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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 = 49

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 = 49
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[214.18093011195, 214.18093011205]见来源署名packomania.com ↗
本站纪录214.180930113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

428.361860226y0
0x428.361860226
已验证构造r = 214.180930113

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 = 49 的当前纪录构型，214.180930113
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 214.180930116 构型出发，将数值统一乘以 10^9。固定目标半径 214180930113，允许每个坐标在原值附近移动至多 64 个整数单位；对包含和分离条件作一阶整数线性化，包含约束预留 2·64² 的最坏二次余量，分离约束利用非负二次余量。整数 MILP 给出 49 个圆的偏移后，按原始整数平方重新核验所有包含约束与 1176 个圆对，得到本证书。继续方向：目标 214180930112 在站内证书的偏移半径至 64，以及 Packomania 原始构型的 96 个旋转取整相位中均未找到修整；这只是局部搜索结果，并非下界证明。

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
    214.180930116→214.180930113
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    214.180930117→214.180930116
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": [
    [
      "221.090939166",
      "204.196072093"
    ],
    [
      "20.148598749",
      "128.434538469"
    ],
    [
      "85.157452440",
      "273.710909842"
    ],
    [
      "175.366659070",
      "120.790859100"
    ],
    [
      "5.598142998",
      "228.953712193"
    ],
    [
      "210.369716039",
      "6.051683241"
    ],
    [
      "24.243986483",
      "296.906923212"
    ],
    [
      "34.808261916",
      "112.760429687"
    ],
    [
      "265.956796712",
      "15.696107706"
    ],
    [
      "128.681028482",
      "399.505930984"
    ],
    [
      "352.807350065",
      "65.851027184"
    ],
    [
      "401.000737020",
      "291.351983998"
    ],
    [
      "13.018192754",
      "211.496816303"
    ],
    [
      "227.277964253",
      "413.932957581"
    ],
    [
      "335.451370391",
      "372.188914831"
    ],
    [
      "88.981961243",
      "219.844606831"
    ],
    [
      "63.418829883",
      "341.267940993"
    ],
    [
      "44.151979177",
      "312.042891863"
    ],
    [
      "407.963895156",
      "191.808549529"
    ],
    [
      "398.612993030",
      "153.606067059"
    ],
    [
      "188.847902163",
      "22.668242664"
    ],
    [
      "92.942473453",
      "65.067729739"
    ],
    [
      "62.317814757",
      "98.044989005"
    ],
    [
      "233.786758103",
      "25.013285714"
    ],
    [
      "305.757947633",
      "139.148675945"
    ],
    [
      "252.122955807",
      "129.977692459"
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    [
      "89.631794535",
      "305.928437825"
    ],
    [
      "325.827048063",
      "243.993489119"
    ],
    [
      "212.779940614",
      "317.776882224"
    ],
    [
      "117.644677620",
      "256.286939245"
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    [
      "142.449868032",
      "46.146369267"
    ],
    [
      "107.785938247",
      "362.066010505"
    ],
    [
      "149.727658411",
      "312.045735953"
    ],
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      "131.743243383",
      "193.859137398"
    ],
    [
      "361.689364618",
      "112.460394006"
    ],
    [
      "389.826902855",
      "244.129821833"
    ],
    [
      "349.331194858",
      "183.391879541"
    ],
    [
      "196.113516082",
      "161.530310088"
    ],
    [
      "270.041189793",
      "281.100604570"
    ],
    [
      "44.623821276",
      "254.043584787"
    ],
    [
      "204.338078579",
      "82.959599439"
    ],
    [
      "129.053964212",
      "117.906732712"
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    [
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      "381.936120873"
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    [
      "269.549886165",
      "198.075450692"
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    [
      "348.053977468",
      "313.527391710"
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    [
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      "291.971407860",
      "67.422559849"
    ],
    [
      "271.526533667",
      "369.088068186"
    ]
  ],
  "radius": "214.180930113"
}
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": 49
}

当前第一名的答案

{
  "centers": [
    [
      "221.090939166",
      "204.196072093"
    ],
    [
      "20.148598749",
      "128.434538469"
    ],
    [
      "85.157452440",
      "273.710909842"
    ],
    [
      "175.366659070",
      "120.790859100"
    ],
    [
      "5.598142998",
      "228.953712193"
    ],
    [
      "210.369716039",
      "6.051683241"
    ],
    [
      "24.243986483",
      "296.906923212"
    ],
    [
      "34.808261916",
      "112.760429687"
    ],
    [
      "265.956796712",
      "15.696107706"
    ],
    [
      "128.681028482",
      "399.505930984"
    ],
    [
      "352.807350065",
      "65.851027184"
    ],
    [
      "401.000737020",
      "291.351983998"
    ],
    [
      "13.018192754",
      "211.496816303"
    ],
    [
      "227.277964253",
      "413.932957581"
    ],
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      "372.188914831"
    ],
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    ],
    [
      "63.418829883",
      "341.267940993"
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    [
      "44.151979177",
      "312.042891863"
    ],
    [
      "407.963895156",
      "191.808549529"
    ],
    [
      "398.612993030",
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    ],
    [
      "188.847902163",
      "22.668242664"
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    [
      "92.942473453",
      "65.067729739"
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    [
      "62.317814757",
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    ],
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      "233.786758103",
      "25.013285714"
    ],
    [
      "305.757947633",
      "139.148675945"
    ],
    [
      "252.122955807",
      "129.977692459"
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    [
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      "129.053964212",
      "117.906732712"
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    [
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      "67.422559849"
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      "271.526533667",
      "369.088068186"
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  ],
  "radius": "214.180930113"
}

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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