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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 = 86

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 = 86
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[0.0555729994115, 0.0555729994125]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 = 0.055572999

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 = 86 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": [
    [
      "0.055572999",
      "0.055572999"
    ],
    [
      "0.248083516",
      "0.055572999"
    ],
    [
      "0.359229515",
      "0.055572999"
    ],
    [
      "0.551740032",
      "0.055572999"
    ],
    [
      "0.744250549",
      "0.055572999"
    ],
    [
      "0.936761066",
      "0.055572999"
    ],
    [
      "0.151828258",
      "0.111145999"
    ],
    [
      "0.840505808",
      "0.111145999"
    ],
    [
      "0.647995291",
      "0.111145999"
    ],
    [
      "0.455484774",
      "0.111145999"
    ],
    [
      "0.944427001",
      "0.166454317"
    ],
    [
      "0.055572999",
      "0.166718998"
    ],
    [
      "0.248083516",
      "0.166718998"
    ],
    [
      "0.744250549",
      "0.166718998"
    ],
    [
      "0.551740032",
      "0.166718998"
    ],
    [
      "0.359229515",
      "0.166718998"
    ],
    [
      "0.848171742",
      "0.222027316"
    ],
    [
      "0.151828258",
      "0.222291998"
    ],
    [
      "0.647995291",
      "0.222291998"
    ],
    [
      "0.455484774",
      "0.222291998"
    ],
    [
      "0.944427001",
      "0.277600315"
    ],
    [
      "0.751916484",
      "0.277600315"
    ],
    [
      "0.055572999",
      "0.277864997"
    ],
    [
      "0.248083516",
      "0.277864997"
    ],
    [
      "0.359229515",
      "0.277864997"
    ],
    [
      "0.551740032",
      "0.277864997"
    ],
    [
      "0.848171742",
      "0.333173315"
    ],
    [
      "0.655661225",
      "0.333173315"
    ],
    [
      "0.151828258",
      "0.333437996"
    ],
    [
      "0.455484774",
      "0.333437996"
    ],
    [
      "0.944427001",
      "0.388746314"
    ],
    [
      "0.751916484",
      "0.388746314"
    ],
    [
      "0.559405967",
      "0.388746314"
    ],
    [
      "0.055572999",
      "0.389010996"
    ],
    [
      "0.248083516",
      "0.389010996"
    ],
    [
      "0.359229515",
      "0.389010996"
    ],
    [
      "0.848171742",
      "0.444319314"
    ],
    [
      "0.655661225",
      "0.444319314"
    ],
    [
      "0.463150708",
      "0.444319314"
    ],
    [
      "0.151828258",
      "0.444583995"
    ],
    [
      "0.944427001",
      "0.499892313"
    ],
    [
      "0.751916484",
      "0.499892313"
    ],
    [
      "0.36689545",
      "0.499892313"
    ],
    [
      "0.559405967",
      "0.499892313"
    ],
    [
      "0.255749177",
      "0.499892607"
    ],
    [
      "0.055572999",
      "0.500156995"
    ],
    [
      "0.848171742",
      "0.555465313"
    ],
    [
      "0.655661225",
      "0.555465313"
    ],
    [
      "0.463150708",
      "0.555465313"
    ],
    [
      "0.156049586",
      "0.555649802"
    ],
    [
      "0.363585117",
      "0.610989004"
    ],
    [
      "0.252439119",
      "0.61098958"
    ],
    [
      "0.944427001",
      "0.611038312"
    ],
    [
      "0.751916484",
      "0.611038312"
    ],
    [
      "0.559405967",
      "0.611038312"
    ],
    [
      "0.059794327",
      "0.611222802"
    ],
    [
      "0.459840376",
      "0.666562004"
    ],
    [
      "0.848171742",
      "0.666611311"
    ],
    [
      "0.655661225",
      "0.666611311"
    ],
    [
      "0.156049586",
      "0.666795801"
    ],
    [
      "0.363585117",
      "0.722135003"
    ],
    [
      "0.556095634",
      "0.722135003"
    ],
    [
      "0.252439119",
      "0.722135579"
    ],
    [
      "0.944427001",
      "0.722184311"
    ],
    [
      "0.751916484",
      "0.722184311"
    ],
    [
      "0.055572999",
      "0.722288608"
    ],
    [
      "0.652350893",
      "0.777708002"
    ],
    [
      "0.459840376",
      "0.777708002"
    ],
    [
      "0.848171742",
      "0.77775731"
    ],
    [
      "0.156049918",
      "0.7779418"
    ],
    [
      "0.748606151",
      "0.833281002"
    ],
    [
      "0.556095634",
      "0.833281002"
    ],
    [
      "0.252439451",
      "0.833281578"
    ],
    [
      "0.36358545",
      "0.833281578"
    ],
    [
      "0.944427001",
      "0.83333031"
    ],
    [
      "0.059704189",
      "0.833357804"
    ],
    [
      "0.84486141",
      "0.888854001"
    ],
    [
      "0.65235056",
      "0.888854577"
    ],
    [
      "0.459661324",
      "0.889164128"
    ],
    [
      "0.15382734",
      "0.892470118"
    ],
    [
      "0.055572999",
      "0.944427001"
    ],
    [
      "0.25208168",
      "0.944427001"
    ],
    [
      "0.363227679",
      "0.944427001"
    ],
    [
      "0.55609497",
      "0.944427001"
    ],
    [
      "0.748606151",
      "0.944427001"
    ],
    [
      "0.941116668",
      "0.944427001"
    ]
  ],
  "radius": "0.055572999"
}
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": 86
}

