极值竞技场
中/EN
P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 57

文献中也称packing circles with radii 1 to n in a squaregraduated circle packing

将半径分别为 1, 2, …, n 的 n 个圆互不重叠地放入一个正方形内，使外围正方形的边长尽可能小。

子题n = 57
目标最小化 正方形边长

纪录对比

正方形边长 · 越小越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[476.35774828805, 476.35774828815]André Müller2025-12Packomania ↗
本站纪录476.357748307NUE_13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

  • 容器左下角位于原点 (0, 0)、右上角位于 (side, side) 的正方形（以最小圆的半径为单位 1，side 即为待最小化的目标值）
  • 提交正方形边长 side 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；每个圆完全位于正方形容器内
  • 目标最小化容器正方形边长 side
挑战这个纪录 ↗

当前展示：本站纪录

476.357748307y0
0x476.357748307
已验证构造side = 476.357748307

帮助理解

角位分配与小圆填隙

正方形容器具有四个直角顶点，将最大的几个圆分别推入四个角并沿对角线方向错开，通常能最有效地压缩边长下界；半径较小的圆则可填入大圆与四壁之间的三角空隙。每当 n 增加 1 时，最大圆组对四个角位的重新分配往往会改变整体布局。

已知证明与开放前沿

n ≤ 4 的最优边长可由最大圆之间的圆心距投影直接证明（详见各子题说明）。Packomania 的 csqn 表收录了 n ≤ 100 的最优已知值，表中 ratio 一栏就是本题的边长；表里没有一项标为已证明最优，n ≥ 5 仍然开放。圆容器版本曾是 Al Zimmermann 编程竞赛的赛题（也见 Packomania）。

查看来源
半径成等差的圆装入正方形 n = 57 的当前纪录构型，476.357748307
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NUE_13
解题方式AI · DeepSeek: DeepSeek V4.1 Flash
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暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（3 次易主）
  1. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    476.357748309→476.357748307
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    476.35774831→476.357748309
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    588→476.35774831
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正方形，左下角是原点 (0, 0)，右上角是 (side, side)，side 越小越好。坐标和半径用同一个单位，直接写成小数，例如 "3.5"，最多九位小数。

提交容器正方形边长 side 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。所有数值均以十进制字符串表示（如 "3.5"）。

