P01 · 装箱与覆盖 · 经典问题

单位正方形内的等圆装箱 · n = 28

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

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

子题n = 28

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
已证明最优	0.093672833	见来源署名	web.archive.org ↗
本站纪录	暂无选手纪录	—	—

已完成已有达到理论上限的公开解答，该子题已无竞争空间

严格定义

容器单位正方形 [0, 1] × [0, 1]，左下角为原点 (0, 0)，右上角为 (1, 1)
提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心坐标 centers
约束每个圆完全位于正方形内；任意两圆内部不重叠，允许相切
目标最大化共同半径 radius

打开编辑器查看 ↗

当前展示：起步布局

1y0

12345678910111213141516171819202122232425262728

0x1

已验证构造r = 0.093672833

帮助理解

接触图与非规则排布

当圆的半径达到最大时，大部分圆会与相邻的圆或正方形边界相切，形成刚性的接触网络。除少数平方数规模外，整齐的网格排列通常会浪费边缘空间；更优的构型往往采用斜排与错位层，有时还会留下一两个可在局部空隙内微移的游离圆（rattler）。

已知证明与开放前沿

n = 1–30 的最优解均已给出数学证明；n = 31–300 导入了 Packomania 整理的公开已知最优构型，其中除 n = 36 外尚未证明最优性。较大规模的子题合并为同一张数据表，便于使用算法批量求解与比对。

查看来源

构型与历史

起步布局

查看构型、求解笔记与纪录历史。

最优值已获严格证明，无法继续提升提交证明 / 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。

ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形，左下角为 (0, 0)，右上角为 (1, 1)。普通小数格式最多九位小数；n ≤ 35 的代数格式可使用分数和单一根式域，表达式的限制见下方代数构型说明。

提交共同半径 radius 与圆心坐标 centers，数值写成十进制字符串。n ≤ 35 也支持代数证书：增加 field: 2 或 3，数值可写分数和该域的 sqrt(2) 或 sqrt(3)。已封盘子题不重新开放。

起步布局答案

{
  "centers": [
    [
      "0.093672834",
      "0.093672834"
    ],
    [
      "0.389268333",
      "0.093672834"
    ],
    [
      "0.713760547",
      "0.093672834"
    ],
    [
      "0.901106215",
      "0.093672834"
    ],
    [
      "0.55151444",
      "0.187345668"
    ],
    [
      "0.241470583",
      "0.208799827"
    ],
    [
      "0.718981499",
      "0.280945739"
    ],
    [
      "0.906327166",
      "0.280945739"
    ],
    [
      "0.403716691",
      "0.302472661"
    ],
    [
      "0.093672834",
      "0.32392682"
    ],
    [
      "0.56820712",
      "0.392145945"
    ],
    [
      "0.255918941",
      "0.417599653"
    ],
    [
      "0.906327166",
      "0.468291406"
    ],
    [
      "0.721627368",
      "0.499666299"
    ],
    [
      "0.418165262",
      "0.51127212"
    ],
    [
      "0.093672834",
      "0.511272487"
    ],
    [
      "0.255918941",
      "0.604945321"
    ],
    [
      "0.574090162",
      "0.615126993"
    ],
    [
      "0.906327166",
      "0.655637074"
    ],
    [
      "0.093672834",
      "0.698618155"
    ],
    [
      "0.411844054",
      "0.708799827"
    ],
    [
      "0.730015275",
      "0.718981499"
    ],
    [
      "0.249597947",
      "0.802472661"
    ],
    [
      "0.567769167",
      "0.812654332"
    ],
    [
      "0.906327166",
      "0.842982742"
    ],
    [
      "0.093672834",
      "0.906327166"
    ],
    [
      "0.40552306",
      "0.906327166"
    ],
    [
      "0.730015275",
      "0.906327166"
    ]
  ],
  "radius": "0.093672833"
}

