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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 = 82

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 = 82
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[0.0565122716495, 0.0565122716505]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 = 0.056512271

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 = 82 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": [
    [
      "0.056512272",
      "0.056512272"
    ],
    [
      "0.643805026",
      "0.056512272"
    ],
    [
      "0.252276523",
      "0.056512272"
    ],
    [
      "0.448040775",
      "0.056512272"
    ],
    [
      "0.839569278",
      "0.056512272"
    ],
    [
      "0.943487728",
      "0.100958902"
    ],
    [
      "0.154394397",
      "0.113024543"
    ],
    [
      "0.350158649",
      "0.113024543"
    ],
    [
      "0.5459229",
      "0.113024543"
    ],
    [
      "0.741687152",
      "0.113024543"
    ],
    [
      "0.056512272",
      "0.169536815"
    ],
    [
      "0.252276523",
      "0.169536815"
    ],
    [
      "0.448040775",
      "0.169536815"
    ],
    [
      "0.643805026",
      "0.169536815"
    ],
    [
      "0.839569278",
      "0.169536815"
    ],
    [
      "0.943487728",
      "0.213983446"
    ],
    [
      "0.154394397",
      "0.226049087"
    ],
    [
      "0.350158649",
      "0.226049087"
    ],
    [
      "0.5459229",
      "0.226049087"
    ],
    [
      "0.741687152",
      "0.226049087"
    ],
    [
      "0.056512272",
      "0.282561358"
    ],
    [
      "0.643805026",
      "0.282561358"
    ],
    [
      "0.252276523",
      "0.282561358"
    ],
    [
      "0.448040775",
      "0.282561358"
    ],
    [
      "0.839569278",
      "0.282561358"
    ],
    [
      "0.943487728",
      "0.327007989"
    ],
    [
      "0.154394397",
      "0.33907363"
    ],
    [
      "0.350158649",
      "0.33907363"
    ],
    [
      "0.5459229",
      "0.33907363"
    ],
    [
      "0.741687152",
      "0.33907363"
    ],
    [
      "0.643805026",
      "0.395585902"
    ],
    [
      "0.056512272",
      "0.395585902"
    ],
    [
      "0.252276523",
      "0.395585902"
    ],
    [
      "0.448040775",
      "0.395585902"
    ],
    [
      "0.839569278",
      "0.395585902"
    ],
    [
      "0.943487728",
      "0.440032532"
    ],
    [
      "0.154394397",
      "0.452098173"
    ],
    [
      "0.350158649",
      "0.452098173"
    ],
    [
      "0.5459229",
      "0.452098173"
    ],
    [
      "0.741687152",
      "0.452098173"
    ],
    [
      "0.056512272",
      "0.508610445"
    ],
    [
      "0.643805026",
      "0.508610445"
    ],
    [
      "0.252276523",
      "0.508610445"
    ],
    [
      "0.448040775",
      "0.508610445"
    ],
    [
      "0.839569278",
      "0.508610445"
    ],
    [
      "0.943487728",
      "0.553057076"
    ],
    [
      "0.154394397",
      "0.565122717"
    ],
    [
      "0.350158649",
      "0.565122717"
    ],
    [
      "0.5459229",
      "0.565122717"
    ],
    [
      "0.741687152",
      "0.565122717"
    ],
    [
      "0.643805026",
      "0.621634988"
    ],
    [
      "0.056512272",
      "0.621634988"
    ],
    [
      "0.252276523",
      "0.621634988"
    ],
    [
      "0.448040775",
      "0.621634988"
    ],
    [
      "0.839569278",
      "0.621634988"
    ],
    [
      "0.943487728",
      "0.666081619"
    ],
    [
      "0.154394397",
      "0.67814726"
    ],
    [
      "0.350158649",
      "0.67814726"
    ],
    [
      "0.5459229",
      "0.67814726"
    ],
    [
      "0.741687152",
      "0.67814726"
    ],
    [
      "0.056512272",
      "0.734659531"
    ],
    [
      "0.643805026",
      "0.734659531"
    ],
    [
      "0.252276523",
      "0.734659531"
    ],
    [
      "0.448040775",
      "0.734659531"
    ],
    [
      "0.839569278",
      "0.734659531"
    ],
    [
      "0.943487728",
      "0.779106162"
    ],
    [
      "0.739431633",
      "0.819323357"
    ],
    [
      "0.169175308",
      "0.819354322"
    ],
    [
      "0.509145083",
      "0.831263651"
    ],
    [
      "0.395046364",
      "0.834490087"
    ],
    [
      "0.282389455",
      "0.843598775"
    ],
    [
      "0.624207371",
      "0.845972061"
    ],
    [
      "0.853036592",
      "0.846878865"
    ],
    [
      "0.056512272",
      "0.847684075"
    ],
    [
      "0.454797409",
      "0.930429443"
    ],
    [
      "0.934634874",
      "0.934562007"
    ],
    [
      "0.116480658",
      "0.943487728"
    ],
    [
      "0.229505201",
      "0.943487728"
    ],
    [
      "0.342529744",
      "0.943487728"
    ],
    [
      "0.567065074",
      "0.943487728"
    ],
    [
      "0.681349667",
      "0.943487728"
    ],
    [
      "0.794374211",
      "0.943487728"
    ]
  ],
  "radius": "0.056512271"
}
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": 82
}

