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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 = 79

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 = 79
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[0.0575084977945, 0.0575084977955]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 = 0.057508497

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 = 79 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": [
    [
      "0.253799351",
      "0.057508498"
    ],
    [
      "0.453014631",
      "0.057508498"
    ],
    [
      "0.652229911",
      "0.057508498"
    ],
    [
      "0.851445191",
      "0.057508498"
    ],
    [
      "0.058083425",
      "0.058083425"
    ],
    [
      "0.353406991",
      "0.115016996"
    ],
    [
      "0.552622271",
      "0.115016996"
    ],
    [
      "0.751837551",
      "0.115016996"
    ],
    [
      "0.157116137",
      "0.119808301"
    ],
    [
      "0.942491502",
      "0.127789921"
    ],
    [
      "0.453014631",
      "0.172525493"
    ],
    [
      "0.652229911",
      "0.172525493"
    ],
    [
      "0.256723777",
      "0.177316799"
    ],
    [
      "0.057508498",
      "0.1773168"
    ],
    [
      "0.842883862",
      "0.185298419"
    ],
    [
      "0.552622271",
      "0.230033991"
    ],
    [
      "0.356331418",
      "0.234825297"
    ],
    [
      "0.157116138",
      "0.234825298"
    ],
    [
      "0.743276222",
      "0.242806917"
    ],
    [
      "0.942491502",
      "0.242806917"
    ],
    [
      "0.455939058",
      "0.292333795"
    ],
    [
      "0.057508498",
      "0.292333796"
    ],
    [
      "0.256723778",
      "0.292333796"
    ],
    [
      "0.643668582",
      "0.300315415"
    ],
    [
      "0.842883862",
      "0.300315415"
    ],
    [
      "0.157116138",
      "0.349842293"
    ],
    [
      "0.356331418",
      "0.349842293"
    ],
    [
      "0.743276222",
      "0.357823913"
    ],
    [
      "0.942491502",
      "0.357823913"
    ],
    [
      "0.546985369",
      "0.362615218"
    ],
    [
      "0.057508498",
      "0.407350791"
    ],
    [
      "0.256723778",
      "0.407350791"
    ],
    [
      "0.842883862",
      "0.41533241"
    ],
    [
      "0.646593009",
      "0.420123716"
    ],
    [
      "0.447377729",
      "0.420123717"
    ],
    [
      "0.157116138",
      "0.464859289"
    ],
    [
      "0.942491502",
      "0.472840908"
    ],
    [
      "0.746200649",
      "0.477632214"
    ],
    [
      "0.347770089",
      "0.477632215"
    ],
    [
      "0.546985369",
      "0.477632215"
    ],
    [
      "0.057508498",
      "0.522367787"
    ],
    [
      "0.845808289",
      "0.535140712"
    ],
    [
      "0.248162449",
      "0.535140713"
    ],
    [
      "0.447377729",
      "0.535140713"
    ],
    [
      "0.646593009",
      "0.535140713"
    ],
    [
      "0.148554809",
      "0.59264921"
    ],
    [
      "0.347770089",
      "0.59264921"
    ],
    [
      "0.546985369",
      "0.59264921"
    ],
    [
      "0.746200649",
      "0.59264921"
    ],
    [
      "0.942491502",
      "0.597440515"
    ],
    [
      "0.248162449",
      "0.650157708"
    ],
    [
      "0.447377729",
      "0.650157708"
    ],
    [
      "0.646593009",
      "0.650157708"
    ],
    [
      "0.842883854",
      "0.654949027"
    ],
    [
      "0.057508498",
      "0.662930634"
    ],
    [
      "0.347770089",
      "0.707666206"
    ],
    [
      "0.546986159",
      "0.707667575"
    ],
    [
      "0.942491502",
      "0.712457511"
    ],
    [
      "0.743276214",
      "0.712457525"
    ],
    [
      "0.157116138",
      "0.720439132"
    ],
    [
      "0.447377729",
      "0.765174704"
    ],
    [
      "0.842883854",
      "0.769966022"
    ],
    [
      "0.643654243",
      "0.769990853"
    ],
    [
      "0.057508498",
      "0.777947629"
    ],
    [
      "0.256723778",
      "0.777947629"
    ],
    [
      "0.942491502",
      "0.827474507"
    ],
    [
      "0.743276214",
      "0.82747452"
    ],
    [
      "0.544045812",
      "0.827497982"
    ],
    [
      "0.356331418",
      "0.835456127"
    ],
    [
      "0.157116138",
      "0.835456127"
    ],
    [
      "0.842855952",
      "0.88503132"
    ],
    [
      "0.642471279",
      "0.887007114"
    ],
    [
      "0.427977201",
      "0.925432741"
    ],
    [
      "0.069010039",
      "0.931637506"
    ],
    [
      "0.19921528",
      "0.942491502"
    ],
    [
      "0.314232276",
      "0.942491502"
    ],
    [
      "0.541722126",
      "0.942491502"
    ],
    [
      "0.743220432",
      "0.942491502"
    ],
    [
      "0.942491471",
      "0.942491502"
    ]
  ],
  "radius": "0.057508497"
}
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": 79
}

