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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 = 52

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 = 52
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[234.23039659045, 234.23039659055]见来源署名packomania.com ↗
本站纪录234.230396592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

468.460793184y0
0x468.460793184
已验证构造r = 234.230396592

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 = 52 的当前纪录构型，234.230396592
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）＋

使用 Packomania ccin52 的单位圆归一化坐标与公开半径比 234.2303965905。固定九位目标 R=234.230396592，取 golden-angle 序列第 24410 个全局旋转，按 (R,R)+R*Rot(theta)*(x,y) 映射并舍入到 1e-9 网格。直接最近舍入剩 5 个约束违例；随后允许每个坐标移动 ±8 个整数单位，解整数可行性模型。容器线性约束预留最坏二次余量，圆间约束利用位移差平方项非负，因此模型解对原二次约束安全。最终用整数平方检查 52 个容器约束和 1326 个圆对约束，全部通过。目标 234.230396591 的 30000 个测试旋转中最近舍入至少仍有 18 个违例；后续可用更高精度原始坐标或连续非线性重解继续。

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
    234.230396596→234.230396592
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    234.230396597→234.230396596
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": [
    [
      "436.513681401",
      "136.739262757"
    ],
    [
      "438.922767542",
      "343.910223333"
    ],
    [
      "124.154523904",
      "210.819841782"
    ],
    [
      "142.608359208",
      "415.629805157"
    ],
    [
      "244.659256046",
      "131.986862208"
    ],
    [
      "15.101008013",
      "271.267102681"
    ],
    [
      "362.744577391",
      "419.181949015"
    ],
    [
      "223.824618942",
      "459.153910497"
    ],
    [
      "64.367078334",
      "381.990986253"
    ],
    [
      "136.069941160",
      "33.356286480"
    ],
    [
      "415.461910422",
      "364.564384321"
    ],
    [
      "18.720971820",
      "179.989848994"
    ],
    [
      "135.322140969",
      "432.119368861"
    ],
    [
      "451.639415756",
      "266.322065259"
    ],
    [
      "437.929379152",
      "153.182933377"
    ],
    [
      "365.483784947",
      "59.882887169"
    ],
    [
      "247.768821658",
      "17.422287311"
    ],
    [
      "272.147309368",
      "228.769801558"
    ],
    [
      "244.820487316",
      "107.573688899"
    ],
    [
      "356.537980780",
      "187.161515398"
    ],
    [
      "131.781909725",
      "291.453826274"
    ],
    [
      "111.460095437",
      "329.348731790"
    ],
    [
      "76.422197382",
      "93.820807509"
    ],
    [
      "33.418925155",
      "296.452358876"
    ],
    [
      "101.776703458",
      "190.244798630"
    ],
    [
      "114.951974461",
      "63.547778617"
    ],
    [
      "83.851668061",
      "288.866949000"
    ],
    [
      "255.242891306",
      "439.361619680"
    ],
    [
      "116.634052562",
      "243.465239041"
    ],
    [
      "61.216171598",
      "342.748288397"
    ],
    [
      "376.247370912",
      "379.604992391"
    ],
    [
      "158.136986120",
      "198.760580379"
    ],
    [
      "176.460041467",
      "261.124557504"
    ],
    [
      "222.667923737",
      "212.608254923"
    ],
    [
      "206.391415090",
      "145.529775995"
    ],
    [
      "55.300743360",
      "148.910590410"
    ],
    [
      "111.154762836",
      "388.347941717"
    ],
    [
      "315.723412591",
      "412.738819054"
    ],
    [
      "250.589837200",
      "371.670371837"
    ],
    [
      "320.503112266",
      "334.885402485"
    ],
    [
      "188.353190300",
      "421.935665857"
    ],
    [
      "130.956870341",
      "129.932960151"
    ],
    [
      "175.906898425",
      "337.810976450"
    ],
    [
      "44.993127927",
      "229.444614321"
    ],
    [
      "249.802551975",
      "287.674061311"
    ],
    [
      "332.613435275",
      "249.742345947"
    ],
    [
      "405.721500755",
      "307.388126840"
    ],
    [
      "419.287850920",
      "213.361778707"
    ],
    [
      "381.567980090",
      "122.861444367"
    ],
    [
      "290.711332415",
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    ],
    [
      "298.541232440",
      "62.656812880"
    ],
    [
      "192.828841515",
      "59.166731036"
    ]
  ],
  "radius": "234.230396592"
}
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": 52
}

当前第一名的答案

{
  "centers": [
    [
      "436.513681401",
      "136.739262757"
    ],
    [
      "438.922767542",
      "343.910223333"
    ],
    [
      "124.154523904",
      "210.819841782"
    ],
    [
      "142.608359208",
      "415.629805157"
    ],
    [
      "244.659256046",
      "131.986862208"
    ],
    [
      "15.101008013",
      "271.267102681"
    ],
    [
      "362.744577391",
      "419.181949015"
    ],
    [
      "223.824618942",
      "459.153910497"
    ],
    [
      "64.367078334",
      "381.990986253"
    ],
    [
      "136.069941160",
      "33.356286480"
    ],
    [
      "415.461910422",
      "364.564384321"
    ],
    [
      "18.720971820",
      "179.989848994"
    ],
    [
      "135.322140969",
      "432.119368861"
    ],
    [
      "451.639415756",
      "266.322065259"
    ],
    [
      "437.929379152",
      "153.182933377"
    ],
    [
      "365.483784947",
      "59.882887169"
    ],
    [
      "247.768821658",
      "17.422287311"
    ],
    [
      "272.147309368",
      "228.769801558"
    ],
    [
      "244.820487316",
      "107.573688899"
    ],
    [
      "356.537980780",
      "187.161515398"
    ],
    [
      "131.781909725",
      "291.453826274"
    ],
    [
      "111.460095437",
      "329.348731790"
    ],
    [
      "76.422197382",
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      "288.866949000"
    ],
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      "439.361619680"
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    [
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      "371.670371837"
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      "249.802551975",
      "287.674061311"
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    [
      "332.613435275",
      "249.742345947"
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    [
      "405.721500755",
      "307.388126840"
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    [
      "419.287850920",
      "213.361778707"
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    [
      "381.567980090",
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      "290.711332415",
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    [
      "298.541232440",
      "62.656812880"
    ],
    [
      "192.828841515",
      "59.166731036"
    ]
  ],
  "radius": "234.230396592"
}

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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