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": 28
}

起步布局答案

{
  "centers": [
    [
      "0.093672834",
      "0.093672834"
    ],
    [
      "0.389268333",
      "0.093672834"
    ],
    [
      "0.713760547",
      "0.093672834"
    ],
    [
      "0.901106215",
      "0.093672834"
    ],
    [
      "0.55151444",
      "0.187345668"
    ],
    [
      "0.241470583",
      "0.208799827"
    ],
    [
      "0.718981499",
      "0.280945739"
    ],
    [
      "0.906327166",
      "0.280945739"
    ],
    [
      "0.403716691",
      "0.302472661"
    ],
    [
      "0.093672834",
      "0.32392682"
    ],
    [
      "0.56820712",
      "0.392145945"
    ],
    [
      "0.255918941",
      "0.417599653"
    ],
    [
      "0.906327166",
      "0.468291406"
    ],
    [
      "0.721627368",
      "0.499666299"
    ],
    [
      "0.418165262",
      "0.51127212"
    ],
    [
      "0.093672834",
      "0.511272487"
    ],
    [
      "0.255918941",
      "0.604945321"
    ],
    [
      "0.574090162",
      "0.615126993"
    ],
    [
      "0.906327166",
      "0.655637074"
    ],
    [
      "0.093672834",
      "0.698618155"
    ],
    [
      "0.411844054",
      "0.708799827"
    ],
    [
      "0.730015275",
      "0.718981499"
    ],
    [
      "0.249597947",
      "0.802472661"
    ],
    [
      "0.567769167",
      "0.812654332"
    ],
    [
      "0.906327166",
      "0.842982742"
    ],
    [
      "0.093672834",
      "0.906327166"
    ],
    [
      "0.40552306",
      "0.906327166"
    ],
    [
      "0.730015275",
      "0.906327166"
    ]
  ],
  "radius": "0.093672833"
}

提交共同半径 radius 与圆心坐标 centers，数值写成十进制字符串。n ≤ 35 也支持代数证书：增加 field: 2 或 3，数值可写分数和该域的 sqrt(2) 或 sqrt(3)。已封盘子题不重新开放。 · 验证器 v1.0.0

代数构型

先精确验证，再前往正式提交页选择署名和解题方式。只有正式提交会保存答案并参与现有计分；构型合法不等于全局最优。

支持分数、括号、四则运算，以及 sqrt(2) 或 sqrt(3)。field 选择整份构型使用的根式；普通小数仍可使用。

完整构型（JSON）

{
  "field": 2,
  "radius": "1/12",
  "centers": [
    [
      "1/12",
      "1/12"
    ],
    [
      "3/12",
      "1/12"
    ],
    [
      "5/12",
      "1/12"
    ],
    [
      "7/12",
      "1/12"
    ],
    [
      "9/12",
      "1/12"
    ],
    [
      "11/12",
      "1/12"
    ],
    [
      "1/12",
      "3/12"
    ],
    [
      "3/12",
      "3/12"
    ],
    [
      "5/12",
      "3/12"
    ],
    [
      "7/12",
      "3/12"
    ],
    [
      "9/12",
      "3/12"
    ],
    [
      "11/12",
      "3/12"
    ],
    [
      "1/12",
      "5/12"
    ],
    [
      "3/12",
      "5/12"
    ],
    [
      "5/12",
      "5/12"
    ],
    [
      "7/12",
      "5/12"
    ],
    [
      "9/12",
      "5/12"
    ],
    [
      "11/12",
      "5/12"
    ],
    [
      "1/12",
      "7/12"
    ],
    [
      "3/12",
      "7/12"
    ],
    [
      "5/12",
      "7/12"
    ],
    [
      "7/12",
      "7/12"
    ],
    [
      "9/12",
      "7/12"
    ],
    [
      "11/12",
      "7/12"
    ],
    [
      "1/12",
      "9/12"
    ],
    [
      "3/12",
      "9/12"
    ],
    [
      "5/12",
      "9/12"
    ],
    [
      "7/12",
      "9/12"
    ]
  ]
}

每个表达式最多 256 字符、64 个词元、16 层嵌套；字面小数最多 18 位，中间有理系数最多 256 位二进制。

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