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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 = 81

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 = 81
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[0.0568699211115, 0.0568699211125]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 = 0.056869920

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 = 81 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": [
    [
      "0.056869921",
      "0.056869921"
    ],
    [
      "0.253873107",
      "0.056869921"
    ],
    [
      "0.450876292",
      "0.056869921"
    ],
    [
      "0.647879478",
      "0.056869921"
    ],
    [
      "0.844882663",
      "0.056869921"
    ],
    [
      "0.155371514",
      "0.113739842"
    ],
    [
      "0.352374699",
      "0.113739842"
    ],
    [
      "0.549377885",
      "0.113739842"
    ],
    [
      "0.746381071",
      "0.113739842"
    ],
    [
      "0.943130079",
      "0.114177836"
    ],
    [
      "0.450876292",
      "0.170609763"
    ],
    [
      "0.253873107",
      "0.170609763"
    ],
    [
      "0.647879478",
      "0.170609763"
    ],
    [
      "0.056869921",
      "0.170609763"
    ],
    [
      "0.844628486",
      "0.171047757"
    ],
    [
      "0.155371514",
      "0.227479684"
    ],
    [
      "0.352374699",
      "0.227479684"
    ],
    [
      "0.549377885",
      "0.227479684"
    ],
    [
      "0.746126893",
      "0.227917678"
    ],
    [
      "0.943130079",
      "0.227917678"
    ],
    [
      "0.056869921",
      "0.284349606"
    ],
    [
      "0.253873107",
      "0.284349606"
    ],
    [
      "0.450876292",
      "0.284349606"
    ],
    [
      "0.647625301",
      "0.284787599"
    ],
    [
      "0.844628486",
      "0.284787599"
    ],
    [
      "0.155371514",
      "0.341219527"
    ],
    [
      "0.352374699",
      "0.341219527"
    ],
    [
      "0.746126893",
      "0.34165752"
    ],
    [
      "0.943130079",
      "0.34165752"
    ],
    [
      "0.549123708",
      "0.34165752"
    ],
    [
      "0.056869921",
      "0.398089448"
    ],
    [
      "0.253873107",
      "0.398089448"
    ],
    [
      "0.450622115",
      "0.398527441"
    ],
    [
      "0.647625301",
      "0.398527441"
    ],
    [
      "0.844628486",
      "0.398527441"
    ],
    [
      "0.155371514",
      "0.454959369"
    ],
    [
      "0.352120522",
      "0.455397362"
    ],
    [
      "0.549123708",
      "0.455397362"
    ],
    [
      "0.746126893",
      "0.455397362"
    ],
    [
      "0.943130079",
      "0.455397362"
    ],
    [
      "0.056869921",
      "0.51182929"
    ],
    [
      "0.844628486",
      "0.512267283"
    ],
    [
      "0.253618929",
      "0.512267283"
    ],
    [
      "0.450622115",
      "0.512267283"
    ],
    [
      "0.647625301",
      "0.512267283"
    ],
    [
      "0.943130079",
      "0.569137204"
    ],
    [
      "0.155117337",
      "0.569137204"
    ],
    [
      "0.352120522",
      "0.569137204"
    ],
    [
      "0.549123708",
      "0.569137204"
    ],
    [
      "0.746126893",
      "0.569137204"
    ],
    [
      "0.253618929",
      "0.626007126"
    ],
    [
      "0.450622115",
      "0.626007126"
    ],
    [
      "0.647625301",
      "0.626007126"
    ],
    [
      "0.844628486",
      "0.626007126"
    ],
    [
      "0.056869921",
      "0.626445119"
    ],
    [
      "0.352120522",
      "0.682877047"
    ],
    [
      "0.549123708",
      "0.682877047"
    ],
    [
      "0.746126893",
      "0.682877047"
    ],
    [
      "0.943130079",
      "0.682877047"
    ],
    [
      "0.155371514",
      "0.68331504"
    ],
    [
      "0.450622115",
      "0.739746968"
    ],
    [
      "0.647625301",
      "0.739746968"
    ],
    [
      "0.844628486",
      "0.739746968"
    ],
    [
      "0.253873107",
      "0.740184961"
    ],
    [
      "0.056869921",
      "0.740184961"
    ],
    [
      "0.943130079",
      "0.796616889"
    ],
    [
      "0.355539594",
      "0.827916258"
    ],
    [
      "0.730427641",
      "0.827916258"
    ],
    [
      "0.125902182",
      "0.830580197"
    ],
    [
      "0.603847203",
      "0.844724251"
    ],
    [
      "0.483523659",
      "0.848624147"
    ],
    [
      "0.237463481",
      "0.852734843"
    ],
    [
      "0.844628486",
      "0.85348681"
    ],
    [
      "0.056869921",
      "0.920975433"
    ],
    [
      "0.934267477",
      "0.934267477"
    ],
    [
      "0.16843122",
      "0.943130079"
    ],
    [
      "0.306495741",
      "0.943130079"
    ],
    [
      "0.420235584",
      "0.943130079"
    ],
    [
      "0.546811735",
      "0.943130079"
    ],
    [
      "0.660882672",
      "0.943130079"
    ],
    [
      "0.774622514",
      "0.943130079"
    ]
  ],
  "radius": "0.05686992"
}
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": 81
}

