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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 = 70

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 = 70
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[0.0605966936305, 0.0605966936315]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 = 0.060596693

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 = 70 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": [
    [
      "0.060596694",
      "0.060596694"
    ],
    [
      "0.270509798",
      "0.060596694"
    ],
    [
      "0.512759418",
      "0.060596694"
    ],
    [
      "0.726076381",
      "0.060596694"
    ],
    [
      "0.93920092",
      "0.060596694"
    ],
    [
      "0.391634608",
      "0.064673156"
    ],
    [
      "0.619417899",
      "0.118144981"
    ],
    [
      "0.832638651",
      "0.118322943"
    ],
    [
      "0.165553246",
      "0.121193387"
    ],
    [
      "0.516250379",
      "0.181739792"
    ],
    [
      "0.729365118",
      "0.181745451"
    ],
    [
      "0.939403306",
      "0.181789912"
    ],
    [
      "0.060596694",
      "0.181790081"
    ],
    [
      "0.273396191",
      "0.181819206"
    ],
    [
      "0.39512557",
      "0.185816254"
    ],
    [
      "0.622806216",
      "0.239477916"
    ],
    [
      "0.834371408",
      "0.242255914"
    ],
    [
      "0.165553246",
      "0.242386775"
    ],
    [
      "0.512762758",
      "0.302882987"
    ],
    [
      "0.726157639",
      "0.302896386"
    ],
    [
      "0.939403306",
      "0.302983299"
    ],
    [
      "0.060596694",
      "0.302983468"
    ],
    [
      "0.270509798",
      "0.302983468"
    ],
    [
      "0.391637948",
      "0.306959449"
    ],
    [
      "0.619598736",
      "0.360628851"
    ],
    [
      "0.834371408",
      "0.363449301"
    ],
    [
      "0.165553246",
      "0.363580162"
    ],
    [
      "0.516308706",
      "0.424024488"
    ],
    [
      "0.729414856",
      "0.424045995"
    ],
    [
      "0.939403306",
      "0.424176686"
    ],
    [
      "0.060596694",
      "0.424176855"
    ],
    [
      "0.273281597",
      "0.424204379"
    ],
    [
      "0.395183896",
      "0.42810095"
    ],
    [
      "0.622855954",
      "0.48177846"
    ],
    [
      "0.834371408",
      "0.484642688"
    ],
    [
      "0.165553246",
      "0.484773549"
    ],
    [
      "0.512766116",
      "0.545166088"
    ],
    [
      "0.726480622",
      "0.545266957"
    ],
    [
      "0.939403306",
      "0.545370074"
    ],
    [
      "0.060596694",
      "0.545370243"
    ],
    [
      "0.270509798",
      "0.545370243"
    ],
    [
      "0.391641306",
      "0.54924255"
    ],
    [
      "0.619567571",
      "0.602927226"
    ],
    [
      "0.834371408",
      "0.605836076"
    ],
    [
      "0.165553246",
      "0.605966936"
    ],
    [
      "0.516268938",
      "0.666308844"
    ],
    [
      "0.729414856",
      "0.666432769"
    ],
    [
      "0.274012619",
      "0.666512999"
    ],
    [
      "0.939403306",
      "0.666563461"
    ],
    [
      "0.060596694",
      "0.66656363"
    ],
    [
      "0.395144128",
      "0.670385306"
    ],
    [
      "0.622808346",
      "0.724077276"
    ],
    [
      "0.834371408",
      "0.727029463"
    ],
    [
      "0.169056067",
      "0.727109692"
    ],
    [
      "0.466204825",
      "0.776678322"
    ],
    [
      "0.341255134",
      "0.778938581"
    ],
    [
      "0.064099515",
      "0.787706386"
    ],
    [
      "0.939403306",
      "0.787756848"
    ],
    [
      "0.702253422",
      "0.825791976"
    ],
    [
      "0.231735659",
      "0.830835803"
    ],
    [
      "0.572744233",
      "0.834446754"
    ],
    [
      "0.834371408",
      "0.84822285"
    ],
    [
      "0.405608132",
      "0.881634874"
    ],
    [
      "0.060596694",
      "0.908849142"
    ],
    [
      "0.930666579",
      "0.930645357"
    ],
    [
      "0.177875337",
      "0.939403306"
    ],
    [
      "0.299068724",
      "0.939403306"
    ],
    [
      "0.51214754",
      "0.939403306"
    ],
    [
      "0.633340927",
      "0.939403306"
    ],
    [
      "0.754534314",
      "0.939403306"
    ]
  ],
  "radius": "0.060596693"
}
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": 70
}

