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P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 54

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 = 54
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[439.84590899675, 439.84590899685]André Müller2025-12Packomania ↗
本站纪录439.845909025NUE_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

439.845909025y0
0x439.845909025
已验证构造side = 439.845909025

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 = 54 的当前纪录构型，439.845909025
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（4 次易主）
  1. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    439.845909027→439.845909025
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    439.845909028→439.845909027
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    439.845909029→439.845909028
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    520→439.845909029
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": [
    [
      "5.343378002",
      "367.765640363"
    ],
    [
      "375.308691401",
      "101.901489460"
    ],
    [
      "47.465690390",
      "319.141391616"
    ],
    [
      "116.724128400",
      "396.112040725"
    ],
    [
      "107.178091654",
      "76.145958129"
    ],
    [
      "155.190650391",
      "370.855921396"
    ],
    [
      "7.056354579",
      "313.342083547"
    ],
    [
      "9.385794723",
      "8.860017129"
    ],
    [
      "429.927462428",
      "95.874342411"
    ],
    [
      "351.100626792",
      "428.253522581"
    ],
    [
      "341.904931198",
      "12.722285969"
    ],
    [
      "104.184033866",
      "12.037350459"
    ],
    [
      "13.000000000",
      "105.497920591"
    ],
    [
      "310.307094026",
      "191.238772132"
    ],
    [
      "162.045414139",
      "106.622011221"
    ],
    [
      "107.061249833",
      "161.663175343"
    ],
    [
      "422.743146819",
      "300.624646312"
    ],
    [
      "199.018927301",
      "421.582512780"
    ],
    [
      "309.841994453",
      "158.242049831"
    ],
    [
      "419.845909022",
      "338.315477488"
    ],
    [
      "238.368613909",
      "418.818722464"
    ],
    [
      "417.845909022",
      "188.871538244"
    ],
    [
      "97.126982002",
      "416.845909010"
    ],
    [
      "283.275435921",
      "122.369632424"
    ],
    [
      "25.000000001",
      "340.201768990"
    ],
    [
      "26.000000001",
      "220.825798088"
    ],
    [
      "222.236234548",
      "27.000000000"
    ],
    [
      "185.448454974",
      "224.873006934"
    ],
    [
      "281.262921751",
      "314.129790804"
    ],
    [
      "168.832097884",
      "282.258374138"
    ],
    [
      "150.531101697",
      "408.845909025"
    ],
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      "112.691013102",
      "113.514621852"
    ],
    [
      "406.839968388",
      "131.550447174"
    ],
    [
      "341.697402565",
      "115.883662533"
    ],
    [
      "35.361388807",
      "281.103193521"
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    [
      "360.102614167",
      "183.420675989"
    ],
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    [
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    ],
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      "160.888191330"
    ],
    [
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      "248.201126145"
    ],
    [
      "361.759501463",
      "319.688366684"
    ],
    [
      "220.305942289",
      "101.175932876"
    ],
    [
      "396.539502509",
      "396.845909024"
    ],
    [
      "314.417258848",
      "249.092955607"
    ],
    [
      "46.495385223",
      "152.848307804"
    ],
    [
      "151.752193796",
      "46.000000001"
    ],
    [
      "249.605256131",
      "185.214913174"
    ],
    [
      "293.846352432",
      "49.293295346"
    ],
    [
      "390.845909024",
      "49.000000002"
    ],
    [
      "303.803317547",
      "389.845909023"
    ],
    [
      "210.004229555",
      "352.392857169"
    ],
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    [
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    ],
    [
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    ]
  ],
  "side": "439.845909025"
}
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": 54
}

当前第一名的答案

{
  "centers": [
    [
      "5.343378002",
      "367.765640363"
    ],
    [
      "375.308691401",
      "101.901489460"
    ],
    [
      "47.465690390",
      "319.141391616"
    ],
    [
      "116.724128400",
      "396.112040725"
    ],
    [
      "107.178091654",
      "76.145958129"
    ],
    [
      "155.190650391",
      "370.855921396"
    ],
    [
      "7.056354579",
      "313.342083547"
    ],
    [
      "9.385794723",
      "8.860017129"
    ],
    [
      "429.927462428",
      "95.874342411"
    ],
    [
      "351.100626792",
      "428.253522581"
    ],
    [
      "341.904931198",
      "12.722285969"
    ],
    [
      "104.184033866",
      "12.037350459"
    ],
    [
      "13.000000000",
      "105.497920591"
    ],
    [
      "310.307094026",
      "191.238772132"
    ],
    [
      "162.045414139",
      "106.622011221"
    ],
    [
      "107.061249833",
      "161.663175343"
    ],
    [
      "422.743146819",
      "300.624646312"
    ],
    [
      "199.018927301",
      "421.582512780"
    ],
    [
      "309.841994453",
      "158.242049831"
    ],
    [
      "419.845909022",
      "338.315477488"
    ],
    [
      "238.368613909",
      "418.818722464"
    ],
    [
      "417.845909022",
      "188.871538244"
    ],
    [
      "97.126982002",
      "416.845909010"
    ],
    [
      "283.275435921",
      "122.369632424"
    ],
    [
      "25.000000001",
      "340.201768990"
    ],
    [
      "26.000000001",
      "220.825798088"
    ],
    [
      "222.236234548",
      "27.000000000"
    ],
    [
      "185.448454974",
      "224.873006934"
    ],
    [
      "281.262921751",
      "314.129790804"
    ],
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      "282.258374138"
    ],
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      "408.845909025"
    ],
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      "115.883662533"
    ],
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      "281.103193521"
    ],
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      "267.800698431"
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    ],
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    ],
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    ],
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      "152.848307804"
    ],
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      "49.000000002"
    ],
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      "303.803317547",
      "389.845909023"
    ],
    [
      "210.004229555",
      "352.392857169"
    ],
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      "105.598047206",
      "229.763035574"
    ],
    [
      "53.000000003",
      "53.000301594"
    ],
    [
      "105.001096015",
      "336.942247789"
    ]
  ],
  "side": "439.845909025"
}

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

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