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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 = 88

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 = 88
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[0.0544066369125, 0.0544066369135]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 = 0.054406636

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 = 88 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": [
    [
      "0.054406637",
      "0.054406637"
    ],
    [
      "0.163219911",
      "0.054406637"
    ],
    [
      "0.272033185",
      "0.054406637"
    ],
    [
      "0.380846458",
      "0.054406637"
    ],
    [
      "0.569316577",
      "0.054406637"
    ],
    [
      "0.757786696",
      "0.054406637"
    ],
    [
      "0.945593363",
      "0.054406637"
    ],
    [
      "0.475081518",
      "0.108813274"
    ],
    [
      "0.663551637",
      "0.108813274"
    ],
    [
      "0.85169003",
      "0.109383837"
    ],
    [
      "0.054406637",
      "0.163219911"
    ],
    [
      "0.163219911",
      "0.163219911"
    ],
    [
      "0.272033185",
      "0.163219911"
    ],
    [
      "0.380846458",
      "0.163219911"
    ],
    [
      "0.569316577",
      "0.163219911"
    ],
    [
      "0.75745497",
      "0.163790474"
    ],
    [
      "0.945593363",
      "0.164361037"
    ],
    [
      "0.475081518",
      "0.217626548"
    ],
    [
      "0.663219911",
      "0.218197111"
    ],
    [
      "0.851358304",
      "0.218767674"
    ],
    [
      "0.163219911",
      "0.272033185"
    ],
    [
      "0.054406637",
      "0.272033185"
    ],
    [
      "0.272033185",
      "0.272033185"
    ],
    [
      "0.380846458",
      "0.272033185"
    ],
    [
      "0.568984851",
      "0.272603748"
    ],
    [
      "0.757123244",
      "0.273174311"
    ],
    [
      "0.945593363",
      "0.273174311"
    ],
    [
      "0.474749792",
      "0.327010384"
    ],
    [
      "0.662888185",
      "0.327580947"
    ],
    [
      "0.851358304",
      "0.327580947"
    ],
    [
      "0.054406637",
      "0.380846458"
    ],
    [
      "0.163219911",
      "0.380846458"
    ],
    [
      "0.272033185",
      "0.380846458"
    ],
    [
      "0.38084058",
      "0.381977542"
    ],
    [
      "0.568653125",
      "0.381987584"
    ],
    [
      "0.757123244",
      "0.381987584"
    ],
    [
      "0.945593363",
      "0.381987584"
    ],
    [
      "0.662888185",
      "0.436394221"
    ],
    [
      "0.851358304",
      "0.436394221"
    ],
    [
      "0.470797195",
      "0.443201133"
    ],
    [
      "0.108813274",
      "0.475081518"
    ],
    [
      "0.218496695",
      "0.475584206"
    ],
    [
      "0.328953626",
      "0.477623099"
    ],
    [
      "0.568653125",
      "0.490800858"
    ],
    [
      "0.757123244",
      "0.490800858"
    ],
    [
      "0.945593363",
      "0.490800858"
    ],
    [
      "0.851358304",
      "0.545207495"
    ],
    [
      "0.662773427",
      "0.545406748"
    ],
    [
      "0.493075757",
      "0.569084892"
    ],
    [
      "0.054406637",
      "0.569316577"
    ],
    [
      "0.163219911",
      "0.569316577"
    ],
    [
      "0.272028198",
      "0.570358336"
    ],
    [
      "0.384285399",
      "0.571317942"
    ],
    [
      "0.757123244",
      "0.599614132"
    ],
    [
      "0.945593363",
      "0.599614132"
    ],
    [
      "0.851358304",
      "0.654020769"
    ],
    [
      "0.556274423",
      "0.657663994"
    ],
    [
      "0.665087696",
      "0.657663994"
    ],
    [
      "0.108813274",
      "0.663551637"
    ],
    [
      "0.447101252",
      "0.667828853"
    ],
    [
      "0.338248599",
      "0.669912777"
    ],
    [
      "0.228864762",
      "0.670244503"
    ],
    [
      "0.945593363",
      "0.708427406"
    ],
    [
      "0.738674557",
      "0.739234288"
    ],
    [
      "0.512136161",
      "0.757123244"
    ],
    [
      "0.620949435",
      "0.757123244"
    ],
    [
      "0.163790474",
      "0.75745497"
    ],
    [
      "0.054406637",
      "0.757786696"
    ],
    [
      "0.847340886",
      "0.762759855"
    ],
    [
      "0.283841962",
      "0.764147837"
    ],
    [
      "0.392655236",
      "0.764147837"
    ],
    [
      "0.941575946",
      "0.817166492"
    ],
    [
      "0.218767674",
      "0.851358304"
    ],
    [
      "0.457729524",
      "0.851358304"
    ],
    [
      "0.566542798",
      "0.851358304"
    ],
    [
      "0.675356072",
      "0.851358304"
    ],
    [
      "0.784169346",
      "0.851358304"
    ],
    [
      "0.109383837",
      "0.85169003"
    ],
    [
      "0.338248599",
      "0.858382896"
    ],
    [
      "0.945593363",
      "0.925905579"
    ],
    [
      "0.054406637",
      "0.945593363"
    ],
    [
      "0.164361037",
      "0.945593363"
    ],
    [
      "0.273174311",
      "0.945593363"
    ],
    [
      "0.403322887",
      "0.945593363"
    ],
    [
      "0.512136161",
      "0.945593363"
    ],
    [
      "0.620949435",
      "0.945593363"
    ],
    [
      "0.729762709",
      "0.945593363"
    ],
    [
      "0.838575983",
      "0.945593363"
    ]
  ],
  "radius": "0.054406636"
}
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": 88
}

