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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 = 83

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 = 83
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[0.0561293730215, 0.0561293730225]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 = 0.056129372

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 = 83 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": [
    [
      "0.056129373",
      "0.056129373"
    ],
    [
      "0.250567225",
      "0.056129373"
    ],
    [
      "0.445005077",
      "0.056129373"
    ],
    [
      "0.639442928",
      "0.056129373"
    ],
    [
      "0.83388078",
      "0.056129373"
    ],
    [
      "0.943870627",
      "0.078585099"
    ],
    [
      "0.153348299",
      "0.112258746"
    ],
    [
      "0.347786151",
      "0.112258746"
    ],
    [
      "0.542224002",
      "0.112258746"
    ],
    [
      "0.736661854",
      "0.112258746"
    ],
    [
      "0.056129373",
      "0.168388119"
    ],
    [
      "0.250567225",
      "0.168388119"
    ],
    [
      "0.445005077",
      "0.168388119"
    ],
    [
      "0.639442928",
      "0.168388119"
    ],
    [
      "0.83388078",
      "0.168388119"
    ],
    [
      "0.943870627",
      "0.190843845"
    ],
    [
      "0.153348299",
      "0.224517492"
    ],
    [
      "0.347786151",
      "0.224517492"
    ],
    [
      "0.542224002",
      "0.224517492"
    ],
    [
      "0.736661854",
      "0.224517492"
    ],
    [
      "0.056129373",
      "0.280646865"
    ],
    [
      "0.250567225",
      "0.280646865"
    ],
    [
      "0.445005077",
      "0.280646865"
    ],
    [
      "0.639442928",
      "0.280646865"
    ],
    [
      "0.83388078",
      "0.280646865"
    ],
    [
      "0.943870627",
      "0.303102591"
    ],
    [
      "0.153348299",
      "0.336776238"
    ],
    [
      "0.347786151",
      "0.336776238"
    ],
    [
      "0.542224002",
      "0.336776238"
    ],
    [
      "0.736661854",
      "0.336776238"
    ],
    [
      "0.83388078",
      "0.392905611"
    ],
    [
      "0.056129373",
      "0.392905611"
    ],
    [
      "0.250567225",
      "0.392905611"
    ],
    [
      "0.445005077",
      "0.392905611"
    ],
    [
      "0.639442928",
      "0.392905611"
    ],
    [
      "0.943870627",
      "0.415361337"
    ],
    [
      "0.153348299",
      "0.449034984"
    ],
    [
      "0.347786151",
      "0.449034984"
    ],
    [
      "0.542224002",
      "0.449034984"
    ],
    [
      "0.736661854",
      "0.449034984"
    ],
    [
      "0.056129373",
      "0.505164357"
    ],
    [
      "0.250567225",
      "0.505164357"
    ],
    [
      "0.445005077",
      "0.505164357"
    ],
    [
      "0.639442928",
      "0.505164357"
    ],
    [
      "0.83388078",
      "0.505164357"
    ],
    [
      "0.943870627",
      "0.527620083"
    ],
    [
      "0.347786151",
      "0.56129373"
    ],
    [
      "0.542224002",
      "0.56129373"
    ],
    [
      "0.736661854",
      "0.56129373"
    ],
    [
      "0.153348299",
      "0.56129373"
    ],
    [
      "0.056129373",
      "0.617423103"
    ],
    [
      "0.250567225",
      "0.617423103"
    ],
    [
      "0.445005077",
      "0.617423103"
    ],
    [
      "0.639442928",
      "0.617423103"
    ],
    [
      "0.83388078",
      "0.617423103"
    ],
    [
      "0.943870627",
      "0.639878829"
    ],
    [
      "0.153348299",
      "0.673552476"
    ],
    [
      "0.347786151",
      "0.673552476"
    ],
    [
      "0.542224002",
      "0.673552476"
    ],
    [
      "0.736661854",
      "0.673552476"
    ],
    [
      "0.056129373",
      "0.729681849"
    ],
    [
      "0.250567225",
      "0.729681849"
    ],
    [
      "0.445005077",
      "0.729681849"
    ],
    [
      "0.639442928",
      "0.729681849"
    ],
    [
      "0.83388078",
      "0.729681849"
    ],
    [
      "0.943870627",
      "0.752137575"
    ],
    [
      "0.545622417",
      "0.797676903"
    ],
    [
      "0.325415913",
      "0.813346059"
    ],
    [
      "0.763479503",
      "0.817121464"
    ],
    [
      "0.201888262",
      "0.8308371"
    ],
    [
      "0.089786734",
      "0.836776238"
    ],
    [
      "0.87346935",
      "0.83957719"
    ],
    [
      "0.653793999",
      "0.841019502"
    ],
    [
      "0.434110003",
      "0.841410643"
    ],
    [
      "0.27673695",
      "0.914501309"
    ],
    [
      "0.943870616",
      "0.927725949"
    ],
    [
      "0.720373065",
      "0.931403428"
    ],
    [
      "0.496947269",
      "0.934434856"
    ],
    [
      "0.056129373",
      "0.943870627"
    ],
    [
      "0.168388119",
      "0.943870627"
    ],
    [
      "0.385085782",
      "0.943870627"
    ],
    [
      "0.608808756",
      "0.943870627"
    ],
    [
      "0.831937373",
      "0.943870627"
    ]
  ],
  "radius": "0.056129372"
}
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": 83
}

