P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 46

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 = 46
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[347.36701443765, 347.36701443775]André Müller2025-12Packomania ↗
本站纪录347.367014456NUE_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

347.367014456y0
0x347.367014456
已验证构造side = 347.367014456

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 = 46 的当前纪录构型,347.367014456
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
    347.367014457→347.367014456
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    347.367014458→347.367014457
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    410→347.367014458
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": [
    [
      "257.284491818",
      "305.985165931"
    ],
    [
      "113.583397481",
      "197.349766082"
    ],
    [
      "5.727770775",
      "154.651370624"
    ],
    [
      "241.453160608",
      "308.569700611"
    ],
    [
      "217.394591313",
      "94.971223837"
    ],
    [
      "341.355899659",
      "281.105791836"
    ],
    [
      "339.473117664",
      "219.478177660"
    ],
    [
      "338.769689576",
      "93.413505217"
    ],
    [
      "9.202105432",
      "276.094169986"
    ],
    [
      "75.614620901",
      "10.018524656"
    ],
    [
      "162.345891863",
      "11.060871808"
    ],
    [
      "268.935449847",
      "58.278697619"
    ],
    [
      "305.617763496",
      "13.024082051"
    ],
    [
      "333.278336827",
      "14.277862951"
    ],
    [
      "160.942203513",
      "84.255109697"
    ],
    [
      "198.482195199",
      "220.548585380"
    ],
    [
      "83.823946476",
      "330.359326301"
    ],
    [
      "165.677600565",
      "329.358333992"
    ],
    [
      "19.014386144",
      "90.070484785"
    ],
    [
      "257.871072038",
      "327.357385708"
    ],
    [
      "21.000000001",
      "179.486633378"
    ],
    [
      "325.342397515",
      "122.219593446"
    ],
    [
      "270.895456279",
      "23.000000002"
    ],
    [
      "124.157270005",
      "322.897594155"
    ],
    [
      "25.010954289",
      "133.661832288"
    ],
    [
      "321.367014456",
      "249.248903952"
    ],
    [
      "104.207356320",
      "226.524912994"
    ],
    [
      "210.553201430",
      "319.156604693"
    ],
    [
      "90.716105661",
      "280.875498602"
    ],
    [
      "68.429335156",
      "99.540514360"
    ],
    [
      "127.741738892",
      "118.324306241"
    ],
    [
      "190.696616815",
      "120.708283188"
    ],
    [
      "314.367014456",
      "58.301892729"
    ],
    [
      "229.495886462",
      "178.734039068"
    ],
    [
      "35.000000000",
      "312.367014450"
    ],
    [
      "311.349595888",
      "310.654300452"
    ],
    [
      "37.000000002",
      "37.000000005"
    ],
    [
      "78.614449554",
      "166.775409869"
    ],
    [
      "39.000000000",
      "236.722985467"
    ],
    [
      "155.656473423",
      "183.606521252"
    ],
    [
      "263.139063195",
      "111.703175580"
    ],
    [
      "305.367014453",
      "183.158057381"
    ],
    [
      "161.356179189",
      "266.947827849"
    ],
    [
      "117.696964013",
      "44.000000000"
    ],
    [
      "206.686159686",
      "45.386742316"
    ],
    [
      "249.697672753",
      "256.141324457"
    ]
  ],
  "side": "347.367014456"
}
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": 46
}

当前第一名的答案

{
  "centers": [
    [
      "257.284491818",
      "305.985165931"
    ],
    [
      "113.583397481",
      "197.349766082"
    ],
    [
      "5.727770775",
      "154.651370624"
    ],
    [
      "241.453160608",
      "308.569700611"
    ],
    [
      "217.394591313",
      "94.971223837"
    ],
    [
      "341.355899659",
      "281.105791836"
    ],
    [
      "339.473117664",
      "219.478177660"
    ],
    [
      "338.769689576",
      "93.413505217"
    ],
    [
      "9.202105432",
      "276.094169986"
    ],
    [
      "75.614620901",
      "10.018524656"
    ],
    [
      "162.345891863",
      "11.060871808"
    ],
    [
      "268.935449847",
      "58.278697619"
    ],
    [
      "305.617763496",
      "13.024082051"
    ],
    [
      "333.278336827",
      "14.277862951"
    ],
    [
      "160.942203513",
      "84.255109697"
    ],
    [
      "198.482195199",
      "220.548585380"
    ],
    [
      "83.823946476",
      "330.359326301"
    ],
    [
      "165.677600565",
      "329.358333992"
    ],
    [
      "19.014386144",
      "90.070484785"
    ],
    [
      "257.871072038",
      "327.357385708"
    ],
    [
      "21.000000001",
      "179.486633378"
    ],
    [
      "325.342397515",
      "122.219593446"
    ],
    [
      "270.895456279",
      "23.000000002"
    ],
    [
      "124.157270005",
      "322.897594155"
    ],
    [
      "25.010954289",
      "133.661832288"
    ],
    [
      "321.367014456",
      "249.248903952"
    ],
    [
      "104.207356320",
      "226.524912994"
    ],
    [
      "210.553201430",
      "319.156604693"
    ],
    [
      "90.716105661",
      "280.875498602"
    ],
    [
      "68.429335156",
      "99.540514360"
    ],
    [
      "127.741738892",
      "118.324306241"
    ],
    [
      "190.696616815",
      "120.708283188"
    ],
    [
      "314.367014456",
      "58.301892729"
    ],
    [
      "229.495886462",
      "178.734039068"
    ],
    [
      "35.000000000",
      "312.367014450"
    ],
    [
      "311.349595888",
      "310.654300452"
    ],
    [
      "37.000000002",
      "37.000000005"
    ],
    [
      "78.614449554",
      "166.775409869"
    ],
    [
      "39.000000000",
      "236.722985467"
    ],
    [
      "155.656473423",
      "183.606521252"
    ],
    [
      "263.139063195",
      "111.703175580"
    ],
    [
      "305.367014453",
      "183.158057381"
    ],
    [
      "161.356179189",
      "266.947827849"
    ],
    [
      "117.696964013",
      "44.000000000"
    ],
    [
      "206.686159686",
      "45.386742316"
    ],
    [
      "249.697672753",
      "256.141324457"
    ]
  ],
  "side": "347.367014456"
}

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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