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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 = 64

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 = 64
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[0.0634589868125, 0.0634589868135]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 = 0.063458986

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 = 64 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": [
    [
      "0.063458987",
      "0.063458987"
    ],
    [
      "0.283287366",
      "0.063458987"
    ],
    [
      "0.503079138",
      "0.063458987"
    ],
    [
      "0.722907506",
      "0.063458987"
    ],
    [
      "0.936541013",
      "0.063458987"
    ],
    [
      "0.173373176",
      "0.126917974"
    ],
    [
      "0.612993322",
      "0.126917983"
    ],
    [
      "0.393183252",
      "0.126949665"
    ],
    [
      "0.82972426",
      "0.132003521"
    ],
    [
      "0.063458987",
      "0.19037696"
    ],
    [
      "0.283269062",
      "0.190408652"
    ],
    [
      "0.503097436",
      "0.190408661"
    ],
    [
      "0.719810076",
      "0.195462517"
    ],
    [
      "0.936541013",
      "0.200548056"
    ],
    [
      "0.173354873",
      "0.253867638"
    ],
    [
      "0.393183247",
      "0.253867648"
    ],
    [
      "0.609914189",
      "0.258953195"
    ],
    [
      "0.826626829",
      "0.264007052"
    ],
    [
      "0.283269057",
      "0.317326634"
    ],
    [
      "0.063458987",
      "0.317358317"
    ],
    [
      "0.5",
      "0.322412182"
    ],
    [
      "0.936541013",
      "0.327466048"
    ],
    [
      "0.716730943",
      "0.32749773"
    ],
    [
      "0.173373171",
      "0.380817312"
    ],
    [
      "0.390085811",
      "0.385871169"
    ],
    [
      "0.606816753",
      "0.390956717"
    ],
    [
      "0.826645127",
      "0.390956726"
    ],
    [
      "0.063458987",
      "0.444276308"
    ],
    [
      "0.280189924",
      "0.449361847"
    ],
    [
      "0.496902564",
      "0.454415704"
    ],
    [
      "0.716730938",
      "0.454415713"
    ],
    [
      "0.936541013",
      "0.454447404"
    ],
    [
      "0.17027574",
      "0.512820843"
    ],
    [
      "0.606816748",
      "0.517874699"
    ],
    [
      "0.387006678",
      "0.517906382"
    ],
    [
      "0.826626824",
      "0.517906391"
    ],
    [
      "0.063458987",
      "0.581365378"
    ],
    [
      "0.277092494",
      "0.581365378"
    ],
    [
      "0.716712634",
      "0.581365378"
    ],
    [
      "0.936541013",
      "0.581365378"
    ],
    [
      "0.496937359",
      "0.581393925"
    ],
    [
      "0.387006688",
      "0.644824355"
    ],
    [
      "0.826626824",
      "0.644824364"
    ],
    [
      "0.607125297",
      "0.645387152"
    ],
    [
      "0.17027574",
      "0.649909912"
    ],
    [
      "0.936541013",
      "0.708283351"
    ],
    [
      "0.496904388",
      "0.708311894"
    ],
    [
      "0.717039487",
      "0.708846139"
    ],
    [
      "0.280189935",
      "0.71336889"
    ],
    [
      "0.063458987",
      "0.718454447"
    ],
    [
      "0.399547154",
      "0.808660331"
    ],
    [
      "0.806966847",
      "0.80937326"
    ],
    [
      "0.145722518",
      "0.815102693"
    ],
    [
      "0.66975437",
      "0.826626824"
    ],
    [
      "0.542836397",
      "0.826626824"
    ],
    [
      "0.936541013",
      "0.835201325"
    ],
    [
      "0.270195905",
      "0.839892767"
    ],
    [
      "0.063458987",
      "0.91175094"
    ],
    [
      "0.187932375",
      "0.936541013"
    ],
    [
      "0.352459436",
      "0.936541013"
    ],
    [
      "0.47937741",
      "0.936541013"
    ],
    [
      "0.606295383",
      "0.936541013"
    ],
    [
      "0.733213357",
      "0.936541013"
    ],
    [
      "0.860131331",
      "0.936541013"
    ]
  ],
  "radius": "0.063458986"
}
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": 64
}

