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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 = 87

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 = 87
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[0.0546952597205, 0.0546952597215]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 = 0.054695259

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 = 87 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": [
    [
      "0.05469526",
      "0.05469526"
    ],
    [
      "0.243209653",
      "0.05469526"
    ],
    [
      "0.431242588",
      "0.05469526"
    ],
    [
      "0.619169458",
      "0.05469526"
    ],
    [
      "0.728559978",
      "0.05469526"
    ],
    [
      "0.945137783",
      "0.05469526"
    ],
    [
      "0.83684888",
      "0.070180712"
    ],
    [
      "0.148952456",
      "0.110209822"
    ],
    [
      "0.33722612",
      "0.11061654"
    ],
    [
      "0.525206023",
      "0.110705603"
    ],
    [
      "0.728726934",
      "0.164085652"
    ],
    [
      "0.94530474",
      "0.164085652"
    ],
    [
      "0.05469526",
      "0.165724385"
    ],
    [
      "0.242968924",
      "0.166131102"
    ],
    [
      "0.619360768",
      "0.166393751"
    ],
    [
      "0.431189555",
      "0.166626884"
    ],
    [
      "0.837015837",
      "0.179571104"
    ],
    [
      "0.148711727",
      "0.221645665"
    ],
    [
      "0.336932359",
      "0.222141446"
    ],
    [
      "0.5253443",
      "0.222315031"
    ],
    [
      "0.728726934",
      "0.273476171"
    ],
    [
      "0.94530474",
      "0.273476171"
    ],
    [
      "0.05469526",
      "0.277566945"
    ],
    [
      "0.242675162",
      "0.277656009"
    ],
    [
      "0.431087104",
      "0.277829594"
    ],
    [
      "0.619434683",
      "0.278111855"
    ],
    [
      "0.837015837",
      "0.288961624"
    ],
    [
      "0.336829907",
      "0.333344156"
    ],
    [
      "0.148658695",
      "0.333577289"
    ],
    [
      "0.525177487",
      "0.333626418"
    ],
    [
      "0.728726934",
      "0.382866691"
    ],
    [
      "0.94530474",
      "0.382866691"
    ],
    [
      "0.619509285",
      "0.389014119"
    ],
    [
      "0.43092029",
      "0.38914098"
    ],
    [
      "0.242813439",
      "0.389265436"
    ],
    [
      "0.05469526",
      "0.389587632"
    ],
    [
      "0.837015837",
      "0.398352143"
    ],
    [
      "0.525252089",
      "0.444528682"
    ],
    [
      "0.336903823",
      "0.44506226"
    ],
    [
      "0.148850004",
      "0.44527578"
    ],
    [
      "0.728726934",
      "0.49225721"
    ],
    [
      "0.94530474",
      "0.49225721"
    ],
    [
      "0.619602642",
      "0.499884431"
    ],
    [
      "0.431235621",
      "0.500449962"
    ],
    [
      "0.05469526",
      "0.500963928"
    ],
    [
      "0.242940387",
      "0.501072604"
    ],
    [
      "0.837015837",
      "0.507742663"
    ],
    [
      "0.525586174",
      "0.555805711"
    ],
    [
      "0.337272186",
      "0.556460306"
    ],
    [
      "0.148785643",
      "0.556760752"
    ],
    [
      "0.728726934",
      "0.60164773"
    ],
    [
      "0.94530474",
      "0.60164773"
    ],
    [
      "0.619775208",
      "0.611435843"
    ],
    [
      "0.431622739",
      "0.611816055"
    ],
    [
      "0.243117441",
      "0.612148454"
    ],
    [
      "0.05469526",
      "0.612557576"
    ],
    [
      "0.837015837",
      "0.617133182"
    ],
    [
      "0.525811773",
      "0.667446186"
    ],
    [
      "0.337467994",
      "0.667504202"
    ],
    [
      "0.149027058",
      "0.667945278"
    ],
    [
      "0.728726934",
      "0.711038249"
    ],
    [
      "0.94530474",
      "0.711038249"
    ],
    [
      "0.620000808",
      "0.723076318"
    ],
    [
      "0.431657029",
      "0.723134334"
    ],
    [
      "0.243377611",
      "0.723301026"
    ],
    [
      "0.05469526",
      "0.723332979"
    ],
    [
      "0.837015837",
      "0.726523701"
    ],
    [
      "0.149045813",
      "0.778688728"
    ],
    [
      "0.525846063",
      "0.778764466"
    ],
    [
      "0.337566646",
      "0.778931158"
    ],
    [
      "0.728726934",
      "0.820428768"
    ],
    [
      "0.94530474",
      "0.820428768"
    ],
    [
      "0.05469526",
      "0.834044477"
    ],
    [
      "0.620196616",
      "0.834120215"
    ],
    [
      "0.243234847",
      "0.83431886"
    ],
    [
      "0.43175568",
      "0.83456129"
    ],
    [
      "0.837015837",
      "0.835914221"
    ],
    [
      "0.148884294",
      "0.889674609"
    ],
    [
      "0.526106233",
      "0.889917039"
    ],
    [
      "0.337423881",
      "0.889948992"
    ],
    [
      "0.728726934",
      "0.929819288"
    ],
    [
      "0.94530474",
      "0.929819288"
    ],
    [
      "0.05469526",
      "0.94530474"
    ],
    [
      "0.243073328",
      "0.94530474"
    ],
    [
      "0.431774434",
      "0.94530474"
    ],
    [
      "0.620438031",
      "0.94530474"
    ],
    [
      "0.837015837",
      "0.94530474"
    ]
  ],
  "radius": "0.054695259"
}
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": 87
}