起步布局答案

{
  "centers": [
    [
      "0.054406637",
      "0.054406637"
    ],
    [
      "0.163219911",
      "0.054406637"
    ],
    [
      "0.272033185",
      "0.054406637"
    ],
    [
      "0.380846458",
      "0.054406637"
    ],
    [
      "0.569316577",
      "0.054406637"
    ],
    [
      "0.757786696",
      "0.054406637"
    ],
    [
      "0.945593363",
      "0.054406637"
    ],
    [
      "0.475081518",
      "0.108813274"
    ],
    [
      "0.663551637",
      "0.108813274"
    ],
    [
      "0.85169003",
      "0.109383837"
    ],
    [
      "0.054406637",
      "0.163219911"
    ],
    [
      "0.163219911",
      "0.163219911"
    ],
    [
      "0.272033185",
      "0.163219911"
    ],
    [
      "0.380846458",
      "0.163219911"
    ],
    [
      "0.569316577",
      "0.163219911"
    ],
    [
      "0.75745497",
      "0.163790474"
    ],
    [
      "0.945593363",
      "0.164361037"
    ],
    [
      "0.475081518",
      "0.217626548"
    ],
    [
      "0.663219911",
      "0.218197111"
    ],
    [
      "0.851358304",
      "0.218767674"
    ],
    [
      "0.163219911",
      "0.272033185"
    ],
    [
      "0.054406637",
      "0.272033185"
    ],
    [
      "0.272033185",
      "0.272033185"
    ],
    [
      "0.380846458",
      "0.272033185"
    ],
    [
      "0.568984851",
      "0.272603748"
    ],
    [
      "0.757123244",
      "0.273174311"
    ],
    [
      "0.945593363",
      "0.273174311"
    ],
    [
      "0.474749792",
      "0.327010384"
    ],
    [
      "0.662888185",
      "0.327580947"
    ],
    [
      "0.851358304",
      "0.327580947"
    ],
    [
      "0.054406637",
      "0.380846458"
    ],
    [
      "0.163219911",
      "0.380846458"
    ],
    [
      "0.272033185",
      "0.380846458"
    ],
    [
      "0.38084058",
      "0.381977542"
    ],
    [
      "0.568653125",
      "0.381987584"
    ],
    [
      "0.757123244",
      "0.381987584"
    ],
    [
      "0.945593363",
      "0.381987584"
    ],
    [
      "0.662888185",
      "0.436394221"
    ],
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      "0.851358304",
      "0.436394221"
    ],
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      "0.470797195",
      "0.443201133"
    ],
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      "0.108813274",
      "0.475081518"
    ],
    [
      "0.218496695",
      "0.475584206"
    ],
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      "0.477623099"
    ],
    [
      "0.568653125",
      "0.490800858"
    ],
    [
      "0.757123244",
      "0.490800858"
    ],
    [
      "0.945593363",
      "0.490800858"
    ],
    [
      "0.851358304",
      "0.545207495"
    ],
    [
      "0.662773427",
      "0.545406748"
    ],
    [
      "0.493075757",
      "0.569084892"
    ],
    [
      "0.054406637",
      "0.569316577"
    ],
    [
      "0.163219911",
      "0.569316577"
    ],
    [
      "0.272028198",
      "0.570358336"
    ],
    [
      "0.384285399",
      "0.571317942"
    ],
    [
      "0.757123244",
      "0.599614132"
    ],
    [
      "0.945593363",
      "0.599614132"
    ],
    [
      "0.851358304",
      "0.654020769"
    ],
    [
      "0.556274423",
      "0.657663994"
    ],
    [
      "0.665087696",
      "0.657663994"
    ],
    [
      "0.108813274",
      "0.663551637"
    ],
    [
      "0.447101252",
      "0.667828853"
    ],
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      "0.338248599",
      "0.669912777"
    ],
    [
      "0.228864762",
      "0.670244503"
    ],
    [
      "0.945593363",
      "0.708427406"
    ],
    [
      "0.738674557",
      "0.739234288"
    ],
    [
      "0.512136161",
      "0.757123244"
    ],
    [
      "0.620949435",
      "0.757123244"
    ],
    [
      "0.163790474",
      "0.75745497"
    ],
    [
      "0.054406637",
      "0.757786696"
    ],
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      "0.847340886",
      "0.762759855"
    ],
    [
      "0.283841962",
      "0.764147837"
    ],
    [
      "0.392655236",
      "0.764147837"
    ],
    [
      "0.941575946",
      "0.817166492"
    ],
    [
      "0.218767674",
      "0.851358304"
    ],
    [
      "0.457729524",
      "0.851358304"
    ],
    [
      "0.566542798",
      "0.851358304"
    ],
    [
      "0.675356072",
      "0.851358304"
    ],
    [
      "0.784169346",
      "0.851358304"
    ],
    [
      "0.109383837",
      "0.85169003"
    ],
    [
      "0.338248599",
      "0.858382896"
    ],
    [
      "0.945593363",
      "0.925905579"
    ],
    [
      "0.054406637",
      "0.945593363"
    ],
    [
      "0.164361037",
      "0.945593363"
    ],
    [
      "0.273174311",
      "0.945593363"
    ],
    [
      "0.403322887",
      "0.945593363"
    ],
    [
      "0.512136161",
      "0.945593363"
    ],
    [
      "0.620949435",
      "0.945593363"
    ],
    [
      "0.729762709",
      "0.945593363"
    ],
    [
      "0.838575983",
      "0.945593363"
    ]
  ],
  "radius": "0.054406636"
}

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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