起步布局答案

{
  "centers": [
    [
      "0.063458987",
      "0.063458987"
    ],
    [
      "0.283287366",
      "0.063458987"
    ],
    [
      "0.503079138",
      "0.063458987"
    ],
    [
      "0.722907506",
      "0.063458987"
    ],
    [
      "0.936541013",
      "0.063458987"
    ],
    [
      "0.173373176",
      "0.126917974"
    ],
    [
      "0.612993322",
      "0.126917983"
    ],
    [
      "0.393183252",
      "0.126949665"
    ],
    [
      "0.82972426",
      "0.132003521"
    ],
    [
      "0.063458987",
      "0.19037696"
    ],
    [
      "0.283269062",
      "0.190408652"
    ],
    [
      "0.503097436",
      "0.190408661"
    ],
    [
      "0.719810076",
      "0.195462517"
    ],
    [
      "0.936541013",
      "0.200548056"
    ],
    [
      "0.173354873",
      "0.253867638"
    ],
    [
      "0.393183247",
      "0.253867648"
    ],
    [
      "0.609914189",
      "0.258953195"
    ],
    [
      "0.826626829",
      "0.264007052"
    ],
    [
      "0.283269057",
      "0.317326634"
    ],
    [
      "0.063458987",
      "0.317358317"
    ],
    [
      "0.5",
      "0.322412182"
    ],
    [
      "0.936541013",
      "0.327466048"
    ],
    [
      "0.716730943",
      "0.32749773"
    ],
    [
      "0.173373171",
      "0.380817312"
    ],
    [
      "0.390085811",
      "0.385871169"
    ],
    [
      "0.606816753",
      "0.390956717"
    ],
    [
      "0.826645127",
      "0.390956726"
    ],
    [
      "0.063458987",
      "0.444276308"
    ],
    [
      "0.280189924",
      "0.449361847"
    ],
    [
      "0.496902564",
      "0.454415704"
    ],
    [
      "0.716730938",
      "0.454415713"
    ],
    [
      "0.936541013",
      "0.454447404"
    ],
    [
      "0.17027574",
      "0.512820843"
    ],
    [
      "0.606816748",
      "0.517874699"
    ],
    [
      "0.387006678",
      "0.517906382"
    ],
    [
      "0.826626824",
      "0.517906391"
    ],
    [
      "0.063458987",
      "0.581365378"
    ],
    [
      "0.277092494",
      "0.581365378"
    ],
    [
      "0.716712634",
      "0.581365378"
    ],
    [
      "0.936541013",
      "0.581365378"
    ],
    [
      "0.496937359",
      "0.581393925"
    ],
    [
      "0.387006688",
      "0.644824355"
    ],
    [
      "0.826626824",
      "0.644824364"
    ],
    [
      "0.607125297",
      "0.645387152"
    ],
    [
      "0.17027574",
      "0.649909912"
    ],
    [
      "0.936541013",
      "0.708283351"
    ],
    [
      "0.496904388",
      "0.708311894"
    ],
    [
      "0.717039487",
      "0.708846139"
    ],
    [
      "0.280189935",
      "0.71336889"
    ],
    [
      "0.063458987",
      "0.718454447"
    ],
    [
      "0.399547154",
      "0.808660331"
    ],
    [
      "0.806966847",
      "0.80937326"
    ],
    [
      "0.145722518",
      "0.815102693"
    ],
    [
      "0.66975437",
      "0.826626824"
    ],
    [
      "0.542836397",
      "0.826626824"
    ],
    [
      "0.936541013",
      "0.835201325"
    ],
    [
      "0.270195905",
      "0.839892767"
    ],
    [
      "0.063458987",
      "0.91175094"
    ],
    [
      "0.187932375",
      "0.936541013"
    ],
    [
      "0.352459436",
      "0.936541013"
    ],
    [
      "0.47937741",
      "0.936541013"
    ],
    [
      "0.606295383",
      "0.936541013"
    ],
    [
      "0.733213357",
      "0.936541013"
    ],
    [
      "0.860131331",
      "0.936541013"
    ]
  ],
  "radius": "0.063458986"
}

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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