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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 = 98

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 = 98
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[0.0520329877025, 0.0520329877035]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 = 0.052032987

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 = 98 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": [
    [
      "0.052032988",
      "0.052032988"
    ],
    [
      "0.230819431",
      "0.052032988"
    ],
    [
      "0.411063642",
      "0.052032988"
    ],
    [
      "0.591311199",
      "0.052032988"
    ],
    [
      "0.771558756",
      "0.052032988"
    ],
    [
      "0.947967012",
      "0.052032988"
    ],
    [
      "0.50118742",
      "0.104065975"
    ],
    [
      "0.681434977",
      "0.104065975"
    ],
    [
      "0.320941537",
      "0.104068873"
    ],
    [
      "0.141426209",
      "0.105311306"
    ],
    [
      "0.859762884",
      "0.107257611"
    ],
    [
      "0.591311199",
      "0.156098963"
    ],
    [
      "0.411065315",
      "0.156101861"
    ],
    [
      "0.231548315",
      "0.157347191"
    ],
    [
      "0.052032988",
      "0.158589623"
    ],
    [
      "0.769639106",
      "0.159290598"
    ],
    [
      "0.947967012",
      "0.162482233"
    ],
    [
      "0.501189093",
      "0.208134848"
    ],
    [
      "0.321672093",
      "0.209380178"
    ],
    [
      "0.142155093",
      "0.210625509"
    ],
    [
      "0.679515327",
      "0.211323586"
    ],
    [
      "0.857843234",
      "0.214515221"
    ],
    [
      "0.411795872",
      "0.261413166"
    ],
    [
      "0.232278872",
      "0.262658496"
    ],
    [
      "0.052032988",
      "0.262661394"
    ],
    [
      "0.589393222",
      "0.263359471"
    ],
    [
      "0.767719456",
      "0.266548209"
    ],
    [
      "0.947967012",
      "0.266548209"
    ],
    [
      "0.32240265",
      "0.314691484"
    ],
    [
      "0.142156766",
      "0.314694381"
    ],
    [
      "0.5",
      "0.316637789"
    ],
    [
      "0.857843234",
      "0.318581197"
    ],
    [
      "0.67759735",
      "0.318584094"
    ],
    [
      "0.052032988",
      "0.366727369"
    ],
    [
      "0.232280544",
      "0.366727369"
    ],
    [
      "0.410606778",
      "0.369916107"
    ],
    [
      "0.947967012",
      "0.370614184"
    ],
    [
      "0.767721128",
      "0.370617082"
    ],
    [
      "0.588204128",
      "0.371862412"
    ],
    [
      "0.142156766",
      "0.418760357"
    ],
    [
      "0.320484673",
      "0.421951992"
    ],
    [
      "0.857844907",
      "0.422650069"
    ],
    [
      "0.678327907",
      "0.4238954"
    ],
    [
      "0.498810907",
      "0.42514073"
    ],
    [
      "0.052032988",
      "0.470793345"
    ],
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      "0.230360894",
      "0.47398498"
    ],
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      "0.947967012",
      "0.474685954"
    ],
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    ],
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      "0.588934685",
      "0.477173717"
    ],
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      "0.408688801",
      "0.477176615"
    ],
    [
      "0.140237116",
      "0.526017967"
    ],
    [
      "0.858573791",
      "0.527964272"
    ],
    [
      "0.679058463",
      "0.529206705"
    ],
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      "0.318565023",
      "0.529209603"
    ],
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      "0.49881258",
      "0.529209603"
    ],
    [
      "0.052032988",
      "0.58124259"
    ],
    [
      "0.228441244",
      "0.58124259"
    ],
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      "0.408688801",
      "0.58124259"
    ],
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      "0.588936358",
      "0.58124259"
    ],
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      "0.769180569",
      "0.58124259"
    ],
    [
      "0.947967012",
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    ],
    [
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    ],
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      "0.633275578"
    ],
    [
      "0.679058463",
      "0.633278475"
    ],
    [
      "0.858573791",
      "0.634520908"
    ],
    [
      "0.140237116",
      "0.636467213"
    ],
    [
      "0.408688801",
      "0.685308566"
    ],
    [
      "0.588934685",
      "0.685311463"
    ],
    [
      "0.768451685",
      "0.686556793"
    ],
    [
      "0.947967012",
      "0.687799226"
    ],
    [
      "0.230360894",
      "0.688500201"
    ],
    [
      "0.052032988",
      "0.691691836"
    ],
    [
      "0.678327907",
      "0.738589781"
    ],
    [
      "0.857844907",
      "0.739835111"
    ],
    [
      "0.320484673",
      "0.740533189"
    ],
    [
      "0.496308901",
      "0.741455207"
    ],
    [
      "0.142156766",
      "0.743724824"
    ],
    [
      "0.588112472",
      "0.790463692"
    ],
    [
      "0.947967012",
      "0.791870996"
    ],
    [
      "0.767626127",
      "0.792027176"
    ],
    [
      "0.052032988",
      "0.795757811"
    ],
    [
      "0.232280544",
      "0.795757811"
    ],
    [
      "0.406872783",
      "0.798557508"
    ],
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      "0.677410692",
      "0.843901086"
    ],
    [
      "0.857841561",
      "0.843901086"
    ],
    [
      "0.499768094",
      "0.845463674"
    ],
    [
      "0.142156766",
      "0.847790799"
    ],
    [
      "0.318668655",
      "0.853782131"
    ],
    [
      "0.947967007",
      "0.896060578"
    ],
    [
      "0.767524518",
      "0.896093102"
    ],
    [
      "0.054672967",
      "0.90431566"
    ],
    [
      "0.579555369",
      "0.912274736"
    ],
    [
      "0.378105225",
      "0.939204735"
    ],
    [
      "0.170342818",
      "0.947967012"
    ],
    [
      "0.274408794",
      "0.947967012"
    ],
    [
      "0.481801655",
      "0.947967012"
    ],
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      "0.677309084",
      "0.947967012"
    ],
    [
      "0.857739953",
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    ]
  ],
  "radius": "0.052032987"
}
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": 98
}

