P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 70

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 = 70
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[644.76519870645, 644.76519870655]André Müller2025-12Packomania ↗
本站纪录644.765198771NUE_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

644.765198771y0
0x644.765198771
已验证构造side = 644.765198771

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 = 70 的当前纪录构型,644.765198771
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
    644.765198772→644.765198771
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    644.765198773→644.765198772
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    756→644.765198773
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

{
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    [
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    ],
    [
      "132.605796630",
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    ],
    [
      "259.283123313",
      "641.743631588"
    ],
    [
      "531.796706156",
      "7.215107187"
    ],
    [
      "82.451850567",
      "10.274516884"
    ],
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      "7.674629148",
      "153.680696513"
    ],
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      "95.155349995",
      "343.394940255"
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    [
      "632.662784031",
      "571.668438585"
    ],
    [
      "570.950946806",
      "634.640579268"
    ],
    [
      "634.693308463",
      "427.188192258"
    ],
    [
      "12.432424119",
      "241.737280956"
    ],
    [
      "238.728828380",
      "632.696201334"
    ],
    [
      "277.070476462",
      "333.177846971"
    ],
    [
      "14.626204158",
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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": 70
}

当前第一名的答案

{
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    [
      "3.240399807",
      "519.926246396"
    ],
    [
      "132.605796630",
      "393.660982644"
    ],
    [
      "259.283123313",
      "641.743631588"
    ],
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      "531.796706156",
      "7.215107187"
    ],
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      "10.274516884"
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      "153.680696513"
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      "343.394940255"
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      "571.668438585"
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      "427.188192258"
    ],
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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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