起步布局答案

{
  "centers": [
    [
      "0.055572999",
      "0.055572999"
    ],
    [
      "0.248083516",
      "0.055572999"
    ],
    [
      "0.359229515",
      "0.055572999"
    ],
    [
      "0.551740032",
      "0.055572999"
    ],
    [
      "0.744250549",
      "0.055572999"
    ],
    [
      "0.936761066",
      "0.055572999"
    ],
    [
      "0.151828258",
      "0.111145999"
    ],
    [
      "0.840505808",
      "0.111145999"
    ],
    [
      "0.647995291",
      "0.111145999"
    ],
    [
      "0.455484774",
      "0.111145999"
    ],
    [
      "0.944427001",
      "0.166454317"
    ],
    [
      "0.055572999",
      "0.166718998"
    ],
    [
      "0.248083516",
      "0.166718998"
    ],
    [
      "0.744250549",
      "0.166718998"
    ],
    [
      "0.551740032",
      "0.166718998"
    ],
    [
      "0.359229515",
      "0.166718998"
    ],
    [
      "0.848171742",
      "0.222027316"
    ],
    [
      "0.151828258",
      "0.222291998"
    ],
    [
      "0.647995291",
      "0.222291998"
    ],
    [
      "0.455484774",
      "0.222291998"
    ],
    [
      "0.944427001",
      "0.277600315"
    ],
    [
      "0.751916484",
      "0.277600315"
    ],
    [
      "0.055572999",
      "0.277864997"
    ],
    [
      "0.248083516",
      "0.277864997"
    ],
    [
      "0.359229515",
      "0.277864997"
    ],
    [
      "0.551740032",
      "0.277864997"
    ],
    [
      "0.848171742",
      "0.333173315"
    ],
    [
      "0.655661225",
      "0.333173315"
    ],
    [
      "0.151828258",
      "0.333437996"
    ],
    [
      "0.455484774",
      "0.333437996"
    ],
    [
      "0.944427001",
      "0.388746314"
    ],
    [
      "0.751916484",
      "0.388746314"
    ],
    [
      "0.559405967",
      "0.388746314"
    ],
    [
      "0.055572999",
      "0.389010996"
    ],
    [
      "0.248083516",
      "0.389010996"
    ],
    [
      "0.359229515",
      "0.389010996"
    ],
    [
      "0.848171742",
      "0.444319314"
    ],
    [
      "0.655661225",
      "0.444319314"
    ],
    [
      "0.463150708",
      "0.444319314"
    ],
    [
      "0.151828258",
      "0.444583995"
    ],
    [
      "0.944427001",
      "0.499892313"
    ],
    [
      "0.751916484",
      "0.499892313"
    ],
    [
      "0.36689545",
      "0.499892313"
    ],
    [
      "0.559405967",
      "0.499892313"
    ],
    [
      "0.255749177",
      "0.499892607"
    ],
    [
      "0.055572999",
      "0.500156995"
    ],
    [
      "0.848171742",
      "0.555465313"
    ],
    [
      "0.655661225",
      "0.555465313"
    ],
    [
      "0.463150708",
      "0.555465313"
    ],
    [
      "0.156049586",
      "0.555649802"
    ],
    [
      "0.363585117",
      "0.610989004"
    ],
    [
      "0.252439119",
      "0.61098958"
    ],
    [
      "0.944427001",
      "0.611038312"
    ],
    [
      "0.751916484",
      "0.611038312"
    ],
    [
      "0.559405967",
      "0.611038312"
    ],
    [
      "0.059794327",
      "0.611222802"
    ],
    [
      "0.459840376",
      "0.666562004"
    ],
    [
      "0.848171742",
      "0.666611311"
    ],
    [
      "0.655661225",
      "0.666611311"
    ],
    [
      "0.156049586",
      "0.666795801"
    ],
    [
      "0.363585117",
      "0.722135003"
    ],
    [
      "0.556095634",
      "0.722135003"
    ],
    [
      "0.252439119",
      "0.722135579"
    ],
    [
      "0.944427001",
      "0.722184311"
    ],
    [
      "0.751916484",
      "0.722184311"
    ],
    [
      "0.055572999",
      "0.722288608"
    ],
    [
      "0.652350893",
      "0.777708002"
    ],
    [
      "0.459840376",
      "0.777708002"
    ],
    [
      "0.848171742",
      "0.77775731"
    ],
    [
      "0.156049918",
      "0.7779418"
    ],
    [
      "0.748606151",
      "0.833281002"
    ],
    [
      "0.556095634",
      "0.833281002"
    ],
    [
      "0.252439451",
      "0.833281578"
    ],
    [
      "0.36358545",
      "0.833281578"
    ],
    [
      "0.944427001",
      "0.83333031"
    ],
    [
      "0.059704189",
      "0.833357804"
    ],
    [
      "0.84486141",
      "0.888854001"
    ],
    [
      "0.65235056",
      "0.888854577"
    ],
    [
      "0.459661324",
      "0.889164128"
    ],
    [
      "0.15382734",
      "0.892470118"
    ],
    [
      "0.055572999",
      "0.944427001"
    ],
    [
      "0.25208168",
      "0.944427001"
    ],
    [
      "0.363227679",
      "0.944427001"
    ],
    [
      "0.55609497",
      "0.944427001"
    ],
    [
      "0.748606151",
      "0.944427001"
    ],
    [
      "0.941116668",
      "0.944427001"
    ]
  ],
  "radius": "0.055572999"
}

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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