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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 = 46

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 = 46
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[195.20390135145, 195.20390135155]见来源署名packomania.com ↗
本站纪录195.203901353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

390.407802706y0
0x390.407802706
已验证构造r = 195.203901353

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 = 46 的当前纪录构型，195.203901353
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 195.203901356 构型出发，把半径和坐标统一乘以 10^9 转成整数。固定目标半径 195203901353，并令每个坐标在原值的 [-8,8] 整数范围内移动；将圆内包含约束和两圆分离约束在原点线性化，包含约束额外预留 2·8² 的最坏二次余量，分离约束的二次余量非负。用整数 MILP 找到全部 46 个圆的偏移后，再对所有包含条件和 1035 个两两距离条件按原始整数平方完整复算，得到本次证书。继续方向：目标 195203901352 在站内构型的安全偏移半径 2/4/8/16/32/64，以及 Packomania 原始构型的 128 个旋转取整相位中均未找到可行修整；这只是局部搜索结果，不是不可能性证明，可继续扩大旋转/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
    195.203901356→195.203901353
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    195.203901357→195.203901356
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": [
    [
      "191.255395033",
      "229.014669801"
    ],
    [
      "97.333781128",
      "34.316214720"
    ],
    [
      "86.012550099",
      "353.299439451"
    ],
    [
      "50.368473630",
      "70.441593518"
    ],
    [
      "45.418999839",
      "78.001439009"
    ],
    [
      "384.086341948",
      "206.205726964"
    ],
    [
      "344.357249587",
      "309.903916733"
    ],
    [
      "284.384897400",
      "359.794088049"
    ],
    [
      "42.011573046",
      "301.052866520"
    ],
    [
      "12.236753313",
      "223.274653209"
    ],
    [
      "370.809098032",
      "139.581134676"
    ],
    [
      "112.145229212",
      "31.923906049"
    ],
    [
      "317.006214820",
      "59.755926247"
    ],
    [
      "216.595102236",
      "15.277034162"
    ],
    [
      "365.253436427",
      "254.841152161"
    ],
    [
      "371.983572044",
      "224.580529652"
    ],
    [
      "28.377592802",
      "132.973086859"
    ],
    [
      "44.662380635",
      "101.724100355"
    ],
    [
      "347.494468983",
      "283.834583751"
    ],
    [
      "256.397505128",
      "358.701047799"
    ],
    [
      "163.351711842",
      "366.471053107"
    ],
    [
      "234.963341049",
      "322.573179315"
    ],
    [
      "304.534288723",
      "173.695017596"
    ],
    [
      "72.228632584",
      "314.316701293"
    ],
    [
      "262.989379943",
      "197.737075629"
    ],
    [
      "210.272037292",
      "363.735534918"
    ],
    [
      "167.058745866",
      "246.081792859"
    ],
    [
      "204.199828793",
      "110.968219422"
    ],
    [
      "75.831745955",
      "136.901801610"
    ],
    [
      "118.787230482",
      "341.671718416"
    ],
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      "358.205440957",
      "179.645430856"
    ],
    [
      "257.424266256",
      "138.665859344"
    ],
    [
      "116.572282125",
      "278.502727129"
    ],
    [
      "34.366992798",
      "184.332532636"
    ],
    [
      "90.043539589",
      "74.346182020"
    ],
    [
      "177.343370890",
      "311.195967568"
    ],
    [
      "140.820446996",
      "125.392295234"
    ],
    [
      "49.686865631",
      "254.683804810"
    ],
    [
      "314.524712345",
      "235.958123452"
    ],
    [
      "296.224544941",
      "313.030386339"
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    [
      "325.900220314",
      "113.366774067"
    ],
    [
      "234.918637366",
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    ],
    [
      "196.682135587",
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    ],
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      "111.626254464",
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    ],
    [
      "257.486656187",
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    ]
  ],
  "radius": "195.203901353"
}
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": 46
}

当前第一名的答案

{
  "centers": [
    [
      "191.255395033",
      "229.014669801"
    ],
    [
      "97.333781128",
      "34.316214720"
    ],
    [
      "86.012550099",
      "353.299439451"
    ],
    [
      "50.368473630",
      "70.441593518"
    ],
    [
      "45.418999839",
      "78.001439009"
    ],
    [
      "384.086341948",
      "206.205726964"
    ],
    [
      "344.357249587",
      "309.903916733"
    ],
    [
      "284.384897400",
      "359.794088049"
    ],
    [
      "42.011573046",
      "301.052866520"
    ],
    [
      "12.236753313",
      "223.274653209"
    ],
    [
      "370.809098032",
      "139.581134676"
    ],
    [
      "112.145229212",
      "31.923906049"
    ],
    [
      "317.006214820",
      "59.755926247"
    ],
    [
      "216.595102236",
      "15.277034162"
    ],
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      "365.253436427",
      "254.841152161"
    ],
    [
      "371.983572044",
      "224.580529652"
    ],
    [
      "28.377592802",
      "132.973086859"
    ],
    [
      "44.662380635",
      "101.724100355"
    ],
    [
      "347.494468983",
      "283.834583751"
    ],
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      "256.397505128",
      "358.701047799"
    ],
    [
      "163.351711842",
      "366.471053107"
    ],
    [
      "234.963341049",
      "322.573179315"
    ],
    [
      "304.534288723",
      "173.695017596"
    ],
    [
      "72.228632584",
      "314.316701293"
    ],
    [
      "262.989379943",
      "197.737075629"
    ],
    [
      "210.272037292",
      "363.735534918"
    ],
    [
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      "246.081792859"
    ],
    [
      "204.199828793",
      "110.968219422"
    ],
    [
      "75.831745955",
      "136.901801610"
    ],
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      "341.671718416"
    ],
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      "184.332532636"
    ],
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      "90.043539589",
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      "177.343370890",
      "311.195967568"
    ],
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      "140.820446996",
      "125.392295234"
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      "49.686865631",
      "254.683804810"
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      "314.524712345",
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      "47.633987756"
    ],
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      "257.486656187",
      "59.621223148"
    ]
  ],
  "radius": "195.203901353"
}

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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