起步布局答案

{
  "centers": [
    [
      "0.056512272",
      "0.056512272"
    ],
    [
      "0.643805026",
      "0.056512272"
    ],
    [
      "0.252276523",
      "0.056512272"
    ],
    [
      "0.448040775",
      "0.056512272"
    ],
    [
      "0.839569278",
      "0.056512272"
    ],
    [
      "0.943487728",
      "0.100958902"
    ],
    [
      "0.154394397",
      "0.113024543"
    ],
    [
      "0.350158649",
      "0.113024543"
    ],
    [
      "0.5459229",
      "0.113024543"
    ],
    [
      "0.741687152",
      "0.113024543"
    ],
    [
      "0.056512272",
      "0.169536815"
    ],
    [
      "0.252276523",
      "0.169536815"
    ],
    [
      "0.448040775",
      "0.169536815"
    ],
    [
      "0.643805026",
      "0.169536815"
    ],
    [
      "0.839569278",
      "0.169536815"
    ],
    [
      "0.943487728",
      "0.213983446"
    ],
    [
      "0.154394397",
      "0.226049087"
    ],
    [
      "0.350158649",
      "0.226049087"
    ],
    [
      "0.5459229",
      "0.226049087"
    ],
    [
      "0.741687152",
      "0.226049087"
    ],
    [
      "0.056512272",
      "0.282561358"
    ],
    [
      "0.643805026",
      "0.282561358"
    ],
    [
      "0.252276523",
      "0.282561358"
    ],
    [
      "0.448040775",
      "0.282561358"
    ],
    [
      "0.839569278",
      "0.282561358"
    ],
    [
      "0.943487728",
      "0.327007989"
    ],
    [
      "0.154394397",
      "0.33907363"
    ],
    [
      "0.350158649",
      "0.33907363"
    ],
    [
      "0.5459229",
      "0.33907363"
    ],
    [
      "0.741687152",
      "0.33907363"
    ],
    [
      "0.643805026",
      "0.395585902"
    ],
    [
      "0.056512272",
      "0.395585902"
    ],
    [
      "0.252276523",
      "0.395585902"
    ],
    [
      "0.448040775",
      "0.395585902"
    ],
    [
      "0.839569278",
      "0.395585902"
    ],
    [
      "0.943487728",
      "0.440032532"
    ],
    [
      "0.154394397",
      "0.452098173"
    ],
    [
      "0.350158649",
      "0.452098173"
    ],
    [
      "0.5459229",
      "0.452098173"
    ],
    [
      "0.741687152",
      "0.452098173"
    ],
    [
      "0.056512272",
      "0.508610445"
    ],
    [
      "0.643805026",
      "0.508610445"
    ],
    [
      "0.252276523",
      "0.508610445"
    ],
    [
      "0.448040775",
      "0.508610445"
    ],
    [
      "0.839569278",
      "0.508610445"
    ],
    [
      "0.943487728",
      "0.553057076"
    ],
    [
      "0.154394397",
      "0.565122717"
    ],
    [
      "0.350158649",
      "0.565122717"
    ],
    [
      "0.5459229",
      "0.565122717"
    ],
    [
      "0.741687152",
      "0.565122717"
    ],
    [
      "0.643805026",
      "0.621634988"
    ],
    [
      "0.056512272",
      "0.621634988"
    ],
    [
      "0.252276523",
      "0.621634988"
    ],
    [
      "0.448040775",
      "0.621634988"
    ],
    [
      "0.839569278",
      "0.621634988"
    ],
    [
      "0.943487728",
      "0.666081619"
    ],
    [
      "0.154394397",
      "0.67814726"
    ],
    [
      "0.350158649",
      "0.67814726"
    ],
    [
      "0.5459229",
      "0.67814726"
    ],
    [
      "0.741687152",
      "0.67814726"
    ],
    [
      "0.056512272",
      "0.734659531"
    ],
    [
      "0.643805026",
      "0.734659531"
    ],
    [
      "0.252276523",
      "0.734659531"
    ],
    [
      "0.448040775",
      "0.734659531"
    ],
    [
      "0.839569278",
      "0.734659531"
    ],
    [
      "0.943487728",
      "0.779106162"
    ],
    [
      "0.739431633",
      "0.819323357"
    ],
    [
      "0.169175308",
      "0.819354322"
    ],
    [
      "0.509145083",
      "0.831263651"
    ],
    [
      "0.395046364",
      "0.834490087"
    ],
    [
      "0.282389455",
      "0.843598775"
    ],
    [
      "0.624207371",
      "0.845972061"
    ],
    [
      "0.853036592",
      "0.846878865"
    ],
    [
      "0.056512272",
      "0.847684075"
    ],
    [
      "0.454797409",
      "0.930429443"
    ],
    [
      "0.934634874",
      "0.934562007"
    ],
    [
      "0.116480658",
      "0.943487728"
    ],
    [
      "0.229505201",
      "0.943487728"
    ],
    [
      "0.342529744",
      "0.943487728"
    ],
    [
      "0.567065074",
      "0.943487728"
    ],
    [
      "0.681349667",
      "0.943487728"
    ],
    [
      "0.794374211",
      "0.943487728"
    ]
  ],
  "radius": "0.056512271"
}

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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