起步布局答案

{
  "centers": [
    [
      "0.056129373",
      "0.056129373"
    ],
    [
      "0.250567225",
      "0.056129373"
    ],
    [
      "0.445005077",
      "0.056129373"
    ],
    [
      "0.639442928",
      "0.056129373"
    ],
    [
      "0.83388078",
      "0.056129373"
    ],
    [
      "0.943870627",
      "0.078585099"
    ],
    [
      "0.153348299",
      "0.112258746"
    ],
    [
      "0.347786151",
      "0.112258746"
    ],
    [
      "0.542224002",
      "0.112258746"
    ],
    [
      "0.736661854",
      "0.112258746"
    ],
    [
      "0.056129373",
      "0.168388119"
    ],
    [
      "0.250567225",
      "0.168388119"
    ],
    [
      "0.445005077",
      "0.168388119"
    ],
    [
      "0.639442928",
      "0.168388119"
    ],
    [
      "0.83388078",
      "0.168388119"
    ],
    [
      "0.943870627",
      "0.190843845"
    ],
    [
      "0.153348299",
      "0.224517492"
    ],
    [
      "0.347786151",
      "0.224517492"
    ],
    [
      "0.542224002",
      "0.224517492"
    ],
    [
      "0.736661854",
      "0.224517492"
    ],
    [
      "0.056129373",
      "0.280646865"
    ],
    [
      "0.250567225",
      "0.280646865"
    ],
    [
      "0.445005077",
      "0.280646865"
    ],
    [
      "0.639442928",
      "0.280646865"
    ],
    [
      "0.83388078",
      "0.280646865"
    ],
    [
      "0.943870627",
      "0.303102591"
    ],
    [
      "0.153348299",
      "0.336776238"
    ],
    [
      "0.347786151",
      "0.336776238"
    ],
    [
      "0.542224002",
      "0.336776238"
    ],
    [
      "0.736661854",
      "0.336776238"
    ],
    [
      "0.83388078",
      "0.392905611"
    ],
    [
      "0.056129373",
      "0.392905611"
    ],
    [
      "0.250567225",
      "0.392905611"
    ],
    [
      "0.445005077",
      "0.392905611"
    ],
    [
      "0.639442928",
      "0.392905611"
    ],
    [
      "0.943870627",
      "0.415361337"
    ],
    [
      "0.153348299",
      "0.449034984"
    ],
    [
      "0.347786151",
      "0.449034984"
    ],
    [
      "0.542224002",
      "0.449034984"
    ],
    [
      "0.736661854",
      "0.449034984"
    ],
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      "0.056129373",
      "0.505164357"
    ],
    [
      "0.250567225",
      "0.505164357"
    ],
    [
      "0.445005077",
      "0.505164357"
    ],
    [
      "0.639442928",
      "0.505164357"
    ],
    [
      "0.83388078",
      "0.505164357"
    ],
    [
      "0.943870627",
      "0.527620083"
    ],
    [
      "0.347786151",
      "0.56129373"
    ],
    [
      "0.542224002",
      "0.56129373"
    ],
    [
      "0.736661854",
      "0.56129373"
    ],
    [
      "0.153348299",
      "0.56129373"
    ],
    [
      "0.056129373",
      "0.617423103"
    ],
    [
      "0.250567225",
      "0.617423103"
    ],
    [
      "0.445005077",
      "0.617423103"
    ],
    [
      "0.639442928",
      "0.617423103"
    ],
    [
      "0.83388078",
      "0.617423103"
    ],
    [
      "0.943870627",
      "0.639878829"
    ],
    [
      "0.153348299",
      "0.673552476"
    ],
    [
      "0.347786151",
      "0.673552476"
    ],
    [
      "0.542224002",
      "0.673552476"
    ],
    [
      "0.736661854",
      "0.673552476"
    ],
    [
      "0.056129373",
      "0.729681849"
    ],
    [
      "0.250567225",
      "0.729681849"
    ],
    [
      "0.445005077",
      "0.729681849"
    ],
    [
      "0.639442928",
      "0.729681849"
    ],
    [
      "0.83388078",
      "0.729681849"
    ],
    [
      "0.943870627",
      "0.752137575"
    ],
    [
      "0.545622417",
      "0.797676903"
    ],
    [
      "0.325415913",
      "0.813346059"
    ],
    [
      "0.763479503",
      "0.817121464"
    ],
    [
      "0.201888262",
      "0.8308371"
    ],
    [
      "0.089786734",
      "0.836776238"
    ],
    [
      "0.87346935",
      "0.83957719"
    ],
    [
      "0.653793999",
      "0.841019502"
    ],
    [
      "0.434110003",
      "0.841410643"
    ],
    [
      "0.27673695",
      "0.914501309"
    ],
    [
      "0.943870616",
      "0.927725949"
    ],
    [
      "0.720373065",
      "0.931403428"
    ],
    [
      "0.496947269",
      "0.934434856"
    ],
    [
      "0.056129373",
      "0.943870627"
    ],
    [
      "0.168388119",
      "0.943870627"
    ],
    [
      "0.385085782",
      "0.943870627"
    ],
    [
      "0.608808756",
      "0.943870627"
    ],
    [
      "0.831937373",
      "0.943870627"
    ]
  ],
  "radius": "0.056129372"
}

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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