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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 = 106

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 = 106
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[0.0500152004255, 0.0500152004265]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 = 0.050015200

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 = 106 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": [
    [
      "0.0500152",
      "0.0500152"
    ],
    [
      "0.223272937",
      "0.0500152"
    ],
    [
      "0.323303338",
      "0.0500152"
    ],
    [
      "0.496561074",
      "0.0500152"
    ],
    [
      "0.669818847",
      "0.0500152"
    ],
    [
      "0.769849248",
      "0.0500152"
    ],
    [
      "0.9499848",
      "0.0500152"
    ],
    [
      "0.862591964",
      "0.099134491"
    ],
    [
      "0.583189961",
      "0.10003037"
    ],
    [
      "0.136644069",
      "0.100030401"
    ],
    [
      "0.409932206",
      "0.100030401"
    ],
    [
      "0.675331819",
      "0.149893568"
    ],
    [
      "0.77536222",
      "0.149893568"
    ],
    [
      "0.948702225",
      "0.150037378"
    ],
    [
      "0.0500152",
      "0.150045601"
    ],
    [
      "0.223272937",
      "0.150045601"
    ],
    [
      "0.323303338",
      "0.150045601"
    ],
    [
      "0.496561074",
      "0.150045601"
    ],
    [
      "0.588702365",
      "0.199908768"
    ],
    [
      "0.861990787",
      "0.19990929"
    ],
    [
      "0.136644069",
      "0.200060802"
    ],
    [
      "0.409932206",
      "0.200060802"
    ],
    [
      "0.675331234",
      "0.249923968"
    ],
    [
      "0.502073514",
      "0.249923998"
    ],
    [
      "0.775361634",
      "0.249923998"
    ],
    [
      "0.9499848",
      "0.250059556"
    ],
    [
      "0.0500152",
      "0.250076002"
    ],
    [
      "0.223272937",
      "0.250076002"
    ],
    [
      "0.323303338",
      "0.250076002"
    ],
    [
      "0.415444646",
      "0.299939198"
    ],
    [
      "0.588702382",
      "0.299939198"
    ],
    [
      "0.862137191",
      "0.299939583"
    ],
    [
      "0.136644069",
      "0.300091203"
    ],
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      "0.675331571",
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    ],
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      "0.349954399"
    ],
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    ],
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      "0.948692711",
      "0.350081612"
    ],
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      "0.0500152",
      "0.350106403"
    ],
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      "0.142156508",
      "0.399969599"
    ],
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      "0.415444646",
      "0.399969599"
    ],
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      "0.399969599"
    ],
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      "0.86199068",
      "0.399969877"
    ],
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      "0.05552764",
      "0.4499848"
    ],
    [
      "0.228785377",
      "0.4499848"
    ],
    [
      "0.328815777",
      "0.4499848"
    ],
    [
      "0.502073514",
      "0.4499848"
    ],
    [
      "0.675331251",
      "0.4499848"
    ],
    [
      "0.775361651",
      "0.4499848"
    ],
    [
      "0.9499848",
      "0.450103668"
    ],
    [
      "0.142156508",
      "0.5"
    ],
    [
      "0.415444646",
      "0.5"
    ],
    [
      "0.588702382",
      "0.5"
    ],
    [
      "0.862117882",
      "0.500000197"
    ],
    [
      "0.0500152",
      "0.549863196"
    ],
    [
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      "0.5500152"
    ],
    [
      "0.328815777",
      "0.5500152"
    ],
    [
      "0.502073514",
      "0.5500152"
    ],
    [
      "0.675331384",
      "0.5500152"
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      "0.415444646",
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    ],
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      "0.861990587",
      "0.600030517"
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    [
      "0.0500152",
      "0.649893597"
    ],
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      "0.649893597"
    ],
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    [
      "0.588702382",
      "0.700060802"
    ],
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      "0.862098247",
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    [
      "0.0500152",
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    ],
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      "0.323303338",
      "0.749923998"
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    ],
    [
      "0.948673204",
      "0.750169329"
    ],
    [
      "0.136644069",
      "0.799939198"
    ],
    [
      "0.409932206",
      "0.799939198"
    ],
    [
      "0.583189943",
      "0.799939198"
    ],
    [
      "0.86199052",
      "0.800091203"
    ],
    [
      "0.0500152",
      "0.849954399"
    ],
    [
      "0.223272937",
      "0.849954399"
    ],
    [
      "0.323303338",
      "0.849954399"
    ],
    [
      "0.496561074",
      "0.849954399"
    ],
    [
      "0.669818811",
      "0.849954399"
    ],
    [
      "0.769849212",
      "0.849954399"
    ],
    [
      "0.9499848",
      "0.850191131"
    ],
    [
      "0.136644069",
      "0.899969599"
    ],
    [
      "0.409932206",
      "0.899969599"
    ],
    [
      "0.583189943",
      "0.899969599"
    ],
    [
      "0.85647808",
      "0.899969599"
    ],
    [
      "0.0500152",
      "0.9499848"
    ],
    [
      "0.223272937",
      "0.9499848"
    ],
    [
      "0.323303338",
      "0.9499848"
    ],
    [
      "0.496561074",
      "0.9499848"
    ],
    [
      "0.669818811",
      "0.9499848"
    ],
    [
      "0.769849212",
      "0.9499848"
    ],
    [
      "0.943106948",
      "0.9499848"
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  ],
  "radius": "0.0500152"
}
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": 106
}

起步布局答案

{
  "centers": [
    [
      "0.0500152",
      "0.0500152"
    ],
    [
      "0.223272937",
      "0.0500152"
    ],
    [
      "0.323303338",
      "0.0500152"
    ],
    [
      "0.496561074",
      "0.0500152"
    ],
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      "0.669818847",
      "0.0500152"
    ],
    [
      "0.769849248",
      "0.0500152"
    ],
    [
      "0.9499848",
      "0.0500152"
    ],
    [
      "0.862591964",
      "0.099134491"
    ],
    [
      "0.583189961",
      "0.10003037"
    ],
    [
      "0.136644069",
      "0.100030401"
    ],
    [
      "0.409932206",
      "0.100030401"
    ],
    [
      "0.675331819",
      "0.149893568"
    ],
    [
      "0.77536222",
      "0.149893568"
    ],
    [
      "0.948702225",
      "0.150037378"
    ],
    [
      "0.0500152",
      "0.150045601"
    ],
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      "0.150045601"
    ],
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    ],
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      "0.199908768"
    ],
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      "0.861990787",
      "0.19990929"
    ],
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      "0.136644069",
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    ],
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    ],
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    ],
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      "0.775361634",
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    [
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      "0.5"
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    [
      "0.415444646",
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    [
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      "0.5"
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    [
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      "0.415444646",
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      "0.849954399"
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      "0.849954399"
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      "0.769849212",
      "0.849954399"
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      "0.136644069",
      "0.899969599"
    ],
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      "0.409932206",
      "0.899969599"
    ],
    [
      "0.583189943",
      "0.899969599"
    ],
    [
      "0.85647808",
      "0.899969599"
    ],
    [
      "0.0500152",
      "0.9499848"
    ],
    [
      "0.223272937",
      "0.9499848"
    ],
    [
      "0.323303338",
      "0.9499848"
    ],
    [
      "0.496561074",
      "0.9499848"
    ],
    [
      "0.669818811",
      "0.9499848"
    ],
    [
      "0.769849212",
      "0.9499848"
    ],
    [
      "0.943106948",
      "0.9499848"
    ]
  ],
  "radius": "0.0500152"
}

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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