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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 = 59

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 = 59
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[500.78137165915, 500.78137165925]André Müller2025-12Packomania ↗
本站纪录500.781371681NUE_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

500.781371681y0
0x500.781371681
已验证构造side = 500.781371681

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 = 59 的当前纪录构型，500.781371681
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
    500.781371683→500.781371681
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    500.781371684→500.781371683
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    500.781371685→500.781371684
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    604→500.781371685
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": [
    [
      "71.699376303",
      "497.486271087"
    ],
    [
      "445.914181453",
      "415.134737437"
    ],
    [
      "244.849057032",
      "6.888875167"
    ],
    [
      "275.622755441",
      "173.490563465"
    ],
    [
      "310.411889116",
      "11.901540916"
    ],
    [
      "490.582970521",
      "196.716132122"
    ],
    [
      "93.047002037",
      "7.758567497"
    ],
    [
      "492.538050866",
      "391.210755444"
    ],
    [
      "441.623234626",
      "491.728475097"
    ],
    [
      "200.888018998",
      "238.991658232"
    ],
    [
      "232.218261659",
      "488.589749473"
    ],
    [
      "12.056841314",
      "101.280800053"
    ],
    [
      "487.781371679",
      "286.290611125"
    ],
    [
      "339.819296414",
      "486.781371680"
    ],
    [
      "168.528390447",
      "320.511678237"
    ],
    [
      "164.114427663",
      "252.371499977"
    ],
    [
      "17.047844226",
      "254.005533222"
    ],
    [
      "274.943271589",
      "371.183800096"
    ],
    [
      "111.633663524",
      "172.130319477"
    ],
    [
      "150.665216622",
      "411.530453896"
    ],
    [
      "149.358384754",
      "288.461041063"
    ],
    [
      "22.000000000",
      "214.826764932"
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    ],
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    ],
    [
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    [
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    [
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    [
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    ],
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    ]
  ],
  "side": "500.781371681"
}
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": 59
}

当前第一名的答案

{
  "centers": [
    [
      "71.699376303",
      "497.486271087"
    ],
    [
      "445.914181453",
      "415.134737437"
    ],
    [
      "244.849057032",
      "6.888875167"
    ],
    [
      "275.622755441",
      "173.490563465"
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      "11.901540916"
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    [
      "490.582970521",
      "196.716132122"
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      "7.758567497"
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      "492.538050866",
      "391.210755444"
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      "491.728475097"
    ],
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      "200.888018998",
      "238.991658232"
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      "232.218261659",
      "488.589749473"
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    ],
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    ],
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      "149.358384754",
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  ],
  "side": "500.781371681"
}

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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