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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 = 56

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 = 56
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[261.41402447675, 261.41402447685]见来源署名packomania.com ↗
本站纪录261.414024479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

522.828048958y0
0x522.828048958
已验证构造r = 261.414024479

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 = 56 的当前纪录构型，261.414024479
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 Packomania ccin56 的公开坐标（连续比值 261.4140244768）出发，按候选半径 261.414024479 缩放。采用黄金角旋转 k=41437，再在每个中心坐标 ±1 个整数单位内求解线性化整数可行性模型；最终 50 个中心发生不超过 1e-9 的修整。用整数平方逐项复核 56 个包含约束和 1540 个两两不重叠约束。更低的 261.414024478 在该旋转的 ±1…±32 修整模型中未得到可行证书，可继续扩大旋转与离散搜索。

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
    261.414024486→261.414024479
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    261.414024487→261.414024486
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": [
    [
      "247.823536607",
      "332.285144996"
    ],
    [
      "287.867280358",
      "5.007132293"
    ],
    [
      "517.431727799",
      "245.267137041"
    ],
    [
      "392.968112747",
      "114.493644771"
    ],
    [
      "197.759733734",
      "65.935742680"
    ],
    [
      "396.293314553",
      "341.654599897"
    ],
    [
      "68.145932463",
      "95.982817102"
    ],
    [
      "469.734669076",
      "230.432508607"
    ],
    [
      "370.854797901",
      "35.057457647"
    ],
    [
      "489.011425590",
      "154.666568050"
    ],
    [
      "67.244006846",
      "419.533785020"
    ],
    [
      "201.107683211",
      "503.427455531"
    ],
    [
      "507.813689823",
      "229.842954242"
    ],
    [
      "299.626132422",
      "505.859385072"
    ],
    [
      "88.320533523",
      "86.032968351"
    ],
    [
      "22.787792328",
      "318.660522782"
    ],
    [
      "28.219864322",
      "188.210889092"
    ],
    [
      "211.635382418",
      "429.486510196"
    ],
    [
      "445.038540205",
      "419.667424970"
    ],
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      "339.812552721",
      "385.058832450"
    ],
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      "275.563080434",
      "173.033606640"
    ],
    [
      "413.711548927",
      "446.134701732"
    ],
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    ],
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    ],
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      "190.398480803",
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      "26.032133332"
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    ],
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      "74.119912191"
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      "212.373376905",
      "315.002461054"
    ],
    [
      "485.629314272",
      "272.408826504"
    ],
    [
      "431.810647043",
      "318.718124250"
    ],
    [
      "169.378853673",
      "466.087378185"
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      "397.998162315",
      "387.374895061"
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    [
      "141.721565759",
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    ]
  ],
  "radius": "261.414024479"
}
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": 56
}

当前第一名的答案

{
  "centers": [
    [
      "247.823536607",
      "332.285144996"
    ],
    [
      "287.867280358",
      "5.007132293"
    ],
    [
      "517.431727799",
      "245.267137041"
    ],
    [
      "392.968112747",
      "114.493644771"
    ],
    [
      "197.759733734",
      "65.935742680"
    ],
    [
      "396.293314553",
      "341.654599897"
    ],
    [
      "68.145932463",
      "95.982817102"
    ],
    [
      "469.734669076",
      "230.432508607"
    ],
    [
      "370.854797901",
      "35.057457647"
    ],
    [
      "489.011425590",
      "154.666568050"
    ],
    [
      "67.244006846",
      "419.533785020"
    ],
    [
      "201.107683211",
      "503.427455531"
    ],
    [
      "507.813689823",
      "229.842954242"
    ],
    [
      "299.626132422",
      "505.859385072"
    ],
    [
      "88.320533523",
      "86.032968351"
    ],
    [
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      "318.660522782"
    ],
    [
      "28.219864322",
      "188.210889092"
    ],
    [
      "211.635382418",
      "429.486510196"
    ],
    [
      "445.038540205",
      "419.667424970"
    ],
    [
      "339.812552721",
      "385.058832450"
    ],
    [
      "275.563080434",
      "173.033606640"
    ],
    [
      "413.711548927",
      "446.134701732"
    ],
    [
      "489.669104868",
      "330.267966594"
    ],
    [
      "239.282469605",
      "263.421844314"
    ],
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      "190.398480803",
      "260.052031439"
    ],
    [
      "257.524520483",
      "26.032133332"
    ],
    [
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    ],
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    ],
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      "375.450002699"
    ],
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      "194.718729642"
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      "266.828638863",
      "383.526410090"
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      "242.516405032",
      "105.706663705"
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      "55.099514824",
      "255.005233081"
    ],
    [
      "404.828371145",
      "230.549447748"
    ]
  ],
  "radius": "261.414024479"
}

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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