230169715 下建立每个中心 ±1 个整数单位的线性化修整模型；得到 59 个中心的可行九位网格构型。随后用整数平方复核 59 个包含约束和 1711 个两两不重叠约束。更低目标仍需更广的旋转与整数搜索，可从本证书继续。

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
    282.230169719→282.230169715
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    282.23016972→282.230169719
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": [
    [
      "122.317762826",
      "293.067410899"
    ],
    [
      "279.615688882",
      "542.495936493"
    ],
    [
      "290.988878463",
      "501.103165714"
    ],
    [
      "281.286850318",
      "4.012731535"
    ],
    [
      "446.190958654",
      "58.700056903"
    ],
    [
      "267.187944930",
      "558.033746829"
    ],
    [
      "383.591286397",
      "234.786348362"
    ],
    [
      "280.869001164",
      "554.883955141"
    ],
    [
      "20.004561545",
      "205.557038296"
    ],
    [
      "58.075892742",
      "127.799041376"
    ],
    [
      "371.109005867",
      "538.452244369"
    ],
    [
      "174.982076820",
      "527.658475741"
    ],
    [
      "457.707254744",
      "486.370309503"
    ],
    [
      "141.428460650",
      "53.927075847"
    ],
    [
      "515.372735459",
      "151.629604463"
    ],
    [
      "547.745550806",
      "262.734383748"
    ],
    [
      "523.847197306",
      "391.629689390"
    ],
    [
      "175.969725045",
      "168.114832015"
    ],
    [
      "441.305031062",
      "197.679847779"
    ],
    [
      "391.316275333",
      "424.394651764"
    ],
    [
      "372.457037889",
      "37.138794581"
    ],
    [
      "58.525122953",
      "415.180511132"
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    ],
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      "26.990551740"
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      "197.062669496"
    ],
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      "353.818747178"
    ],
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      "364.141551382"
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      "154.877152154",
      "312.276496905"
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      "363.320256342"
    ],
    [
      "317.880014585",
      "35.330108811"
    ],
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      "306.774327945",
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      "133.439948829"
    ],
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      "57.354244197",
      "179.499575734"
    ],
    [
      "88.945738664",
      "309.268388655"
    ],
    [
      "324.164393175",
      "523.848374826"
    ],
    [
      "272.801787678",
      "89.928449343"
    ],
    [
      "40.929528102",
      "251.653776591"
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    [
      "382.987609031",
      "187.777164057"
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    [
      "210.091344572",
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    [
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      "104.760643724"
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      "213.972017319"
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    [
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      "424.163987060"
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      "387.374007474"
    ],
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      "237.413231262",
      "509.075565114"
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    [
      "452.822359293",
      "127.620222099"
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    [
      "288.486275412",
      "179.566593258"
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      "119.375262256"
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      "394.238913497",
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      "236.513517269",
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    ],
    [
      "311.858300688",
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    ]
  ],
  "radius": "282.230169715"
}
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": 59
}

当前第一名的答案

{
  "centers": [
    [
      "122.317762826",
      "293.067410899"
    ],
    [
      "279.615688882",
      "542.495936493"
    ],
    [
      "290.988878463",
      "501.103165714"
    ],
    [
      "281.286850318",
      "4.012731535"
    ],
    [
      "446.190958654",
      "58.700056903"
    ],
    [
      "267.187944930",
      "558.033746829"
    ],
    [
      "383.591286397",
      "234.786348362"
    ],
    [
      "280.869001164",
      "554.883955141"
    ],
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      "20.004561545",
      "205.557038296"
    ],
    [
      "58.075892742",
      "127.799041376"
    ],
    [
      "371.109005867",
      "538.452244369"
    ],
    [
      "174.982076820",
      "527.658475741"
    ],
    [
      "457.707254744",
      "486.370309503"
    ],
    [
      "141.428460650",
      "53.927075847"
    ],
    [
      "515.372735459",
      "151.629604463"
    ],
    [
      "547.745550806",
      "262.734383748"
    ],
    [
      "523.847197306",
      "391.629689390"
    ],
    [
      "175.969725045",
      "168.114832015"
    ],
    [
      "441.305031062",
      "197.679847779"
    ],
    [
      "391.316275333",
      "424.394651764"
    ],
    [
      "372.457037889",
      "37.138794581"
    ],
    [
      "58.525122953",
      "415.180511132"
    ],
    [
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      "451.428702113"
    ],
    [
      "25.817568629",
      "312.814455797"
    ],
    [
      "413.838914019",
      "61.217729716"
    ],
    [
      "260.065162011",
      "26.990551740"
    ],
    [
      "210.422920908",
      "197.062669496"
    ],
    [
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      "353.818747178"
    ],
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      "278.308760373"
    ],
    [
      "311.858300688",
      "428.094507834"
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  ],
  "radius": "282.230169715"
}

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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