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P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 89

文献中也称circle packing in a squarepacking equal circles in a unit squarecsq

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆，使共同半径尽可能大。

子题n = 89
目标最大化 共同半径

纪录对比

共同半径 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.0539470408575, 0.0539470408585]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器单位正方形：左下角是原点 (0, 0)，右上角是 (1, 1)
  • 提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心 centers
  • 约束每个圆完整落在容器内；两两内部不重叠，相切允许
  • 目标让共同半径尽可能大
放大来摆，然后提交 ↗

当前展示：起步布局

1y0
0x1
已验证构造r = 0.053947040

帮助理解

哪里有优化空间

最优构形是「卡死」的接触结构：圆彼此顶住、顶住边界，常出现斜排、错位、以及不碰任何邻居的游离圆。规整的网格摆法几乎从不最优。

前沿在哪里

n = 1–30 已证明最优，适合观察经典接触结构；n = 31–300 接入 Packomania 的公开前沿，除 n = 36 外最优性仍未知。大规模部分集中成一张表，供算法批量挑战。

查看来源
单位正方形内的等圆装箱 n = 89 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。
ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标和长度用同一个单位，直接写成小数，例如 "0.5"，最多九位小数。

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。

起步布局答案

{
  "centers": [
    [
      "0.167776968",
      "0.053947041"
    ],
    [
      "0.27567105",
      "0.053947041"
    ],
    [
      "0.383565131",
      "0.053947041"
    ],
    [
      "0.491513698",
      "0.053947041"
    ],
    [
      "0.618795532",
      "0.053947041"
    ],
    [
      "0.730264796",
      "0.053947041"
    ],
    [
      "0.838158877",
      "0.053947041"
    ],
    [
      "0.946052959",
      "0.053947041"
    ],
    [
      "0.053952617",
      "0.053952617"
    ],
    [
      "0.555154615",
      "0.141073194"
    ],
    [
      "0.674530164",
      "0.146330934"
    ],
    [
      "0.437539415",
      "0.147370323"
    ],
    [
      "0.113829927",
      "0.147386057"
    ],
    [
      "0.221724009",
      "0.147386057"
    ],
    [
      "0.329618091",
      "0.147386057"
    ],
    [
      "0.78440128",
      "0.147495176"
    ],
    [
      "0.892295362",
      "0.147495176"
    ],
    [
      "0.610889247",
      "0.233457087"
    ],
    [
      "0.501180331",
      "0.234496476"
    ],
    [
      "0.053947041",
      "0.237136671"
    ],
    [
      "0.728481798",
      "0.239767297"
    ],
    [
      "0.383592374",
      "0.240809339"
    ],
    [
      "0.27567105",
      "0.240825072"
    ],
    [
      "0.946052959",
      "0.241043311"
    ],
    [
      "0.837069475",
      "0.242766099"
    ],
    [
      "0.16325618",
      "0.243294343"
    ],
    [
      "0.555706722",
      "0.327598601"
    ],
    [
      "0.663600803",
      "0.327598601"
    ],
    [
      "0.447233291",
      "0.327935492"
    ],
    [
      "0.329645333",
      "0.334248354"
    ],
    [
      "0.221751256",
      "0.334279814"
    ],
    [
      "0.770893321",
      "0.338976146"
    ],
    [
      "0.053947041",
      "0.345030753"
    ],
    [
      "0.946052959",
      "0.348937393"
    ],
    [
      "0.854906472",
      "0.406673437"
    ],
    [
      "0.142172685",