起步布局答案

{
  "centers": [
    [
      "0.052032988",
      "0.052032988"
    ],
    [
      "0.230819431",
      "0.052032988"
    ],
    [
      "0.411063642",
      "0.052032988"
    ],
    [
      "0.591311199",
      "0.052032988"
    ],
    [
      "0.771558756",
      "0.052032988"
    ],
    [
      "0.947967012",
      "0.052032988"
    ],
    [
      "0.50118742",
      "0.104065975"
    ],
    [
      "0.681434977",
      "0.104065975"
    ],
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      "0.320941537",
      "0.104068873"
    ],
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    ],
    [
      "0.859762884",
      "0.107257611"
    ],
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      "0.156098963"
    ],
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      "0.411065315",
      "0.156101861"
    ],
    [
      "0.231548315",
      "0.157347191"
    ],
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      "0.052032988",
      "0.158589623"
    ],
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      "0.159290598"
    ],
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      "0.947967012",
      "0.162482233"
    ],
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      "0.501189093",
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    ],
    [
      "0.321672093",
      "0.209380178"
    ],
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      "0.142155093",
      "0.210625509"
    ],
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      "0.679515327",
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      "0.261413166"
    ],
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      "0.262658496"
    ],
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      "0.589393222",
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      "0.266548209"
    ],
    [
      "0.947967012",
      "0.266548209"
    ],
    [
      "0.32240265",
      "0.314691484"
    ],
    [
      "0.142156766",
      "0.314694381"
    ],
    [
      "0.5",
      "0.316637789"
    ],
    [
      "0.857843234",
      "0.318581197"
    ],
    [
      "0.67759735",
      "0.318584094"
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      "0.370614184"
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      "0.947967012",
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      "0.052032988",
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      "0.847790799"
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      "0.318668655",
      "0.853782131"
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      "0.947967007",
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      "0.767524518",
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      "0.054672967",
      "0.90431566"
    ],
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      "0.579555369",
      "0.912274736"
    ],
    [
      "0.378105225",
      "0.939204735"
    ],
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      "0.170342818",
      "0.947967012"
    ],
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      "0.274408794",
      "0.947967012"
    ],
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      "0.481801655",
      "0.947967012"
    ],
    [
      "0.677309084",
      "0.947967012"
    ],
    [
      "0.857739953",
      "0.947967012"
    ]
  ],
  "radius": "0.052032987"
}

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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