起步布局答案

{
  "centers": [
    [
      "0.05469526",
      "0.05469526"
    ],
    [
      "0.243209653",
      "0.05469526"
    ],
    [
      "0.431242588",
      "0.05469526"
    ],
    [
      "0.619169458",
      "0.05469526"
    ],
    [
      "0.728559978",
      "0.05469526"
    ],
    [
      "0.945137783",
      "0.05469526"
    ],
    [
      "0.83684888",
      "0.070180712"
    ],
    [
      "0.148952456",
      "0.110209822"
    ],
    [
      "0.33722612",
      "0.11061654"
    ],
    [
      "0.525206023",
      "0.110705603"
    ],
    [
      "0.728726934",
      "0.164085652"
    ],
    [
      "0.94530474",
      "0.164085652"
    ],
    [
      "0.05469526",
      "0.165724385"
    ],
    [
      "0.242968924",
      "0.166131102"
    ],
    [
      "0.619360768",
      "0.166393751"
    ],
    [
      "0.431189555",
      "0.166626884"
    ],
    [
      "0.837015837",
      "0.179571104"
    ],
    [
      "0.148711727",
      "0.221645665"
    ],
    [
      "0.336932359",
      "0.222141446"
    ],
    [
      "0.5253443",
      "0.222315031"
    ],
    [
      "0.728726934",
      "0.273476171"
    ],
    [
      "0.94530474",
      "0.273476171"
    ],
    [
      "0.05469526",
      "0.277566945"
    ],
    [
      "0.242675162",
      "0.277656009"
    ],
    [
      "0.431087104",
      "0.277829594"
    ],
    [
      "0.619434683",
      "0.278111855"
    ],
    [
      "0.837015837",
      "0.288961624"
    ],
    [
      "0.336829907",
      "0.333344156"
    ],
    [
      "0.148658695",
      "0.333577289"
    ],
    [
      "0.525177487",
      "0.333626418"
    ],
    [
      "0.728726934",
      "0.382866691"
    ],
    [
      "0.94530474",
      "0.382866691"
    ],
    [
      "0.619509285",
      "0.389014119"
    ],
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      "0.43092029",
      "0.38914098"
    ],
    [
      "0.242813439",
      "0.389265436"
    ],
    [
      "0.05469526",
      "0.389587632"
    ],
    [
      "0.837015837",
      "0.398352143"
    ],
    [
      "0.525252089",
      "0.444528682"
    ],
    [
      "0.336903823",
      "0.44506226"
    ],
    [
      "0.148850004",
      "0.44527578"
    ],
    [
      "0.728726934",
      "0.49225721"
    ],
    [
      "0.94530474",
      "0.49225721"
    ],
    [
      "0.619602642",
      "0.499884431"
    ],
    [
      "0.431235621",
      "0.500449962"
    ],
    [
      "0.05469526",
      "0.500963928"
    ],
    [
      "0.242940387",
      "0.501072604"
    ],
    [
      "0.837015837",
      "0.507742663"
    ],
    [
      "0.525586174",
      "0.555805711"
    ],
    [
      "0.337272186",
      "0.556460306"
    ],
    [
      "0.148785643",
      "0.556760752"
    ],
    [
      "0.728726934",
      "0.60164773"
    ],
    [
      "0.94530474",
      "0.60164773"
    ],
    [
      "0.619775208",
      "0.611435843"
    ],
    [
      "0.431622739",
      "0.611816055"
    ],
    [
      "0.243117441",
      "0.612148454"
    ],
    [
      "0.05469526",
      "0.612557576"
    ],
    [
      "0.837015837",
      "0.617133182"
    ],
    [
      "0.525811773",
      "0.667446186"
    ],
    [
      "0.337467994",
      "0.667504202"
    ],
    [
      "0.149027058",
      "0.667945278"
    ],
    [
      "0.728726934",
      "0.711038249"
    ],
    [
      "0.94530474",
      "0.711038249"
    ],
    [
      "0.620000808",
      "0.723076318"
    ],
    [
      "0.431657029",
      "0.723134334"
    ],
    [
      "0.243377611",
      "0.723301026"
    ],
    [
      "0.05469526",
      "0.723332979"
    ],
    [
      "0.837015837",
      "0.726523701"
    ],
    [
      "0.149045813",
      "0.778688728"
    ],
    [
      "0.525846063",
      "0.778764466"
    ],
    [
      "0.337566646",
      "0.778931158"
    ],
    [
      "0.728726934",
      "0.820428768"
    ],
    [
      "0.94530474",
      "0.820428768"
    ],
    [
      "0.05469526",
      "0.834044477"
    ],
    [
      "0.620196616",
      "0.834120215"
    ],
    [
      "0.243234847",
      "0.83431886"
    ],
    [
      "0.43175568",
      "0.83456129"
    ],
    [
      "0.837015837",
      "0.835914221"
    ],
    [
      "0.148884294",
      "0.889674609"
    ],
    [
      "0.526106233",
      "0.889917039"
    ],
    [
      "0.337423881",
      "0.889948992"
    ],
    [
      "0.728726934",
      "0.929819288"
    ],
    [
      "0.94530474",
      "0.929819288"
    ],
    [
      "0.05469526",
      "0.94530474"
    ],
    [
      "0.243073328",
      "0.94530474"
    ],
    [
      "0.431774434",
      "0.94530474"
    ],
    [
      "0.620438031",
      "0.94530474"
    ],
    [
      "0.837015837",
      "0.94530474"
    ]
  ],
  "radius": "0.054695259"
}

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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