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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 = 48

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 = 48
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[207.80084340725, 207.80084340735]见来源署名packomania.com ↗
本站纪录207.800843409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

415.601686818y0
0x415.601686818
已验证构造r = 207.800843409

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 = 48 的当前纪录构型，207.800843409
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 207.800843413 构型出发，把半径和坐标乘以 10^9 转成整数。固定目标半径 207800843409，每个坐标仅允许在原值的 [-8,8] 内移动；线性化圆内包含和两圆分离约束，包含约束预留 2·8² 的最坏二次余量，分离约束则利用非负二次余量。整数 MILP 找到 48 个圆的偏移后，按原始整数平方式重新核验全部包含约束和 1128 个圆对，得到本证书。继续方向：更低目标 207800843408 在 Packomania 原始构型的 96 个旋转取整相位中没有找到修整；这不是不可能性证明，可扩大相位、局部偏移或采用非线性整数搜索。

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
    207.800843413→207.800843409
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    207.800843414→207.800843413
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": [
    [
      "206.991830566",
      "195.976366910"
    ],
    [
      "11.666110416",
      "249.017059532"
    ],
    [
      "406.691445279",
      "159.141464940"
    ],
    [
      "142.539512152",
      "400.773554755"
    ],
    [
      "220.845229910",
      "5.534525169"
    ],
    [
      "77.637112554",
      "361.948530055"
    ],
    [
      "355.723711092",
      "343.440886627"
    ],
    [
      "9.832815095",
      "181.463737725"
    ],
    [
      "263.130537936",
      "398.746909730"
    ],
    [
      "402.975234385",
      "175.688128485"
    ],
    [
      "18.755530249",
      "262.353212162"
    ],
    [
      "394.081217798",
      "268.113299810"
    ],
    [
      "69.349700789",
      "344.729262016"
    ],
    [
      "168.470402029",
      "397.564096179"
    ],
    [
      "40.086445772",
      "113.037868823"
    ],
    [
      "255.636463698",
      "263.720215112"
    ],
    [
      "285.925708852",
      "33.727677054"
    ],
    [
      "315.902849532",
      "51.793419128"
    ],
    [
      "73.451220411",
      "130.496484639"
    ],
    [
      "198.070579813",
      "20.252239694"
    ],
    [
      "102.777472938",
      "53.318977958"
    ],
    [
      "326.875543969",
      "350.408833349"
    ],
    [
      "285.226464124",
      "375.600202709"
    ],
    [
      "94.617958767",
      "232.575009268"
    ],
    [
      "70.986528587",
      "86.565569622"
    ],
    [
      "243.136500649",
      "29.475188204"
    ],
    [
      "34.765017725",
      "155.383785050"
    ],
    [
      "129.262066931",
      "271.353686007"
    ],
    [
      "376.086065465",
      "147.387056561"
    ],
    [
      "151.745760108",
      "39.067421844"
    ],
    [
      "89.649676659",
      "177.799867343"
    ],
    [
      "204.376746629",
      "72.622395163"
    ],
    [
      "341.811301158",
      "95.722363281"
    ],
    [
      "302.049268651",
      "245.123131021"
    ],
    [
      "239.865708998",
      "215.219871852"
    ],
    [
      "36.325073504",
      "218.364466422"
    ],
    [
      "272.330361107",
      "86.137858373"
    ],
    [
      "352.747521469",
      "296.247363644"
    ],
    [
      "277.832068644",
      "314.042726189"
    ],
    [
      "219.849421244",
      "142.940218943"
    ],
    [
      "62.999608199",
      "290.596517472"
    ],
    [
      "373.168692216",
      "218.327094152"
    ],
    [
      "219.029978182",
      "371.194402958"
    ],
    [
      "199.814578807",
      "285.718332470"
    ],
    [
      "160.475237557",
      "205.363247461"
    ],
    [
      "303.082850733",
      "164.578061153"
    ],
    [
      "137.544416373",
      "114.746493166"
    ],
    [
      "131.045747812",
      "347.961340176"
    ]
  ],
  "radius": "207.800843409"
}
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": 48
}

当前第一名的答案

{
  "centers": [
    [
      "206.991830566",
      "195.976366910"
    ],
    [
      "11.666110416",
      "249.017059532"
    ],
    [
      "406.691445279",
      "159.141464940"
    ],
    [
      "142.539512152",
      "400.773554755"
    ],
    [
      "220.845229910",
      "5.534525169"
    ],
    [
      "77.637112554",
      "361.948530055"
    ],
    [
      "355.723711092",
      "343.440886627"
    ],
    [
      "9.832815095",
      "181.463737725"
    ],
    [
      "263.130537936",
      "398.746909730"
    ],
    [
      "402.975234385",
      "175.688128485"
    ],
    [
      "18.755530249",
      "262.353212162"
    ],
    [
      "394.081217798",
      "268.113299810"
    ],
    [
      "69.349700789",
      "344.729262016"
    ],
    [
      "168.470402029",
      "397.564096179"
    ],
    [
      "40.086445772",
      "113.037868823"
    ],
    [
      "255.636463698",
      "263.720215112"
    ],
    [
      "285.925708852",
      "33.727677054"
    ],
    [
      "315.902849532",
      "51.793419128"
    ],
    [
      "73.451220411",
      "130.496484639"
    ],
    [
      "198.070579813",
      "20.252239694"
    ],
    [
      "102.777472938",
      "53.318977958"
    ],
    [
      "326.875543969",
      "350.408833349"
    ],
    [
      "285.226464124",
      "375.600202709"
    ],
    [
      "94.617958767",
      "232.575009268"
    ],
    [
      "70.986528587",
      "86.565569622"
    ],
    [
      "243.136500649",
      "29.475188204"
    ],
    [
      "34.765017725",
      "155.383785050"
    ],
    [
      "129.262066931",
      "271.353686007"
    ],
    [
      "376.086065465",
      "147.387056561"
    ],
    [
      "151.745760108",
      "39.067421844"
    ],
    [
      "89.649676659",
      "177.799867343"
    ],
    [
      "204.376746629",
      "72.622395163"
    ],
    [
      "341.811301158",
      "95.722363281"
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    [
      "302.049268651",
      "245.123131021"
    ],
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      "239.865708998",
      "215.219871852"
    ],
    [
      "36.325073504",
      "218.364466422"
    ],
    [
      "272.330361107",
      "86.137858373"
    ],
    [
      "352.747521469",
      "296.247363644"
    ],
    [
      "277.832068644",
      "314.042726189"
    ],
    [
      "219.849421244",
      "142.940218943"
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    [
      "62.999608199",
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    [
      "373.168692216",
      "218.327094152"
    ],
    [
      "219.029978182",
      "371.194402958"
    ],
    [
      "199.814578807",
      "285.718332470"
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    [
      "160.475237557",
      "205.363247461"
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      "303.082850733",
      "164.578061153"
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      "137.544416373",
      "114.746493166"
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    [
      "131.045747812",
      "347.961340176"
    ]
  ],
  "radius": "207.800843409"
}

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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