P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 44

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 = 44
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[324.88069895365, 324.88069895375]André Müller2025-12Packomania ↗
本站纪录324.880698972NUE_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

324.880698972y0
0x324.880698972
已验证构造side = 324.880698972

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 = 44 的当前纪录构型,324.880698972
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
    324.880698973→324.880698972
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    324.880698974→324.880698973
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    388→324.880698974
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": [
    [
      "131.484127105",
      "79.115169403"
    ],
    [
      "47.464802931",
      "3.703092990"
    ],
    [
      "135.669246989",
      "321.567779550"
    ],
    [
      "130.369986849",
      "313.150525967"
    ],
    [
      "268.938634878",
      "5.013086127"
    ],
    [
      "9.505223143",
      "269.102290833"
    ],
    [
      "317.337161622",
      "61.940092323"
    ],
    [
      "56.889173986",
      "316.674991697"
    ],
    [
      "315.879577491",
      "126.506079037"
    ],
    [
      "314.728681808",
      "208.533899271"
    ],
    [
      "167.163358572",
      "38.871244176"
    ],
    [
      "12.008883109",
      "57.978276991"
    ],
    [
      "80.422572580",
      "229.756810792"
    ],
    [
      "163.940631679",
      "14.024499122"
    ],
    [
      "15.000000000",
      "85.090540304"
    ],
    [
      "250.987355670",
      "16.032796446"
    ],
    [
      "64.745171125",
      "17.106287808"
    ],
    [
      "268.808538899",
      "306.849200097"
    ],
    [
      "305.847450162",
      "305.465519418"
    ],
    [
      "112.767581304",
      "223.214608496"
    ],
    [
      "21.000000003",
      "183.659963301"
    ],
    [
      "243.976751514",
      "112.119587979"
    ],
    [
      "156.939091606",
      "301.840450411"
    ],
    [
      "24.000000004",
      "24.000000001"
    ],
    [
      "215.369049790",
      "151.968260215"
    ],
    [
      "203.087333704",
      "26.000000001"
    ],
    [
      "27.047248306",
      "297.828954042"
    ],
    [
      "245.030450827",
      "60.011394069"
    ],
    [
      "55.078205523",
      "66.931866278"
    ],
    [
      "294.828200855",
      "30.224079215"
    ],
    [
      "293.880698969",
      "93.096938599"
    ],
    [
      "159.188118950",
      "246.648210116"
    ],
    [
      "33.000000003",
      "129.587731229"
    ],
    [
      "34.000000004",
      "237.101520168"
    ],
    [
      "76.957162019",
      "181.469981644"
    ],
    [
      "218.223327885",
      "212.901445730"
    ],
    [
      "101.882167647",
      "113.465608121"
    ],
    [
      "216.471160430",
      "286.880698969"
    ],
    [
      "117.076680075",
      "39.000000000"
    ],
    [
      "284.880698972",
      "249.408909081"
    ],
    [
      "152.465593451",
      "172.839988409"
    ],
    [
      "94.665202461",
      "282.880698966"
    ],
    [
      "281.880698970",
      "166.117483657"
    ],
    [
      "180.685769573",
      "92.318699665"
    ]
  ],
  "side": "324.880698972"
}
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": 44
}

当前第一名的答案

{
  "centers": [
    [
      "131.484127105",
      "79.115169403"
    ],
    [
      "47.464802931",
      "3.703092990"
    ],
    [
      "135.669246989",
      "321.567779550"
    ],
    [
      "130.369986849",
      "313.150525967"
    ],
    [
      "268.938634878",
      "5.013086127"
    ],
    [
      "9.505223143",
      "269.102290833"
    ],
    [
      "317.337161622",
      "61.940092323"
    ],
    [
      "56.889173986",
      "316.674991697"
    ],
    [
      "315.879577491",
      "126.506079037"
    ],
    [
      "314.728681808",
      "208.533899271"
    ],
    [
      "167.163358572",
      "38.871244176"
    ],
    [
      "12.008883109",
      "57.978276991"
    ],
    [
      "80.422572580",
      "229.756810792"
    ],
    [
      "163.940631679",
      "14.024499122"
    ],
    [
      "15.000000000",
      "85.090540304"
    ],
    [
      "250.987355670",
      "16.032796446"
    ],
    [
      "64.745171125",
      "17.106287808"
    ],
    [
      "268.808538899",
      "306.849200097"
    ],
    [
      "305.847450162",
      "305.465519418"
    ],
    [
      "112.767581304",
      "223.214608496"
    ],
    [
      "21.000000003",
      "183.659963301"
    ],
    [
      "243.976751514",
      "112.119587979"
    ],
    [
      "156.939091606",
      "301.840450411"
    ],
    [
      "24.000000004",
      "24.000000001"
    ],
    [
      "215.369049790",
      "151.968260215"
    ],
    [
      "203.087333704",
      "26.000000001"
    ],
    [
      "27.047248306",
      "297.828954042"
    ],
    [
      "245.030450827",
      "60.011394069"
    ],
    [
      "55.078205523",
      "66.931866278"
    ],
    [
      "294.828200855",
      "30.224079215"
    ],
    [
      "293.880698969",
      "93.096938599"
    ],
    [
      "159.188118950",
      "246.648210116"
    ],
    [
      "33.000000003",
      "129.587731229"
    ],
    [
      "34.000000004",
      "237.101520168"
    ],
    [
      "76.957162019",
      "181.469981644"
    ],
    [
      "218.223327885",
      "212.901445730"
    ],
    [
      "101.882167647",
      "113.465608121"
    ],
    [
      "216.471160430",
      "286.880698969"
    ],
    [
      "117.076680075",
      "39.000000000"
    ],
    [
      "284.880698972",
      "249.408909081"
    ],
    [
      "152.465593451",
      "172.839988409"
    ],
    [
      "94.665202461",
      "282.880698966"
    ],
    [
      "281.880698970",
      "166.117483657"
    ],
    [
      "180.685769573",
      "92.318699665"
    ]
  ],
  "side": "324.880698972"
}

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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