P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 65

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 = 65
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[577.88919969095, 577.88919969105]André Müller2025-12Packomania ↗
本站纪录577.889199731NUE_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

577.889199731y0
0x577.889199731
已验证构造side = 577.889199731

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 = 65 的当前纪录构型,577.889199731
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
    577.889199732→577.889199731
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    577.889199733→577.889199732
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    658→577.889199733
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": [
    [
      "142.829361805",
      "386.263885896"
    ],
    [
      "571.693730096",
      "83.635311658"
    ],
    [
      "454.348016965",
      "571.506124881"
    ],
    [
      "566.086949227",
      "487.803222825"
    ],
    [
      "284.005488699",
      "111.708556635"
    ],
    [
      "47.400470054",
      "414.983031459"
    ],
    [
      "568.620195721",
      "334.714644169"
    ],
    [
      "149.252358307",
      "568.130085436"
    ],
    [
      "12.514879968",
      "79.822034675"
    ],
    [
      "163.173926322",
      "10.068899570"
    ],
    [
      "248.017918206",
      "11.176467311"
    ],
    [
      "235.678993155",
      "565.806520239"
    ],
    [
      "564.886020912",
      "105.257073113"
    ],
    [
      "474.974173481",
      "563.889199695"
    ],
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      "15.103349345",
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    ],
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    ],
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    ],
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      "421.087975213"
    ],
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    ],
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      "556.889199714",
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    ]
  ],
  "side": "577.889199731"
}
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": 65
}

当前第一名的答案

{
  "centers": [
    [
      "142.829361805",
      "386.263885896"
    ],
    [
      "571.693730096",
      "83.635311658"
    ],
    [
      "454.348016965",
      "571.506124881"
    ],
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      "566.086949227",
      "487.803222825"
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      "111.708556635"
    ],
    [
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      "414.983031459"
    ],
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      "568.620195721",
      "334.714644169"
    ],
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      "149.252358307",
      "568.130085436"
    ],
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      "79.822034675"
    ],
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      "10.068899570"
    ],
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      "199.783286690"
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      "561.889199710"
    ],
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      "421.087975213"
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    ],
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      "259.537915955"
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      "121.180151234"
    ],
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  "side": "577.889199731"
}

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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