起步布局答案

{
  "centers": [
    [
      "0.060596694",
      "0.060596694"
    ],
    [
      "0.270509798",
      "0.060596694"
    ],
    [
      "0.512759418",
      "0.060596694"
    ],
    [
      "0.726076381",
      "0.060596694"
    ],
    [
      "0.93920092",
      "0.060596694"
    ],
    [
      "0.391634608",
      "0.064673156"
    ],
    [
      "0.619417899",
      "0.118144981"
    ],
    [
      "0.832638651",
      "0.118322943"
    ],
    [
      "0.165553246",
      "0.121193387"
    ],
    [
      "0.516250379",
      "0.181739792"
    ],
    [
      "0.729365118",
      "0.181745451"
    ],
    [
      "0.939403306",
      "0.181789912"
    ],
    [
      "0.060596694",
      "0.181790081"
    ],
    [
      "0.273396191",
      "0.181819206"
    ],
    [
      "0.39512557",
      "0.185816254"
    ],
    [
      "0.622806216",
      "0.239477916"
    ],
    [
      "0.834371408",
      "0.242255914"
    ],
    [
      "0.165553246",
      "0.242386775"
    ],
    [
      "0.512762758",
      "0.302882987"
    ],
    [
      "0.726157639",
      "0.302896386"
    ],
    [
      "0.939403306",
      "0.302983299"
    ],
    [
      "0.060596694",
      "0.302983468"
    ],
    [
      "0.270509798",
      "0.302983468"
    ],
    [
      "0.391637948",
      "0.306959449"
    ],
    [
      "0.619598736",
      "0.360628851"
    ],
    [
      "0.834371408",
      "0.363449301"
    ],
    [
      "0.165553246",
      "0.363580162"
    ],
    [
      "0.516308706",
      "0.424024488"
    ],
    [
      "0.729414856",
      "0.424045995"
    ],
    [
      "0.939403306",
      "0.424176686"
    ],
    [
      "0.060596694",
      "0.424176855"
    ],
    [
      "0.273281597",
      "0.424204379"
    ],
    [
      "0.395183896",
      "0.42810095"
    ],
    [
      "0.622855954",
      "0.48177846"
    ],
    [
      "0.834371408",
      "0.484642688"
    ],
    [
      "0.165553246",
      "0.484773549"
    ],
    [
      "0.512766116",
      "0.545166088"
    ],
    [
      "0.726480622",
      "0.545266957"
    ],
    [
      "0.939403306",
      "0.545370074"
    ],
    [
      "0.060596694",
      "0.545370243"
    ],
    [
      "0.270509798",
      "0.545370243"
    ],
    [
      "0.391641306",
      "0.54924255"
    ],
    [
      "0.619567571",
      "0.602927226"
    ],
    [
      "0.834371408",
      "0.605836076"
    ],
    [
      "0.165553246",
      "0.605966936"
    ],
    [
      "0.516268938",
      "0.666308844"
    ],
    [
      "0.729414856",
      "0.666432769"
    ],
    [
      "0.274012619",
      "0.666512999"
    ],
    [
      "0.939403306",
      "0.666563461"
    ],
    [
      "0.060596694",
      "0.66656363"
    ],
    [
      "0.395144128",
      "0.670385306"
    ],
    [
      "0.622808346",
      "0.724077276"
    ],
    [
      "0.834371408",
      "0.727029463"
    ],
    [
      "0.169056067",
      "0.727109692"
    ],
    [
      "0.466204825",
      "0.776678322"
    ],
    [
      "0.341255134",
      "0.778938581"
    ],
    [
      "0.064099515",
      "0.787706386"
    ],
    [
      "0.939403306",
      "0.787756848"
    ],
    [
      "0.702253422",
      "0.825791976"
    ],
    [
      "0.231735659",
      "0.830835803"
    ],
    [
      "0.572744233",
      "0.834446754"
    ],
    [
      "0.834371408",
      "0.84822285"
    ],
    [
      "0.405608132",
      "0.881634874"
    ],
    [
      "0.060596694",
      "0.908849142"
    ],
    [
      "0.930666579",
      "0.930645357"
    ],
    [
      "0.177875337",
      "0.939403306"
    ],
    [
      "0.299068724",
      "0.939403306"
    ],
    [
      "0.51214754",
      "0.939403306"
    ],
    [
      "0.633340927",
      "0.939403306"
    ],
    [
      "0.754534314",
      "0.939403306"
    ]
  ],
  "radius": "0.060596693"
}

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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