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P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 38

文献中也称circles in a circlepacking equal circles in a circlecci

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆，使共同半径尽可能大。

子题n = 38
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[0.1436392180732898324324959244875, 0.1436392180732898324324959244885]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器半径 1 的圆，圆心在 (1, 1)，两个坐标都在 0 到 2 之间
  • 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
挑战这个纪录 ↗

当前展示：起步布局

2y0
0x2
已验证构造r = 0.143639217

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..14 与 19 已证明；其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型，大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 38 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

容器是半径 1 的圆，圆心在 (1, 1)，所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位，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 "0.2"。

起步布局答案

{
  "centers": [
    [
      "1.091932621",
      "0.148588124"
    ],
    [
      "0.805187954",
      "0.166092298"
    ],
    [
      "1.368331533",
      "0.226898667"
    ],
    [
      "0.540366709",
      "0.277441334"
    ],
    [
      "1.603279788",
      "0.39221115"
    ],
    [
      "1.074049698",
      "0.435309419"
    ],
    [
      "0.787305031",
      "0.452813593"
    ],
    [
      "0.327270879",
      "0.470104427"
    ],
    [
      "1.347572221",
      "0.523140673"
    ],
    [
      "1.770337187",
      "0.625921924"
    ],
    [
      "0.563744064",
      "0.633228099"
    ],
    [
      "0.95517893",
      "0.696840757"
    ],
    [
      "0.189881476",
      "0.722399991"
    ],
    [
      "1.514629619",
      "0.756851446"
    ],
    [
      "1.228701454",
      "0.78467201"
    ],
    [
      "0.726116898",
      "0.870217472"
    ],
    [
      "0.426354661",
      "0.885523663"
    ],
    [
      "1.850703712",
      "0.901730048"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.143659789",
      "1.005935624"
    ],
    [
      "1.593848684",
      "1.032991337"
    ],
    [
      "1.307920518",
      "1.060811902"
    ],
    [
      "0.588727494",
      "1.122513036"
    ],
    [
      "1.835335214",
      "1.188597106"
    ],
    [
      "0.840136358",
      "1.261518371"
    ],
    [
      "1.126564153",
      "1.283609498"
    ],
    [
      "0.193807435",
      "1.288803285"
    ],
    [
      "1.480249421",
      "1.300689778"
    ],
    [
      "0.4655359",
      "1.382037083"
    ],
    [
      "1.725961208",
      "1.45424015"
    ],
    [
      "1.28865108",
      "1.520794507"
    ],
    [
      "0.716944764",
      "1.521042418"
    ],
    [
      "1.00382569",
      "1.567619176"
    ],
    [
      "1.534890235",
      "1.668764701"
    ],
    [
      "0.465799021",
      "1.669315399"
    ],
    [
      "1.28362471",
      "1.808028968"
    ],
    [
      "0.717207885",
      "1.808320734"
    ],
    [
      "1.000441118",
      "1.856360668"
    ]
  ],
  "radius": "0.143639217"
}
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": 38
}

起步布局答案

{
  "centers": [
    [
      "1.091932621",
      "0.148588124"
    ],
    [
      "0.805187954",
      "0.166092298"
    ],
    [
      "1.368331533",
      "0.226898667"
    ],
    [
      "0.540366709",
      "0.277441334"
    ],
    [
      "1.603279788",
      "0.39221115"
    ],
    [
      "1.074049698",
      "0.435309419"
    ],
    [
      "0.787305031",
      "0.452813593"
    ],
    [
      "0.327270879",
      "0.470104427"
    ],
    [
      "1.347572221",
      "0.523140673"
    ],
    [
      "1.770337187",
      "0.625921924"
    ],
    [
      "0.563744064",
      "0.633228099"
    ],
    [
      "0.95517893",
      "0.696840757"
    ],
    [
      "0.189881476",
      "0.722399991"
    ],
    [
      "1.514629619",
      "0.756851446"
    ],
    [
      "1.228701454",
      "0.78467201"
    ],
    [
      "0.726116898",
      "0.870217472"
    ],
    [
      "0.426354661",
      "0.885523663"
    ],
    [
      "1.850703712",
      "0.901730048"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.143659789",
      "1.005935624"
    ],
    [
      "1.593848684",
      "1.032991337"
    ],
    [
      "1.307920518",
      "1.060811902"
    ],
    [
      "0.588727494",
      "1.122513036"
    ],
    [
      "1.835335214",
      "1.188597106"
    ],
    [
      "0.840136358",
      "1.261518371"
    ],
    [
      "1.126564153",
      "1.283609498"
    ],
    [
      "0.193807435",
      "1.288803285"
    ],
    [
      "1.480249421",
      "1.300689778"
    ],
    [
      "0.4655359",
      "1.382037083"
    ],
    [
      "1.725961208",
      "1.45424015"
    ],
    [
      "1.28865108",
      "1.520794507"
    ],
    [
      "0.716944764",
      "1.521042418"
    ],
    [
      "1.00382569",
      "1.567619176"
    ],
    [
      "1.534890235",
      "1.668764701"
    ],
    [
      "0.465799021",
      "1.669315399"
    ],
    [
      "1.28362471",
      "1.808028968"
    ],
    [
      "0.717207885",
      "1.808320734"
    ],
    [
      "1.000441118",
      "1.856360668"
    ]
  ],
  "radius": "0.143639217"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。 · 验证器 v1.0.0

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