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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 = 47

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 = 47
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[201.48619333075, 201.48619333085]见来源署名packomania.com ↗
本站纪录201.486193333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

402.972386666y0
0x402.972386666
已验证构造r = 201.486193333

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 = 47 的当前纪录构型，201.486193333
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）＋

从站内公开的 201.486193339 构型出发，将全部数值乘以 10^9 后在整数格求解。固定目标半径 201486193333，允许每个坐标在原值附近移动至多 16 个整数单位；对圆内包含和两圆分离条件作一阶整数线性化，其中包含约束预留 2·16² 的最坏二次余量，而分离约束的二次余量天然非负。整数 MILP 给出 47 个圆的坐标偏移后，重新用原始整数平方式逐项核验全部包含约束及 1081 个圆对，得到提交证书。继续方向：201486193331 和 201486193332 在 Packomania 原始构型各 96 个旋转取整相位中未找到可行修整；这是局部搜索结果而非下界证明，可扩大旋转、偏移或做非线性整数搜索。

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
    201.486193339→201.486193333
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    201.48619334→201.486193339
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": [
    [
      "216.466431320",
      "211.869955957"
    ],
    [
      "124.720103781",
      "17.410671982"
    ],
    [
      "281.337921710",
      "19.844403749"
    ],
    [
      "241.172249412",
      "78.050586677"
    ],
    [
      "57.062371520",
      "68.369007807"
    ],
    [
      "396.321251225",
      "217.253764248"
    ],
    [
      "218.113942735",
      "395.102558815"
    ],
    [
      "66.734739891",
      "340.236354524"
    ],
    [
      "377.293498700",
      "279.861738009"
    ],
    [
      "334.648474738",
      "63.882526783"
    ],
    [
      "15.553383178",
      "160.205555433"
    ],
    [
      "51.220669993",
      "86.050973582"
    ],
    [
      "22.504103276",
      "260.583202698"
    ],
    [
      "253.643632553",
      "21.401007312"
    ],
    [
      "142.279151691",
      "24.648339570"
    ],
    [
      "165.217720321",
      "249.355036486"
    ],
    [
      "289.373747738",
      "363.683215579"
    ],
    [
      "318.833203283",
      "342.542431761"
    ],
    [
      "383.875903757",
      "195.552873151"
    ],
    [
      "134.097242735",
      "369.931744934"
    ],
    [
      "343.626505165",
      "195.485885081"
    ],
    [
      "224.154639926",
      "156.169512540"
    ],
    [
      "97.689503912",
      "346.681440830"
    ],
    [
      "215.163510611",
      "324.754854836"
    ],
    [
      "240.670194790",
      "200.225999389"
    ],
    [
      "235.011776526",
      "109.058476600"
    ],
    [
      "343.184672707",
      "99.884378461"
    ],
    [
      "244.724280570",
      "369.497880692"
    ],
    [
      "366.374610684",
      "150.857182746"
    ],
    [
      "368.129199265",
      "241.953749195"
    ],
    [
      "185.750142600",
      "371.244606451"
    ],
    [
      "340.205895272",
      "297.340572981"
    ],
    [
      "48.467900137",
      "130.966697534"
    ],
    [
      "117.592617906",
      "234.103004019"
    ],
    [
      "176.173301344",
      "125.153004809"
    ],
    [
      "273.928549470",
      "312.549564843"
    ],
    [
      "178.720762123",
      "197.107924333"
    ],
    [
      "112.300437725",
      "162.274886755"
    ],
    [
      "218.838060332",
      "261.657148840"
    ],
    [
      "284.677722311",
      "65.592468854"
    ],
    [
      "67.702978739",
      "290.130819470"
    ],
    [
      "296.248468912",
      "237.811213264"
    ],
    [
      "150.114332930",
      "306.390238293"
    ],
    [
      "44.128399360",
      "207.844319330"
    ],
    [
      "292.374621667",
      "150.897501414"
    ],
    [
      "199.192123384",
      "46.600871387"
    ],
    [
      "109.570886969",
      "77.318724220"
    ]
  ],
  "radius": "201.486193333"
}
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": 47
}

当前第一名的答案

{
  "centers": [
    [
      "216.466431320",
      "211.869955957"
    ],
    [
      "124.720103781",
      "17.410671982"
    ],
    [
      "281.337921710",
      "19.844403749"
    ],
    [
      "241.172249412",
      "78.050586677"
    ],
    [
      "57.062371520",
      "68.369007807"
    ],
    [
      "396.321251225",
      "217.253764248"
    ],
    [
      "218.113942735",
      "395.102558815"
    ],
    [
      "66.734739891",
      "340.236354524"
    ],
    [
      "377.293498700",
      "279.861738009"
    ],
    [
      "334.648474738",
      "63.882526783"
    ],
    [
      "15.553383178",
      "160.205555433"
    ],
    [
      "51.220669993",
      "86.050973582"
    ],
    [
      "22.504103276",
      "260.583202698"
    ],
    [
      "253.643632553",
      "21.401007312"
    ],
    [
      "142.279151691",
      "24.648339570"
    ],
    [
      "165.217720321",
      "249.355036486"
    ],
    [
      "289.373747738",
      "363.683215579"
    ],
    [
      "318.833203283",
      "342.542431761"
    ],
    [
      "383.875903757",
      "195.552873151"
    ],
    [
      "134.097242735",
      "369.931744934"
    ],
    [
      "343.626505165",
      "195.485885081"
    ],
    [
      "224.154639926",
      "156.169512540"
    ],
    [
      "97.689503912",
      "346.681440830"
    ],
    [
      "215.163510611",
      "324.754854836"
    ],
    [
      "240.670194790",
      "200.225999389"
    ],
    [
      "235.011776526",
      "109.058476600"
    ],
    [
      "343.184672707",
      "99.884378461"
    ],
    [
      "244.724280570",
      "369.497880692"
    ],
    [
      "366.374610684",
      "150.857182746"
    ],
    [
      "368.129199265",
      "241.953749195"
    ],
    [
      "185.750142600",
      "371.244606451"
    ],
    [
      "340.205895272",
      "297.340572981"
    ],
    [
      "48.467900137",
      "130.966697534"
    ],
    [
      "117.592617906",
      "234.103004019"
    ],
    [
      "176.173301344",
      "125.153004809"
    ],
    [
      "273.928549470",
      "312.549564843"
    ],
    [
      "178.720762123",
      "197.107924333"
    ],
    [
      "112.300437725",
      "162.274886755"
    ],
    [
      "218.838060332",
      "261.657148840"
    ],
    [
      "284.677722311",
      "65.592468854"
    ],
    [
      "67.702978739",
      "290.130819470"
    ],
    [
      "296.248468912",
      "237.811213264"
    ],
    [
      "150.114332930",
      "306.390238293"
    ],
    [
      "44.128399360",
      "207.844319330"
    ],
    [
      "292.374621667",
      "150.897501414"
    ],
    [
      "199.192123384",
      "46.600871387"
    ],
    [
      "109.570886969",
      "77.318724220"
    ]
  ],
  "radius": "201.486193333"
}

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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