P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 49

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 = 49
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[381.11431270115, 381.11431270125]André Müller2025-12Packomania ↗
本站纪录381.114312723NUE_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

381.114312723y0
0x381.114312723
已验证构造side = 381.114312723

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 = 49 的当前纪录构型,381.114312723
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
    381.114312724→381.114312723
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    381.114312725→381.114312724
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    448→381.114312725
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": [
    [
      "17.529575934",
      "231.427319212"
    ],
    [
      "162.271287046",
      "378.627236191"
    ],
    [
      "99.094520744",
      "278.863632422"
    ],
    [
      "340.370479272",
      "309.182086583"
    ],
    [
      "8.963500883",
      "5.643963503"
    ],
    [
      "67.181476623",
      "331.924636777"
    ],
    [
      "7.983284423",
      "234.254702526"
    ],
    [
      "313.851183913",
      "8.647018271"
    ],
    [
      "79.992716181",
      "9.051997899"
    ],
    [
      "10.000000000",
      "316.623953557"
    ],
    [
      "171.063842447",
      "368.917346982"
    ],
    [
      "73.526821656",
      "369.056321011"
    ],
    [
      "44.102070037",
      "94.910826019"
    ],
    [
      "367.092407656",
      "150.111770092"
    ],
    [
      "15.129509153",
      "91.188189446"
    ],
    [
      "365.067921756",
      "79.269902533"
    ],
    [
      "237.255555617",
      "186.221883586"
    ],
    [
      "224.487280521",
      "18.000000000"
    ],
    [
      "173.891153695",
      "287.186734778"
    ],
    [
      "361.114312723",
      "116.315704470"
    ],
    [
      "360.053424579",
      "293.468625337"
    ],
    [
      "22.012893007",
      "208.854739309"
    ],
    [
      "219.704720057",
      "59.489680534"
    ],
    [
      "354.062605154",
      "185.871335023"
    ],
    [
      "285.622849916",
      "356.114312721"
    ],
    [
      "69.779848277",
      "203.910257212"
    ],
    [
      "323.530970215",
      "144.537900812"
    ],
    [
      "179.587391877",
      "28.000000001"
    ],
    [
      "320.699362195",
      "88.609535947"
    ],
    [
      "351.114312722",
      "240.278276838"
    ],
    [
      "33.651550550",
      "350.114312721"
    ],
    [
      "89.090737471",
      "95.920678898"
    ],
    [
      "348.114312721",
      "33.000000001"
    ],
    [
      "346.384571977",
      "346.978518408"
    ],
    [
      "116.977488503",
      "35.000000000"
    ],
    [
      "113.089513490",
      "159.545035477"
    ],
    [
      "184.421804087",
      "175.059684002"
    ],
    [
      "276.794067847",
      "38.000000001"
    ],
    [
      "39.000000003",
      "146.659921486"
    ],
    [
      "291.593959508",
      "203.436350262"
    ],
    [
      "41.000000003",
      "41.000000001"
    ],
    [
      "218.979068953",
      "246.100485276"
    ],
    [
      "164.135537485",
      "97.414571710"
    ],
    [
      "294.271393177",
      "287.658467828"
    ],
    [
      "132.281790015",
      "238.434760484"
    ],
    [
      "217.799550081",
      "335.114312719"
    ],
    [
      "47.000000003",
      "273.264986778"
    ],
    [
      "251.733722932",
      "122.854836379"
    ],
    [
      "122.846930312",
      "332.114312719"
    ]
  ],
  "side": "381.114312723"
}
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": 49
}

当前第一名的答案

{
  "centers": [
    [
      "17.529575934",
      "231.427319212"
    ],
    [
      "162.271287046",
      "378.627236191"
    ],
    [
      "99.094520744",
      "278.863632422"
    ],
    [
      "340.370479272",
      "309.182086583"
    ],
    [
      "8.963500883",
      "5.643963503"
    ],
    [
      "67.181476623",
      "331.924636777"
    ],
    [
      "7.983284423",
      "234.254702526"
    ],
    [
      "313.851183913",
      "8.647018271"
    ],
    [
      "79.992716181",
      "9.051997899"
    ],
    [
      "10.000000000",
      "316.623953557"
    ],
    [
      "171.063842447",
      "368.917346982"
    ],
    [
      "73.526821656",
      "369.056321011"
    ],
    [
      "44.102070037",
      "94.910826019"
    ],
    [
      "367.092407656",
      "150.111770092"
    ],
    [
      "15.129509153",
      "91.188189446"
    ],
    [
      "365.067921756",
      "79.269902533"
    ],
    [
      "237.255555617",
      "186.221883586"
    ],
    [
      "224.487280521",
      "18.000000000"
    ],
    [
      "173.891153695",
      "287.186734778"
    ],
    [
      "361.114312723",
      "116.315704470"
    ],
    [
      "360.053424579",
      "293.468625337"
    ],
    [
      "22.012893007",
      "208.854739309"
    ],
    [
      "219.704720057",
      "59.489680534"
    ],
    [
      "354.062605154",
      "185.871335023"
    ],
    [
      "285.622849916",
      "356.114312721"
    ],
    [
      "69.779848277",
      "203.910257212"
    ],
    [
      "323.530970215",
      "144.537900812"
    ],
    [
      "179.587391877",
      "28.000000001"
    ],
    [
      "320.699362195",
      "88.609535947"
    ],
    [
      "351.114312722",
      "240.278276838"
    ],
    [
      "33.651550550",
      "350.114312721"
    ],
    [
      "89.090737471",
      "95.920678898"
    ],
    [
      "348.114312721",
      "33.000000001"
    ],
    [
      "346.384571977",
      "346.978518408"
    ],
    [
      "116.977488503",
      "35.000000000"
    ],
    [
      "113.089513490",
      "159.545035477"
    ],
    [
      "184.421804087",
      "175.059684002"
    ],
    [
      "276.794067847",
      "38.000000001"
    ],
    [
      "39.000000003",
      "146.659921486"
    ],
    [
      "291.593959508",
      "203.436350262"
    ],
    [
      "41.000000003",
      "41.000000001"
    ],
    [
      "218.979068953",
      "246.100485276"
    ],
    [
      "164.135537485",
      "97.414571710"
    ],
    [
      "294.271393177",
      "287.658467828"
    ],
    [
      "132.281790015",
      "238.434760484"
    ],
    [
      "217.799550081",
      "335.114312719"
    ],
    [
      "47.000000003",
      "273.264986778"
    ],
    [
      "251.733722932",
      "122.854836379"
    ],
    [
      "122.846930312",
      "332.114312719"
    ]
  ],
  "side": "381.114312723"
}

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

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖;发言使用与排行榜一致的公开署名,并附带唯一注册编号(#UID)以防冒名。新发言经审核后公开。

暂无讨论,欢迎发表第一篇帖子。