      "0.407138471"
    ],
    [
      "0.501759681",
      "0.421037617"
    ],
    [
      "0.609653762",
      "0.421037617"
    ],
    [
      "0.39328625",
      "0.421374507"
    ],
    [
      "0.253197643",
      "0.440473866"
    ],
    [
      "0.748020494",
      "0.445287805"
    ],
    [
      "0.946052959",
      "0.464409481"
    ],
    [
      "0.053947041",
      "0.469246189"
    ],
    [
      "0.44781264",
      "0.514476632"
    ],
    [
      "0.555706722",
      "0.514476632"
    ],
    [
      "0.663600803",
      "0.514476632"
    ],
    [
      "0.850478835",
      "0.514476632"
    ],
    [
      "0.338770294",
      "0.514482743"
    ],
    [
      "0.15305111",
      "0.514482743"
    ],
    [
      "0.757039819",
      "0.568423673"
    ],
    [
      "0.245910702",
      "0.569421154"
    ],
    [
      "0.946052959",
      "0.572303563"
    ],
    [
      "0.053947041",
      "0.57714027"
    ],
    [
      "0.339918558",
      "0.622370714"
    ],
    [
      "0.663600803",
      "0.622370714"
    ],
    [
      "0.151902846",
      "0.622370714"
    ],
    [
      "0.44781264",
      "0.622370714"
    ],
    [
      "0.555706722",
      "0.622370714"
    ],
    [
      "0.850478835",
      "0.622370714"
    ],
    [
      "0.757039819",
      "0.676317755"
    ],
    [
      "0.245910702",
      "0.677315236"
    ],
    [
      "0.946052959",
      "0.680197645"
    ],
    [
      "0.053947041",
      "0.685034352"
    ],
    [
      "0.151902846",
      "0.730264796"
    ],
    [
      "0.339918558",
      "0.730264796"
    ],
    [
      "0.44781264",
      "0.730264796"
    ],
    [
      "0.555706722",
      "0.730264796"
    ],
    [
      "0.663600803",
      "0.730264796"
    ],
    [
      "0.850478835",
      "0.730264796"
    ],
    [
      "0.757039819",
      "0.784211837"
    ],
    [
      "0.245910702",
      "0.785209317"
    ],
    [
      "0.946052959",
      "0.788091726"
    ],
    [
      "0.053947041",
      "0.792928434"
    ],
    [
      "0.151902846",
      "0.838158877"
    ],
    [
      "0.339918558",
      "0.838158877"
    ],
    [
      "0.44781264",
      "0.838158877"
    ],
    [
      "0.555706722",
      "0.838158877"
    ],
    [
      "0.663600803",
      "0.838158877"
    ],
    [
      "0.850478835",
      "0.838158877"
    ],
    [
      "0.757039819",
      "0.892105918"
    ],
    [
      "0.245910702",
      "0.893103399"
    ],
    [
      "0.946052959",
      "0.895985808"
    ],
    [
      "0.053947041",
      "0.900822515"
    ],
    [
      "0.151902846",
      "0.946052959"
    ],
    [
      "0.339918558",
      "0.946052959"
    ],
    [
      "0.44781264",
      "0.946052959"
    ],
    [
      "0.555706722",
      "0.946052959"
    ],
    [
      "0.663600803",
      "0.946052959"
    ],
    [
      "0.850478835",
      "0.946052959"
    ]
  ],
  "radius": "0.05394704"
}
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": 89
}

