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P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 55

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 = 55
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[451.78422596915, 451.78422596925]André Müller2025-12Packomania ↗
本站纪录451.784225991NUE_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

451.784225991y0
0x451.784225991
已验证构造side = 451.784225991

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 = 55 的当前纪录构型，451.784225991
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（4 次易主）
  1. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    451.784225994→451.784225991
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    451.784225995→451.784225994
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    451.784225996→451.784225995
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    550→451.784225996
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": [
    [
      "366.892698077",
      "450.221998239"
    ],
    [
      "447.990684128",
      "38.814128989"
    ],
    [
      "98.618656720",
      "329.235171970"
    ],
    [
      "8.508830406",
      "96.167183648"
    ],
    [
      "217.764460404",
      "445.674686643"
    ],
    [
      "144.379328637",
      "6.024831402"
    ],
    [
      "273.727066119",
      "113.538872591"
    ],
    [
      "8.492777168",
      "10.361225767"
    ],
    [
      "357.248077615",
      "441.150291640"
    ],
    [
      "440.790801117",
      "353.523278015"
    ],
    [
      "11.204689104",
      "364.595675223"
    ],
    [
      "213.889052916",
      "12.024610273"
    ],
    [
      "317.112022662",
      "13.000000000"
    ],
    [
      "124.257456725",
      "55.065264370"
    ],
    [
      "248.224199063",
      "383.868512760"
    ],
    [
      "75.656908433",
      "122.928581145"
    ],
    [
      "200.318621870",
      "86.289948602"
    ],
    [
      "281.132760103",
      "379.079616235"
    ],
    [
      "120.078247213",
      "19.059841566"
    ],
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      "431.743194157",
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    ],
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      "178.195924115"
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    [
      "429.784225988",
      "22.000000012"
    ],
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    ],
    [
      "110.472202688",
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    ],
    [
      "358.667884211",
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    ],
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    ],
    [
      "284.158918876",
      "204.588166983"
    ],
    [
      "29.110067124",
      "198.867930119"
    ],
    [
      "105.889387407",
      "371.717732310"
    ],
    [
      "31.000000000",
      "137.583291296"
    ],
    [
      "83.864442871",
      "171.852091689"
    ],
    [
      "344.082308081",
      "215.998137469"
    ],
    [
      "319.034172833",
      "417.665856297"
    ],
    [
      "296.256288728",
      "266.415779286"
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    [
      "80.822728106",
      "239.784028007"
    ],
    [
      "172.677557405",
      "38.718873544"
    ],
    [
      "412.670470744",
      "234.346541844"
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    [
      "412.766265795",
      "120.167508633"
    ],
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      "312.341506879"
    ],
    [
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    ],
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      "203.286197582"
    ],
    [
      "43.000000009",
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    ],
    [
      "242.968081822",
      "324.869471863"
    ],
    [
      "330.915301209",
      "338.518156878"
    ],
    [
      "137.004734193",
      "113.959507899"
    ],
    [
      "365.861460841",
      "47.978454479"
    ],
    [
      "48.098780517",
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    [
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    ],
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    ],
    [
      "227.066587606",
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    ],
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      "182.936593285",
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    ],
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    ]
  ],
  "side": "451.784225991"
}
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": 55
}

当前第一名的答案

{
  "centers": [
    [
      "366.892698077",
      "450.221998239"
    ],
    [
      "447.990684128",
      "38.814128989"
    ],
    [
      "98.618656720",
      "329.235171970"
    ],
    [
      "8.508830406",
      "96.167183648"
    ],
    [
      "217.764460404",
      "445.674686643"
    ],
    [
      "144.379328637",
      "6.024831402"
    ],
    [
      "273.727066119",
      "113.538872591"
    ],
    [
      "8.492777168",
      "10.361225767"
    ],
    [
      "357.248077615",
      "441.150291640"
    ],
    [
      "440.790801117",
      "353.523278015"
    ],
    [
      "11.204689104",
      "364.595675223"
    ],
    [
      "213.889052916",
      "12.024610273"
    ],
    [
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      "13.000000000"
    ],
    [
      "124.257456725",
      "55.065264370"
    ],
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      "75.656908433",
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      "379.079616235"
    ],
    [
      "120.078247213",
      "19.059841566"
    ],
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    ],
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      "22.000000012"
    ],
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      "251.511436761"
    ],
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      "385.796312217",
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    ],
    [
      "110.472202688",
      "426.705565254"
    ],
    [
      "358.667884211",
      "273.166843676"
    ],
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    ],
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    ],
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      "149.985367506"
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      "182.936593285",
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      "299.254127197"
    ],
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      "55.000000006",
      "55.000000001"
    ]
  ],
  "side": "451.784225991"
}

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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