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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 = 53

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 = 53
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[240.91314227505, 240.91314227515]见来源署名packomania.com ↗
本站纪录240.913142277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

481.826284554y0
0x481.826284554
已验证构造r = 240.913142277

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 = 53 的当前纪录构型，240.913142277
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 ccin53 的单位圆归一化坐标与公开半径比 240.9131422751。固定九位目标 R=240.913142277，取 golden-angle 序列第 5200 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 2 个约束违例；随后允许每个坐标移动 ±4 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不等式安全。最终用整数平方检查 53 个容器约束和 1378 个圆对约束，全部通过。目标 240.913142276 的 30000 个测试旋转中最近舍入至少仍有 10 个违例，小邻域安全整数模型未找到可行解；可从更高精度坐标或连续重解继续。

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
    240.913142281→240.913142277
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    240.913142282→240.913142281
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": [
    [
      "414.689026191",
      "406.318815747"
    ],
    [
      "287.569700545",
      "6.605176256"
    ],
    [
      "55.816817391",
      "390.373440673"
    ],
    [
      "302.526417324",
      "84.358794345"
    ],
    [
      "267.940605580",
      "475.260156157"
    ],
    [
      "36.615164614",
      "356.841337751"
    ],
    [
      "65.008109171",
      "394.377166267"
    ],
    [
      "9.022492658",
      "219.258290330"
    ],
    [
      "130.893478348",
      "445.042403702"
    ],
    [
      "406.690233279",
      "80.205077182"
    ],
    [
      "38.751951762",
      "131.468939144"
    ],
    [
      "95.916651136",
      "63.806602866"
    ],
    [
      "457.281611287",
      "312.530780053"
    ],
    [
      "319.014627598",
      "27.875671458"
    ],
    [
      "202.320037669",
      "18.359614790"
    ],
    [
      "449.087868863",
      "155.770880271"
    ],
    [
      "108.237438715",
      "218.360073617"
    ],
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      "22.362382301",
      "284.797471068"
    ],
    [
      "59.233558887",
      "368.341442486"
    ],
    [
      "456.131152528",
      "191.075159831"
    ],
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      "21.147775429",
      "245.651828831"
    ],
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      "392.456999010",
      "398.892328716"
    ],
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      "335.461084311"
    ],
    [
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      "322.876696888"
    ],
    [
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      "183.251149177"
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    [
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      "123.318045195"
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    ],
    [
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      "341.986916882"
    ],
    [
      "196.849998141",
      "377.061235885"
    ],
    [
      "223.064785184",
      "135.937835010"
    ],
    [
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    ],
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  ],
  "radius": "240.913142277"
}
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": 53
}

当前第一名的答案

{
  "centers": [
    [
      "414.689026191",
      "406.318815747"
    ],
    [
      "287.569700545",
      "6.605176256"
    ],
    [
      "55.816817391",
      "390.373440673"
    ],
    [
      "302.526417324",
      "84.358794345"
    ],
    [
      "267.940605580",
      "475.260156157"
    ],
    [
      "36.615164614",
      "356.841337751"
    ],
    [
      "65.008109171",
      "394.377166267"
    ],
    [
      "9.022492658",
      "219.258290330"
    ],
    [
      "130.893478348",
      "445.042403702"
    ],
    [
      "406.690233279",
      "80.205077182"
    ],
    [
      "38.751951762",
      "131.468939144"
    ],
    [
      "95.916651136",
      "63.806602866"
    ],
    [
      "457.281611287",
      "312.530780053"
    ],
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      "27.875671458"
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      "108.237438715",
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    ],
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      "284.797471068"
    ],
    [
      "59.233558887",
      "368.341442486"
    ],
    [
      "456.131152528",
      "191.075159831"
    ],
    [
      "21.147775429",
      "245.651828831"
    ],
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      "398.892328716"
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      "159.693771755",
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      "337.484151133",
      "293.402230459"
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    [
      "232.293806474",
      "290.468271654"
    ]
  ],
  "radius": "240.913142277"
}

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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