P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 42

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 = 42
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[303.92254496385, 303.92254496395]André Müller2025-12Packomania ↗
本站纪录303.922544989NUE_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

303.922544989y0
0x303.922544989
已验证构造side = 303.922544989

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 = 42 的当前纪录构型,303.922544989
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
    303.922544992→303.922544989
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    303.922544993→303.922544992
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    368→303.922544993
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": [
    [
      "69.492277000",
      "302.663970770"
    ],
    [
      "203.622506536",
      "258.029966974"
    ],
    [
      "121.265793407",
      "58.354671934"
    ],
    [
      "176.847642073",
      "6.464608960"
    ],
    [
      "201.890674006",
      "297.154223523"
    ],
    [
      "62.379437722",
      "295.051769035"
    ],
    [
      "296.863547620",
      "255.279444214"
    ],
    [
      "8.000000010",
      "133.557819483"
    ],
    [
      "113.900013137",
      "9.390704201"
    ],
    [
      "137.820452242",
      "269.000965443"
    ],
    [
      "56.312332263",
      "240.670670298"
    ],
    [
      "134.723522048",
      "291.902435796"
    ],
    [
      "75.606067744",
      "98.004497502"
    ],
    [
      "107.951146726",
      "153.862607193"
    ],
    [
      "176.527066358",
      "128.626031653"
    ],
    [
      "287.922544986",
      "90.560768061"
    ],
    [
      "17.000000001",
      "56.299465540"
    ],
    [
      "18.093661871",
      "19.528702885"
    ],
    [
      "284.922544989",
      "169.550442045"
    ],
    [
      "253.258944601",
      "100.278525445"
    ],
    [
      "282.902114743",
      "128.674784459"
    ],
    [
      "204.517892492",
      "22.000000009"
    ],
    [
      "280.922544964",
      "280.922544972"
    ],
    [
      "74.216110173",
      "136.371257647"
    ],
    [
      "25.000000007",
      "219.361501528"
    ],
    [
      "140.276869507",
      "233.084868572"
    ],
    [
      "173.962513305",
      "276.922544981"
    ],
    [
      "229.156840852",
      "275.776848756"
    ],
    [
      "29.000000000",
      "164.020911917"
    ],
    [
      "31.934878474",
      "273.922544966"
    ],
    [
      "31.144784132",
      "102.168028655"
    ],
    [
      "197.923875940",
      "223.007305010"
    ],
    [
      "94.863409369",
      "270.922544973"
    ],
    [
      "149.818715162",
      "34.000000000"
    ],
    [
      "152.886607792",
      "173.402421396"
    ],
    [
      "200.083458938",
      "82.718123259"
    ],
    [
      "266.922544981",
      "222.578736380"
    ],
    [
      "85.138944743",
      "200.591649996"
    ],
    [
      "128.074173270",
      "103.686260472"
    ],
    [
      "263.847480390",
      "40.000000018"
    ],
    [
      "226.669281328",
      "155.178341541"
    ],
    [
      "74.240940649",
      "42.000000019"
    ]
  ],
  "side": "303.922544989"
}
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": 42
}

当前第一名的答案

{
  "centers": [
    [
      "69.492277000",
      "302.663970770"
    ],
    [
      "203.622506536",
      "258.029966974"
    ],
    [
      "121.265793407",
      "58.354671934"
    ],
    [
      "176.847642073",
      "6.464608960"
    ],
    [
      "201.890674006",
      "297.154223523"
    ],
    [
      "62.379437722",
      "295.051769035"
    ],
    [
      "296.863547620",
      "255.279444214"
    ],
    [
      "8.000000010",
      "133.557819483"
    ],
    [
      "113.900013137",
      "9.390704201"
    ],
    [
      "137.820452242",
      "269.000965443"
    ],
    [
      "56.312332263",
      "240.670670298"
    ],
    [
      "134.723522048",
      "291.902435796"
    ],
    [
      "75.606067744",
      "98.004497502"
    ],
    [
      "107.951146726",
      "153.862607193"
    ],
    [
      "176.527066358",
      "128.626031653"
    ],
    [
      "287.922544986",
      "90.560768061"
    ],
    [
      "17.000000001",
      "56.299465540"
    ],
    [
      "18.093661871",
      "19.528702885"
    ],
    [
      "284.922544989",
      "169.550442045"
    ],
    [
      "253.258944601",
      "100.278525445"
    ],
    [
      "282.902114743",
      "128.674784459"
    ],
    [
      "204.517892492",
      "22.000000009"
    ],
    [
      "280.922544964",
      "280.922544972"
    ],
    [
      "74.216110173",
      "136.371257647"
    ],
    [
      "25.000000007",
      "219.361501528"
    ],
    [
      "140.276869507",
      "233.084868572"
    ],
    [
      "173.962513305",
      "276.922544981"
    ],
    [
      "229.156840852",
      "275.776848756"
    ],
    [
      "29.000000000",
      "164.020911917"
    ],
    [
      "31.934878474",
      "273.922544966"
    ],
    [
      "31.144784132",
      "102.168028655"
    ],
    [
      "197.923875940",
      "223.007305010"
    ],
    [
      "94.863409369",
      "270.922544973"
    ],
    [
      "149.818715162",
      "34.000000000"
    ],
    [
      "152.886607792",
      "173.402421396"
    ],
    [
      "200.083458938",
      "82.718123259"
    ],
    [
      "266.922544981",
      "222.578736380"
    ],
    [
      "85.138944743",
      "200.591649996"
    ],
    [
      "128.074173270",
      "103.686260472"
    ],
    [
      "263.847480390",
      "40.000000018"
    ],
    [
      "226.669281328",
      "155.178341541"
    ],
    [
      "74.240940649",
      "42.000000019"
    ]
  ],
  "side": "303.922544989"
}

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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