P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 66

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 = 66
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[590.99741225825, 590.99741225835]André Müller2025-12Packomania ↗
本站纪录590.997412298NUE_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

590.997412298y0
0x590.997412298
已验证构造side = 590.997412298

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 = 66 的当前纪录构型,590.997412298
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
    590.997412299→590.997412298
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    590.997412301→590.997412299
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    590.997412302→590.997412301
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    672→590.997412302
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": [
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      "336.222064505"
    ],
    [
      "580.474184146",
      "578.974932165"
    ],
    [
      "253.476083590",
      "572.610398036"
    ],
    [
      "172.895381530",
      "440.257097397"
    ],
    [
      "96.443950951",
      "197.736874047"
    ],
    [
      "89.909952245",
      "295.644705275"
    ],
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      "575.062598819",
      "295.565411264"
    ],
    [
      "379.788396376",
      "11.229451037"
    ],
    [
      "581.379838141",
      "368.550201750"
    ],
    [
      "10.069543961",
      "491.032620003"
    ],
    [
      "112.900103511",
      "11.040049629"
    ],
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      "104.547903974",
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    ],
    [
      "532.782502460",
      "496.323088378"
    ],
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      "14.050972255",
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}
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": 66
}

当前第一名的答案

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      "126.316353470",
      "336.222064505"
    ],
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    ],
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      "572.610398036"
    ],
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      "440.257097397"
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      "197.736874047"
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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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