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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 = 36

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 = 36
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[0.1482194297611193353695455256785, 0.1482194297611193353695455256795]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 = 0.148219429

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 = 36 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": [
    [
      "1.119019053",
      "0.156575667"
    ],
    [
      "0.822758781",
      "0.166863943"
    ],
    [
      "1.400863773",
      "0.2484427"
    ],
    [
      "0.547965913",
      "0.278061412"
    ],
    [
      "1.63415599",
      "0.431338131"
    ],
    [
      "1.077273072",
      "0.45006037"
    ],
    [
      "0.781044674",
      "0.461228792"
    ],
    [
      "0.327923282",
      "0.476699872"
    ],
    [
      "1.359117791",
      "0.541927402"
    ],
    [
      "0.561002042",
      "0.659867251"
    ],
    [
      "1.790639419",
      "0.683109722"
    ],
    [
      "0.938850151",
      "0.7122527"
    ],
    [
      "0.189282384",
      "0.738720289"
    ],
    [
      "1.517960176",
      "0.799393976"
    ],
    [
      "1.221565657",
      "0.804521019"
    ],
    [
      "0.718597931",
      "0.910943999"
    ],
    [
      "0.422361144",
      "0.921887669"
    ],
    [
      "1.851360835",
      "0.973262969"
    ],
    [
      "1.000646281",
      "1.002183932"
    ],
    [
      "0.149004893",
      "1.036571394"
    ],
    [
      "1.578681592",
      "1.089547224"
    ],
    [
      "1.282287073",
      "1.094674267"
    ],
    [
      "0.78275794",
      "1.203183106"
    ],
    [
      "0.486430776",
      "1.211320009"
    ],
    [
      "1.803460456",
      "1.282809894"
    ],
    [
      "1.064398731",
      "1.295673442"
    ],
    [
      "0.213074524",
      "1.326003735"
    ],
    [
      "1.418337936",
      "1.363890215"
    ],
    [
      "0.648294308",
      "1.467371541"
    ],
    [
      "1.657880377",
      "1.541039323"
    ],
    [
      "1.18919149",
      "1.564565176"
    ],
    [
      "0.372456338",
      "1.575950599"
    ],
    [
      "0.899071244",
      "1.625444074"
    ],
    [
      "1.432618021",
      "1.733738229"
    ],
    [
      "0.640269803",
      "1.771178163"
    ],
    [
      "1.096854039",
      "1.846256128"
    ]
  ],
  "radius": "0.148219429"
}
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": 36
}

起步布局答案

{
  "centers": [
    [
      "1.119019053",
      "0.156575667"
    ],
    [
      "0.822758781",
      "0.166863943"
    ],
    [
      "1.400863773",
      "0.2484427"
    ],
    [
      "0.547965913",
      "0.278061412"
    ],
    [
      "1.63415599",
      "0.431338131"
    ],
    [
      "1.077273072",
      "0.45006037"
    ],
    [
      "0.781044674",
      "0.461228792"
    ],
    [
      "0.327923282",
      "0.476699872"
    ],
    [
      "1.359117791",
      "0.541927402"
    ],
    [
      "0.561002042",
      "0.659867251"
    ],
    [
      "1.790639419",
      "0.683109722"
    ],
    [
      "0.938850151",
      "0.7122527"
    ],
    [
      "0.189282384",
      "0.738720289"
    ],
    [
      "1.517960176",
      "0.799393976"
    ],
    [
      "1.221565657",
      "0.804521019"
    ],
    [
      "0.718597931",
      "0.910943999"
    ],
    [
      "0.422361144",
      "0.921887669"
    ],
    [
      "1.851360835",
      "0.973262969"
    ],
    [
      "1.000646281",
      "1.002183932"
    ],
    [
      "0.149004893",
      "1.036571394"
    ],
    [
      "1.578681592",
      "1.089547224"
    ],
    [
      "1.282287073",
      "1.094674267"
    ],
    [
      "0.78275794",
      "1.203183106"
    ],
    [
      "0.486430776",
      "1.211320009"
    ],
    [
      "1.803460456",
      "1.282809894"
    ],
    [
      "1.064398731",
      "1.295673442"
    ],
    [
      "0.213074524",
      "1.326003735"
    ],
    [
      "1.418337936",
      "1.363890215"
    ],
    [
      "0.648294308",
      "1.467371541"
    ],
    [
      "1.657880377",
      "1.541039323"
    ],
    [
      "1.18919149",
      "1.564565176"
    ],
    [
      "0.372456338",
      "1.575950599"
    ],
    [
      "0.899071244",
      "1.625444074"
    ],
    [
      "1.432618021",
      "1.733738229"
    ],
    [
      "0.640269803",
      "1.771178163"
    ],
    [
      "1.096854039",
      "1.846256128"
    ]
  ],
  "radius": "0.148219429"
}

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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