起步布局答案

{
  "centers": [
    [
      "0.056869921",
      "0.056869921"
    ],
    [
      "0.253873107",
      "0.056869921"
    ],
    [
      "0.450876292",
      "0.056869921"
    ],
    [
      "0.647879478",
      "0.056869921"
    ],
    [
      "0.844882663",
      "0.056869921"
    ],
    [
      "0.155371514",
      "0.113739842"
    ],
    [
      "0.352374699",
      "0.113739842"
    ],
    [
      "0.549377885",
      "0.113739842"
    ],
    [
      "0.746381071",
      "0.113739842"
    ],
    [
      "0.943130079",
      "0.114177836"
    ],
    [
      "0.450876292",
      "0.170609763"
    ],
    [
      "0.253873107",
      "0.170609763"
    ],
    [
      "0.647879478",
      "0.170609763"
    ],
    [
      "0.056869921",
      "0.170609763"
    ],
    [
      "0.844628486",
      "0.171047757"
    ],
    [
      "0.155371514",
      "0.227479684"
    ],
    [
      "0.352374699",
      "0.227479684"
    ],
    [
      "0.549377885",
      "0.227479684"
    ],
    [
      "0.746126893",
      "0.227917678"
    ],
    [
      "0.943130079",
      "0.227917678"
    ],
    [
      "0.056869921",
      "0.284349606"
    ],
    [
      "0.253873107",
      "0.284349606"
    ],
    [
      "0.450876292",
      "0.284349606"
    ],
    [
      "0.647625301",
      "0.284787599"
    ],
    [
      "0.844628486",
      "0.284787599"
    ],
    [
      "0.155371514",
      "0.341219527"
    ],
    [
      "0.352374699",
      "0.341219527"
    ],
    [
      "0.746126893",
      "0.34165752"
    ],
    [
      "0.943130079",
      "0.34165752"
    ],
    [
      "0.549123708",
      "0.34165752"
    ],
    [
      "0.056869921",
      "0.398089448"
    ],
    [
      "0.253873107",
      "0.398089448"
    ],
    [
      "0.450622115",
      "0.398527441"
    ],
    [
      "0.647625301",
      "0.398527441"
    ],
    [
      "0.844628486",
      "0.398527441"
    ],
    [
      "0.155371514",
      "0.454959369"
    ],
    [
      "0.352120522",
      "0.455397362"
    ],
    [
      "0.549123708",
      "0.455397362"
    ],
    [
      "0.746126893",
      "0.455397362"
    ],
    [
      "0.943130079",
      "0.455397362"
    ],
    [
      "0.056869921",
      "0.51182929"
    ],
    [
      "0.844628486",
      "0.512267283"
    ],
    [
      "0.253618929",
      "0.512267283"
    ],
    [
      "0.450622115",
      "0.512267283"
    ],
    [
      "0.647625301",
      "0.512267283"
    ],
    [
      "0.943130079",
      "0.569137204"
    ],
    [
      "0.155117337",
      "0.569137204"
    ],
    [
      "0.352120522",
      "0.569137204"
    ],
    [
      "0.549123708",
      "0.569137204"
    ],
    [
      "0.746126893",
      "0.569137204"
    ],
    [
      "0.253618929",
      "0.626007126"
    ],
    [
      "0.450622115",
      "0.626007126"
    ],
    [
      "0.647625301",
      "0.626007126"
    ],
    [
      "0.844628486",
      "0.626007126"
    ],
    [
      "0.056869921",
      "0.626445119"
    ],
    [
      "0.352120522",
      "0.682877047"
    ],
    [
      "0.549123708",
      "0.682877047"
    ],
    [
      "0.746126893",
      "0.682877047"
    ],
    [
      "0.943130079",
      "0.682877047"
    ],
    [
      "0.155371514",
      "0.68331504"
    ],
    [
      "0.450622115",
      "0.739746968"
    ],
    [
      "0.647625301",
      "0.739746968"
    ],
    [
      "0.844628486",
      "0.739746968"
    ],
    [
      "0.253873107",
      "0.740184961"
    ],
    [
      "0.056869921",
      "0.740184961"
    ],
    [
      "0.943130079",
      "0.796616889"
    ],
    [
      "0.355539594",
      "0.827916258"
    ],
    [
      "0.730427641",
      "0.827916258"
    ],
    [
      "0.125902182",
      "0.830580197"
    ],
    [
      "0.603847203",
      "0.844724251"
    ],
    [
      "0.483523659",
      "0.848624147"
    ],
    [
      "0.237463481",
      "0.852734843"
    ],
    [
      "0.844628486",
      "0.85348681"
    ],
    [
      "0.056869921",
      "0.920975433"
    ],
    [
      "0.934267477",
      "0.934267477"
    ],
    [
      "0.16843122",
      "0.943130079"
    ],
    [
      "0.306495741",
      "0.943130079"
    ],
    [
      "0.420235584",
      "0.943130079"
    ],
    [
      "0.546811735",
      "0.943130079"
    ],
    [
      "0.660882672",
      "0.943130079"
    ],
    [
      "0.774622514",
      "0.943130079"
    ]
  ],
  "radius": "0.05686992"
}

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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