P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 53

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 = 53
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[427.88700534505, 427.88700534515]André Müller2025-12Packomania ↗
本站纪录427.887005367NUE_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

427.887005367y0
0x427.887005367
已验证构造side = 427.887005367

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 = 53 的当前纪录构型,427.887005367
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
    427.887005368→427.887005367
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    427.887005369→427.887005368
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    506→427.887005369
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": [
    [
      "4.023110003",
      "423.863909727"
    ],
    [
      "13.342618281",
      "90.546672154"
    ],
    [
      "78.969102941",
      "6.867892443"
    ],
    [
      "246.646826874",
      "335.098571476"
    ],
    [
      "263.616927048",
      "418.471263239"
    ],
    [
      "6.125373994",
      "202.393228952"
    ],
    [
      "55.532161733",
      "97.472054026"
    ],
    [
      "416.820635930",
      "250.398983895"
    ],
    [
      "418.660993602",
      "75.579712751"
    ],
    [
      "341.206752225",
      "417.826296936"
    ],
    [
      "416.886690659",
      "153.187981835"
    ],
    [
      "415.879717440",
      "337.869060420"
    ],
    [
      "104.647184469",
      "49.073483631"
    ],
    [
      "295.189225215",
      "15.614215729"
    ],
    [
      "187.843201552",
      "321.820091839"
    ],
    [
      "99.048712163",
      "254.820751835"
    ],
    [
      "339.023494563",
      "17.005245885"
    ],
    [
      "101.253459329",
      "18.000000014"
    ],
    [
      "339.940454602",
      "139.649409987"
    ],
    [
      "20.000000006",
      "177.492231830"
    ],
    [
      "206.157572281",
      "21.127487694"
    ],
    [
      "22.000000003",
      "297.657971028"
    ],
    [
      "62.968853655",
      "172.688798759"
    ],
    [
      "24.000000004",
      "403.887005356"
    ],
    [
      "148.928570174",
      "402.887005366"
    ],
    [
      "277.048435297",
      "335.008174834"
    ],
    [
      "255.478289277",
      "27.000000000"
    ],
    [
      "331.048426905",
      "334.978063171"
    ],
    [
      "29.000000003",
      "348.175294792"
    ],
    [
      "229.810227901",
      "237.036885375"
    ],
    [
      "31.000000008",
      "127.692633435"
    ],
    [
      "230.250282958",
      "300.745470064"
    ],
    [
      "145.184815424",
      "344.840829745"
    ],
    [
      "112.398904000",
      "202.785206018"
    ],
    [
      "151.453060926",
      "35.000000009"
    ],
    [
      "301.767175810",
      "391.886858550"
    ],
    [
      "390.637798857",
      "112.999332839"
    ],
    [
      "389.887005363",
      "38.000000013"
    ],
    [
      "83.565261300",
      "307.596527950"
    ],
    [
      "85.819405796",
      "387.323157133"
    ],
    [
      "386.828655675",
      "293.501856280"
    ],
    [
      "184.715514451",
      "179.411976600"
    ],
    [
      "43.000000005",
      "236.143745028"
    ],
    [
      "383.887005366",
      "383.887005367"
    ],
    [
      "45.000000000",
      "46.364560896"
    ],
    [
      "159.827604300",
      "267.209723000"
    ],
    [
      "301.269297745",
      "266.143453658"
    ],
    [
      "218.210602480",
      "379.887005366"
    ],
    [
      "312.335572605",
      "77.430638834"
    ],
    [
      "214.395689038",
      "92.124649390"
    ],
    [
      "277.344112883",
      "171.108799295"
    ],
    [
      "375.887005366",
      "201.080424907"
    ],
    [
      "114.154282143",
      "115.802916739"
    ]
  ],
  "side": "427.887005367"
}
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": [
    [
      "4.023110003",
      "423.863909727"
    ],
    [
      "13.342618281",
      "90.546672154"
    ],
    [
      "78.969102941",
      "6.867892443"
    ],
    [
      "246.646826874",
      "335.098571476"
    ],
    [
      "263.616927048",
      "418.471263239"
    ],
    [
      "6.125373994",
      "202.393228952"
    ],
    [
      "55.532161733",
      "97.472054026"
    ],
    [
      "416.820635930",
      "250.398983895"
    ],
    [
      "418.660993602",
      "75.579712751"
    ],
    [
      "341.206752225",
      "417.826296936"
    ],
    [
      "416.886690659",
      "153.187981835"
    ],
    [
      "415.879717440",
      "337.869060420"
    ],
    [
      "104.647184469",
      "49.073483631"
    ],
    [
      "295.189225215",
      "15.614215729"
    ],
    [
      "187.843201552",
      "321.820091839"
    ],
    [
      "99.048712163",
      "254.820751835"
    ],
    [
      "339.023494563",
      "17.005245885"
    ],
    [
      "101.253459329",
      "18.000000014"
    ],
    [
      "339.940454602",
      "139.649409987"
    ],
    [
      "20.000000006",
      "177.492231830"
    ],
    [
      "206.157572281",
      "21.127487694"
    ],
    [
      "22.000000003",
      "297.657971028"
    ],
    [
      "62.968853655",
      "172.688798759"
    ],
    [
      "24.000000004",
      "403.887005356"
    ],
    [
      "148.928570174",
      "402.887005366"
    ],
    [
      "277.048435297",
      "335.008174834"
    ],
    [
      "255.478289277",
      "27.000000000"
    ],
    [
      "331.048426905",
      "334.978063171"
    ],
    [
      "29.000000003",
      "348.175294792"
    ],
    [
      "229.810227901",
      "237.036885375"
    ],
    [
      "31.000000008",
      "127.692633435"
    ],
    [
      "230.250282958",
      "300.745470064"
    ],
    [
      "145.184815424",
      "344.840829745"
    ],
    [
      "112.398904000",
      "202.785206018"
    ],
    [
      "151.453060926",
      "35.000000009"
    ],
    [
      "301.767175810",
      "391.886858550"
    ],
    [
      "390.637798857",
      "112.999332839"
    ],
    [
      "389.887005363",
      "38.000000013"
    ],
    [
      "83.565261300",
      "307.596527950"
    ],
    [
      "85.819405796",
      "387.323157133"
    ],
    [
      "386.828655675",
      "293.501856280"
    ],
    [
      "184.715514451",
      "179.411976600"
    ],
    [
      "43.000000005",
      "236.143745028"
    ],
    [
      "383.887005366",
      "383.887005367"
    ],
    [
      "45.000000000",
      "46.364560896"
    ],
    [
      "159.827604300",
      "267.209723000"
    ],
    [
      "301.269297745",
      "266.143453658"
    ],
    [
      "218.210602480",
      "379.887005366"
    ],
    [
      "312.335572605",
      "77.430638834"
    ],
    [
      "214.395689038",
      "92.124649390"
    ],
    [
      "277.344112883",
      "171.108799295"
    ],
    [
      "375.887005366",
      "201.080424907"
    ],
    [
      "114.154282143",
      "115.802916739"
    ]
  ],
  "side": "427.887005367"
}

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。