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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 = 57

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 = 57
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[268.23567910475, 268.23567910485]见来源署名packomania.com ↗
本站纪录268.23567911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

536.47135822y0
0x536.47135822
已验证构造r = 268.23567911

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 = 57 的当前纪录构型，268.23567911
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 ccin57 的公开原始坐标出发，使用高精度十进制定点计算，在零旋转下按容器半径缩放并四舍五入到 1e-9 网格。对整数半径做局部二分，得到 268.235679110 可行，而同一坐标构造在前一个整数网格点不可行；最终用整数平方复核 57 个包含约束和 1596 个两两不重叠约束。这只是公开构型的网格化改进，不代表全局最优；后续可从该证书继续做连续局部优化或尝试其他旋转。

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
    268.358130035→268.23567911
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    268.235679111→268.358130035
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": [
    [
      "135.154632331",
      "135.311484784"
    ],
    [
      "200.910663129",
      "399.682875206"
    ],
    [
      "431.464107473",
      "59.188729669"
    ],
    [
      "393.484130679",
      "309.864135794"
    ],
    [
      "413.159049988",
      "128.458988435"
    ],
    [
      "63.985772895",
      "432.651311441"
    ],
    [
      "203.295397340",
      "521.231827121"
    ],
    [
      "100.391008602",
      "316.987579695"
    ],
    [
      "285.474104556",
      "526.848761581"
    ],
    [
      "407.753633144",
      "50.997417954"
    ],
    [
      "373.169666343",
      "503.030128588"
    ],
    [
      "80.856084996",
      "442.758292597"
    ],
    [
      "85.425981728",
      "90.218932750"
    ],
    [
      "180.403567785",
      "29.665262295"
    ],
    [
      "294.924710008",
      "16.479076641"
    ],
    [
      "24.797619064",
      "333.940165458"
    ],
    [
      "26.174586454",
      "200.961658271"
    ],
    [
      "390.187283330",
      "126.964114111"
    ],
    [
      "208.095937832",
      "110.870120197"
    ],
    [
      "492.473235028",
      "162.005963309"
    ],
    [
      "470.318729340",
      "125.800002981"
    ],
    [
      "481.164141215",
      "391.902488288"
    ],
    [
      "454.291304037",
      "427.997510413"
    ],
    [
      "180.463740777",
      "496.153306051"
    ],
    [
      "193.248540044",
      "235.875672831"
    ],
    [
      "238.175695260",
      "208.933933047"
    ],
    [
      "509.329516991",
      "276.506967474"
    ],
    [
      "268.635651512",
      "438.869850729"
    ],
    [
      "368.806366892",
      "224.498837722"
    ],
    [
      "135.571685462",
      "226.389573356"
    ],
    [
      "164.792762773",
      "388.202607550"
    ],
    [
      "427.855716467",
      "94.084194022"
    ],
    [
      "205.165652044",
      "438.059204518"
    ],
    [
      "496.757928744",
      "216.816472677"
    ],
    [
      "491.167314871",
      "335.787102547"
    ],
    [
      "348.722037815",
      "161.556503657"
    ],
    [
      "241.458806625",
      "497.915757465"
    ],
    [
      "301.528587535",
      "218.554558456"
    ],
    [
      "244.940622598",
      "273.610651318"
    ],
    [
      "133.617012835",
      "452.543410659"
    ],
    [
      "396.613357317",
      "455.732947353"
    ],
    [
      "320.079994425",
      "488.420064977"
    ],
    [
      "330.462477958",
      "403.963640226"
    ],
    [
      "181.347493008",
      "167.909761672"
    ],
    [
      "428.572413650",
      "176.919912277"
    ],
    [
      "267.467838438",
      "141.770019834"
    ],
    [
      "148.713897311",
      "82.064334950"
    ],
    [
      "414.946739715",
      "368.641690098"
    ],
    [
      "240.010296205",
      "50.824523108"
    ],
    [
      "432.471086754",
      "271.839880400"
    ],
    [
      "244.275661419",
      "363.719395754"
    ],
    [
      "159.569805444",
      "305.119754426"
    ],
    [
      "81.603122474",
      "375.448954244"
    ],
    [
      "83.719931415",
      "159.374000564"
    ],
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      "333.183682347",
      "306.001427883"
    ],
    [
      "343.296738030",
      "69.716609697"
    ],
    [
      "57.000000001",
      "268.235679110"
    ]
  ],
  "radius": "268.235679110"
}
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": 57
}

当前第一名的答案

{
  "centers": [
    [
      "135.154632331",
      "135.311484784"
    ],
    [
      "200.910663129",
      "399.682875206"
    ],
    [
      "431.464107473",
      "59.188729669"
    ],
    [
      "393.484130679",
      "309.864135794"
    ],
    [
      "413.159049988",
      "128.458988435"
    ],
    [
      "63.985772895",
      "432.651311441"
    ],
    [
      "203.295397340",
      "521.231827121"
    ],
    [
      "100.391008602",
      "316.987579695"
    ],
    [
      "285.474104556",
      "526.848761581"
    ],
    [
      "407.753633144",
      "50.997417954"
    ],
    [
      "373.169666343",
      "503.030128588"
    ],
    [
      "80.856084996",
      "442.758292597"
    ],
    [
      "85.425981728",
      "90.218932750"
    ],
    [
      "180.403567785",
      "29.665262295"
    ],
    [
      "294.924710008",
      "16.479076641"
    ],
    [
      "24.797619064",
      "333.940165458"
    ],
    [
      "26.174586454",
      "200.961658271"
    ],
    [
      "390.187283330",
      "126.964114111"
    ],
    [
      "208.095937832",
      "110.870120197"
    ],
    [
      "492.473235028",
      "162.005963309"
    ],
    [
      "470.318729340",
      "125.800002981"
    ],
    [
      "481.164141215",
      "391.902488288"
    ],
    [
      "454.291304037",
      "427.997510413"
    ],
    [
      "180.463740777",
      "496.153306051"
    ],
    [
      "193.248540044",
      "235.875672831"
    ],
    [
      "238.175695260",
      "208.933933047"
    ],
    [
      "509.329516991",
      "276.506967474"
    ],
    [
      "268.635651512",
      "438.869850729"
    ],
    [
      "368.806366892",
      "224.498837722"
    ],
    [
      "135.571685462",
      "226.389573356"
    ],
    [
      "164.792762773",
      "388.202607550"
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    [
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      "496.757928744",
      "216.816472677"
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    [
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      "161.556503657"
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      "241.458806625",
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      "141.770019834"
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      "82.064334950"
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      "159.569805444",
      "305.119754426"
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      "81.603122474",
      "375.448954244"
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    [
      "83.719931415",
      "159.374000564"
    ],
    [
      "333.183682347",
      "306.001427883"
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    [
      "343.296738030",
      "69.716609697"
    ],
    [
      "57.000000001",
      "268.235679110"
    ]
  ],
  "radius": "268.235679110"
}

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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