起步布局答案

{
  "centers": [
    [
      "0.253799351",
      "0.057508498"
    ],
    [
      "0.453014631",
      "0.057508498"
    ],
    [
      "0.652229911",
      "0.057508498"
    ],
    [
      "0.851445191",
      "0.057508498"
    ],
    [
      "0.058083425",
      "0.058083425"
    ],
    [
      "0.353406991",
      "0.115016996"
    ],
    [
      "0.552622271",
      "0.115016996"
    ],
    [
      "0.751837551",
      "0.115016996"
    ],
    [
      "0.157116137",
      "0.119808301"
    ],
    [
      "0.942491502",
      "0.127789921"
    ],
    [
      "0.453014631",
      "0.172525493"
    ],
    [
      "0.652229911",
      "0.172525493"
    ],
    [
      "0.256723777",
      "0.177316799"
    ],
    [
      "0.057508498",
      "0.1773168"
    ],
    [
      "0.842883862",
      "0.185298419"
    ],
    [
      "0.552622271",
      "0.230033991"
    ],
    [
      "0.356331418",
      "0.234825297"
    ],
    [
      "0.157116138",
      "0.234825298"
    ],
    [
      "0.743276222",
      "0.242806917"
    ],
    [
      "0.942491502",
      "0.242806917"
    ],
    [
      "0.455939058",
      "0.292333795"
    ],
    [
      "0.057508498",
      "0.292333796"
    ],
    [
      "0.256723778",
      "0.292333796"
    ],
    [
      "0.643668582",
      "0.300315415"
    ],
    [
      "0.842883862",
      "0.300315415"
    ],
    [
      "0.157116138",
      "0.349842293"
    ],
    [
      "0.356331418",
      "0.349842293"
    ],
    [
      "0.743276222",
      "0.357823913"
    ],
    [
      "0.942491502",
      "0.357823913"
    ],
    [
      "0.546985369",
      "0.362615218"
    ],
    [
      "0.057508498",
      "0.407350791"
    ],
    [
      "0.256723778",
      "0.407350791"
    ],
    [
      "0.842883862",
      "0.41533241"
    ],
    [
      "0.646593009",
      "0.420123716"
    ],
    [
      "0.447377729",
      "0.420123717"
    ],
    [
      "0.157116138",
      "0.464859289"
    ],
    [
      "0.942491502",
      "0.472840908"
    ],
    [
      "0.746200649",
      "0.477632214"
    ],
    [
      "0.347770089",
      "0.477632215"
    ],
    [
      "0.546985369",
      "0.477632215"
    ],
    [
      "0.057508498",
      "0.522367787"
    ],
    [
      "0.845808289",
      "0.535140712"
    ],
    [
      "0.248162449",
      "0.535140713"
    ],
    [
      "0.447377729",
      "0.535140713"
    ],
    [
      "0.646593009",
      "0.535140713"
    ],
    [
      "0.148554809",
      "0.59264921"
    ],
    [
      "0.347770089",
      "0.59264921"
    ],
    [
      "0.546985369",
      "0.59264921"
    ],
    [
      "0.746200649",
      "0.59264921"
    ],
    [
      "0.942491502",
      "0.597440515"
    ],
    [
      "0.248162449",
      "0.650157708"
    ],
    [
      "0.447377729",
      "0.650157708"
    ],
    [
      "0.646593009",
      "0.650157708"
    ],
    [
      "0.842883854",
      "0.654949027"
    ],
    [
      "0.057508498",
      "0.662930634"
    ],
    [
      "0.347770089",
      "0.707666206"
    ],
    [
      "0.546986159",
      "0.707667575"
    ],
    [
      "0.942491502",
      "0.712457511"
    ],
    [
      "0.743276214",
      "0.712457525"
    ],
    [
      "0.157116138",
      "0.720439132"
    ],
    [
      "0.447377729",
      "0.765174704"
    ],
    [
      "0.842883854",
      "0.769966022"
    ],
    [
      "0.643654243",
      "0.769990853"
    ],
    [
      "0.057508498",
      "0.777947629"
    ],
    [
      "0.256723778",
      "0.777947629"
    ],
    [
      "0.942491502",
      "0.827474507"
    ],
    [
      "0.743276214",
      "0.82747452"
    ],
    [
      "0.544045812",
      "0.827497982"
    ],
    [
      "0.356331418",
      "0.835456127"
    ],
    [
      "0.157116138",
      "0.835456127"
    ],
    [
      "0.842855952",
      "0.88503132"
    ],
    [
      "0.642471279",
      "0.887007114"
    ],
    [
      "0.427977201",
      "0.925432741"
    ],
    [
      "0.069010039",
      "0.931637506"
    ],
    [
      "0.19921528",
      "0.942491502"
    ],
    [
      "0.314232276",
      "0.942491502"
    ],
    [
      "0.541722126",
      "0.942491502"
    ],
    [
      "0.743220432",
      "0.942491502"
    ],
    [
      "0.942491471",
      "0.942491502"
    ]
  ],
  "radius": "0.057508497"
}

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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