起步布局答案

{
  "centers": [
    [
      "0.167776968",
      "0.053947041"
    ],
    [
      "0.27567105",
      "0.053947041"
    ],
    [
      "0.383565131",
      "0.053947041"
    ],
    [
      "0.491513698",
      "0.053947041"
    ],
    [
      "0.618795532",
      "0.053947041"
    ],
    [
      "0.730264796",
      "0.053947041"
    ],
    [
      "0.838158877",
      "0.053947041"
    ],
    [
      "0.946052959",
      "0.053947041"
    ],
    [
      "0.053952617",
      "0.053952617"
    ],
    [
      "0.555154615",
      "0.141073194"
    ],
    [
      "0.674530164",
      "0.146330934"
    ],
    [
      "0.437539415",
      "0.147370323"
    ],
    [
      "0.113829927",
      "0.147386057"
    ],
    [
      "0.221724009",
      "0.147386057"
    ],
    [
      "0.329618091",
      "0.147386057"
    ],
    [
      "0.78440128",
      "0.147495176"
    ],
    [
      "0.892295362",
      "0.147495176"
    ],
    [
      "0.610889247",
      "0.233457087"
    ],
    [
      "0.501180331",
      "0.234496476"
    ],
    [
      "0.053947041",
      "0.237136671"
    ],
    [
      "0.728481798",
      "0.239767297"
    ],
    [
      "0.383592374",
      "0.240809339"
    ],
    [
      "0.27567105",
      "0.240825072"
    ],
    [
      "0.946052959",
      "0.241043311"
    ],
    [
      "0.837069475",
      "0.242766099"
    ],
    [
      "0.16325618",
      "0.243294343"
    ],
    [
      "0.555706722",
      "0.327598601"
    ],
    [
      "0.663600803",
      "0.327598601"
    ],
    [
      "0.447233291",
      "0.327935492"
    ],
    [
      "0.329645333",
      "0.334248354"
    ],
    [
      "0.221751256",
      "0.334279814"
    ],
    [
      "0.770893321",
      "0.338976146"
    ],
    [
      "0.053947041",
      "0.345030753"
    ],
    [
      "0.946052959",
      "0.348937393"
    ],
    [
      "0.854906472",
      "0.406673437"
    ],
    [
      "0.142172685",
      "0.407138471"
    ],
    [
      "0.501759681",
      "0.421037617"
    ],
    [
      "0.609653762",
      "0.421037617"
    ],
    [
      "0.39328625",
      "0.421374507"
    ],
    [
      "0.253197643",
      "0.440473866"
    ],
    [
      "0.748020494",
      "0.445287805"
    ],
    [
      "0.946052959",
      "0.464409481"
    ],
    [
      "0.053947041",
      "0.469246189"
    ],
    [
      "0.44781264",
      "0.514476632"
    ],
    [
      "0.555706722",
      "0.514476632"
    ],
    [
      "0.663600803",
      "0.514476632"
    ],
    [
      "0.850478835",
      "0.514476632"
    ],
    [
      "0.338770294",
      "0.514482743"
    ],
    [
      "0.15305111",
      "0.514482743"
    ],
    [
      "0.757039819",
      "0.568423673"
    ],
    [
      "0.245910702",
      "0.569421154"
    ],
    [
      "0.946052959",
      "0.572303563"
    ],
    [
      "0.053947041",
      "0.57714027"
    ],
    [
      "0.339918558",
      "0.622370714"
    ],
    [
      "0.663600803",
      "0.622370714"
    ],
    [
      "0.151902846",
      "0.622370714"
    ],
    [
      "0.44781264",
      "0.622370714"
    ],
    [
      "0.555706722",
      "0.622370714"
    ],
    [
      "0.850478835",
      "0.622370714"
    ],
    [
      "0.757039819",
      "0.676317755"
    ],
    [
      "0.245910702",
      "0.677315236"
    ],
    [
      "0.946052959",
      "0.680197645"
    ],
    [
      "0.053947041",
      "0.685034352"
    ],
    [
      "0.151902846",
      "0.730264796"
    ],
    [
      "0.339918558",
      "0.730264796"
    ],
    [
      "0.44781264",
      "0.730264796"
    ],
    [
      "0.555706722",
      "0.730264796"
    ],
    [
      "0.663600803",
      "0.730264796"
    ],
    [
      "0.850478835",
      "0.730264796"
    ],
    [
      "0.757039819",
      "0.784211837"
    ],
    [
      "0.245910702",
      "0.785209317"
    ],
    [
      "0.946052959",
      "0.788091726"
    ],
    [
      "0.053947041",
      "0.792928434"
    ],
    [
      "0.151902846",
      "0.838158877"
    ],
    [
      "0.339918558",
      "0.838158877"
    ],
    [
      "0.44781264",
      "0.838158877"
    ],
    [
      "0.555706722",
      "0.838158877"
    ],
    [
      "0.663600803",
      "0.838158877"
    ],
    [
      "0.850478835",
      "0.838158877"
    ],
    [
      "0.757039819",
      "0.892105918"
    ],
    [
      "0.245910702",
      "0.893103399"
    ],
    [
      "0.946052959",
      "0.895985808"
    ],
    [
      "0.053947041",
      "0.900822515"
    ],
    [
      "0.151902846",
      "0.946052959"
    ],
    [
      "0.339918558",
      "0.946052959"
    ],
    [
      "0.44781264",
      "0.946052959"
    ],
    [
      "0.555706722",
      "0.946052959"
    ],
    [
      "0.663600803",
      "0.946052959"
    ],
    [
      "0.850478835",
      "0.946052959"
    ]
  ],
  "radius": "0.05394704"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。 · 验证器 v1.0.0

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