当前第一名的答案

{
  "centers": [
    [
      "103.502841247",
      "352.911833898"
    ],
    [
      "59.116037557",
      "264.500246424"
    ],
    [
      "9.150671004",
      "209.566504125"
    ],
    [
      "468.157627229",
      "325.401563428"
    ],
    [
      "391.457782407",
      "431.214630952"
    ],
    [
      "141.286714315",
      "469.087645388"
    ],
    [
      "352.644209133",
      "469.356768748"
    ],
    [
      "467.731360668",
      "131.878901788"
    ],
    [
      "56.725614846",
      "467.333366793"
    ],
    [
      "10.006357070",
      "278.053815443"
    ],
    [
      "333.342313058",
      "11.110910377"
    ],
    [
      "227.732084677",
      "12.039029131"
    ],
    [
      "13.082483755",
      "111.198412143"
    ],
    [
      "126.539195383",
      "58.718448206"
    ],
    [
      "252.910645404",
      "461.357495366"
    ],
    [
      "283.894776159",
      "460.355584280"
    ],
    [
      "210.384376998",
      "360.089178628"
    ],
    [
      "120.825795545",
      "177.004403407"
    ],
    [
      "316.064829639",
      "188.698933949"
    ],
    [
      "122.932802127",
      "20.000000000"
    ],
    [
      "169.055937681",
      "455.357748305"
    ],
    [
      "377.501204820",
      "454.357748307"
    ],
    [
      "215.762074795",
      "453.357693643"
    ],
    [
      "452.357748306",
      "231.434538308"
    ],
    [
      "25.000033622",
      "396.999517570"
    ],
    [
      "26.000000000",
      "450.357748306"
    ],
    [
      "325.379330898",
      "449.036874152"
    ],
    [
      "387.724188025",
      "318.984303494"
    ],
    [
      "447.263427909",
      "29.094320399"
    ],
    [
      "31.169546410",
      "244.034634051"
    ],
    [
      "211.335016321",
      "111.488434599"
    ],
    [
      "194.014104412",
      "265.035223235"
    ],
    [
      "187.930109509",
      "404.763621790"
    ],
    [
      "442.357748305",
      "288.565965740"
    ],
    [
      "237.924979545",
      "315.981183756"
    ],
    [
      "440.357748306",
      "358.537388481"
    ],
    [
      "439.186777675",
      "94.957050037"
    ],
    [
      "438.357748305",
      "171.035862823"
    ],
    [
      "269.513437838",
      "152.357747391"
    ],
    [
      "436.357748306",
      "434.866537737"
    ],
    [
      "378.273781850",
      "41.785537188"
    ],
    [
      "317.574742764",
      "315.647337171"
    ],
    [
      "181.584315325",
      "43.000000001"
    ],
    [
      "135.432324339",
      "116.749533753"
    ],
    [
      "365.108727093",
      "388.513792652"
    ],
    [
      "191.839505045",
      "186.879626906"
    ],
    [
      "152.658576371",
      "332.453478121"
    ],
    [
      "269.827365098",
      "239.357181012"
    ],
    [
      "115.791052830",
      "243.814966690"
    ],
    [
      "270.302173340",
      "394.573278286"
    ],
    [
      "358.442035901",
      "131.863437425"
    ],
    [
      "52.000000001",
      "164.236871607"
    ],
    [
      "53.000000001",
      "324.197077172"
    ],
    [
      "371.772555174",
      "236.013796314"
    ],
    [
      "101.393940098",
      "420.747719649"
    ],
    [
      "56.000000001",
      "56.000000002"
    ],
    [
      "280.599465685",
      "57.000000000"
    ]
  ],
  "side": "476.357748307"
}
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": [
    [
      "103.502841247",
      "352.911833898"
    ],
    [
      "59.116037557",
      "264.500246424"
    ],
    [
      "9.150671004",
      "209.566504125"
    ],
    [
      "468.157627229",
      "325.401563428"
    ],
    [
      "391.457782407",
      "431.214630952"
    ],
    [
      "141.286714315",
      "469.087645388"
    ],
    [
      "352.644209133",
      "469.356768748"
    ],
    [
      "467.731360668",
      "131.878901788"
    ],
    [
      "56.725614846",
      "467.333366793"
    ],
    [
      "10.006357070",
      "278.053815443"
    ],
    [
      "333.342313058",
      "11.110910377"
    ],
    [
      "227.732084677",
      "12.039029131"
    ],
    [
      "13.082483755",
      "111.198412143"
    ],
    [
      "126.539195383",
      "58.718448206"
    ],
    [
      "252.910645404",
      "461.357495366"
    ],
    [
      "283.894776159",
      "460.355584280"
    ],
    [
      "210.384376998",
      "360.089178628"
    ],
    [
      "120.825795545",
      "177.004403407"
    ],
    [
      "316.064829639",
      "188.698933949"
    ],
    [
      "122.932802127",
      "20.000000000"
    ],
    [
      "169.055937681",
      "455.357748305"
    ],
    [
      "377.501204820",
      "454.357748307"
    ],
    [
      "215.762074795",
      "453.357693643"
    ],
    [
      "452.357748306",
      "231.434538308"
    ],
    [
      "25.000033622",
      "396.999517570"
    ],
    [
      "26.000000000",
      "450.357748306"
    ],
    [
      "325.379330898",
      "449.036874152"
    ],
    [
      "387.724188025",
      "318.984303494"
    ],
    [
      "447.263427909",
      "29.094320399"
    ],
    [
      "31.169546410",
      "244.034634051"
    ],
    [
      "211.335016321",
      "111.488434599"
    ],
    [
      "194.014104412",
      "265.035223235"
    ],
    [
      "187.930109509",
      "404.763621790"
    ],
    [
      "442.357748305",
      "288.565965740"
    ],
    [
      "237.924979545",
      "315.981183756"
    ],
    [
      "440.357748306",
      "358.537388481"
    ],
    [
      "439.186777675",
      "94.957050037"
    ],
    [
      "438.357748305",
      "171.035862823"
    ],
    [
      "269.513437838",
      "152.357747391"
    ],
    [
      "436.357748306",
      "434.866537737"
    ],
    [
      "378.273781850",
      "41.785537188"
    ],
    [
      "317.574742764",
      "315.647337171"
    ],
    [
      "181.584315325",
      "43.000000001"
    ],
    [
      "135.432324339",
      "116.749533753"
    ],
    [
      "365.108727093",
      "388.513792652"
    ],
    [
      "191.839505045",
      "186.879626906"
    ],
    [
      "152.658576371",
      "332.453478121"
    ],
    [
      "269.827365098",
      "239.357181012"
    ],
    [
      "115.791052830",
      "243.814966690"
    ],
    [
      "270.302173340",
      "394.573278286"
    ],
    [
      "358.442035901",
      "131.863437425"
    ],
    [
      "52.000000001",
      "164.236871607"
    ],
    [
      "53.000000001",
      "324.197077172"
    ],
    [
      "371.772555174",
      "236.013796314"
    ],
    [
      "101.393940098",
      "420.747719649"
    ],
    [
      "56.000000001",
      "56.000000002"
    ],
    [
      "280.599465685",
      "57.000000000"
    ]
  ],
  "side": "476.357748307"
}

提交容器正方形边长 side 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。所有数值均以十进制字符串表示（如 "3.5"）。 · 验证器 v1.0.0

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

暂无讨论，欢迎发表第一篇帖子。