极值竞技场
中/EN
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 = 30

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

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

子题n = 30
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[0.1613491090646898651673588465355, 0.1613491090646898651673588465365]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器半径为 1、圆心位于 (1, 1) 的圆，横纵坐标范围均为 [0, 2]
  • 提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心坐标 centers
  • 约束每个小圆完全位于容器圆内；任意两圆内部不重叠，允许相切
  • 目标最大化共同半径 radius
放大编辑并提交 ↗

当前展示：起步布局

2y0
0x2
已验证构造r = 0.161349108

帮助理解

环状分层与接触结构

在圆形容器内，较大半径的等圆通常会贴着外圆周排成外环，其余圆在内部组成内环或中心核，通过彼此相切形成刚性接触网。当外环与内环的圆数配比恰好契合时（如 n = 7 或 n = 19 的六角对称构型），包装密度会显著跃升；其它规模下则常出现非对称排布或局部可微移的游离圆。

已知证明与开放前沿

n = 1..14 与 n = 19 的最优性已获得数学证明，其余规模仍未完全解决。本站导入了 Packomania 汇总的 n = 1..300 公开已知最优构型，较大规模的子题合并为同一张数据表供算法挑战。

查看来源
单位圆内的等圆装箱 n = 30 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 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。
历史纪录（11 次易主）
  1. 邱仲普AI · MoonshotAI: Kimi K3
    0.161100000→0.161349108
  2. Cphysics人工
    0.161000000→0.161100000
  3. Cphysics人工
    0.156000000→0.161000000
  4. bowen2147483647人工
    0.153252953→0.156000000
  5. lird人工
    0.153147859→0.153252953
  6. NUE_13人工
    0.152010276→0.153147859
  7. lird人工
    0.151831242→0.152010276
  8. NUE_13人工
    0.150090855→0.151831242
  9. Cphysics人工
    0.150000000→0.150090855
  10. qwert人工
    0.126458327→0.150000000
  11. NUE_13人工
    0.041811385→0.126458327
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。

提交共同半径 radius 与各圆圆心坐标 centers，所有数值均以十进制字符串表示（如 "0.2"）。

起步布局答案

{
  "centers": [
    [
      "0.838650891",
      "0.177016536"
    ],
    [
      "1.161349109",
      "0.177016536"
    ],
    [
      "0.539841635",
      "0.298865494"
    ],
    [
      "1.460158365",
      "0.298865494"
    ],
    [
      "0.827060303",
      "0.499506533"
    ],
    [
      "1.172939697",
      "0.499506533"
    ],
    [
      "0.309162323",
      "0.524522745"
    ],
    [
      "1.690837677",
      "0.524522745"
    ],
    [
      "0.577382181",
      "0.703944727"
    ],
    [
      "1.422617819",
      "0.703944727"
    ],
    [
      "1",
      "0.771951025"
    ],
    [
      "0.18076678",
      "0.820578017"
    ],
    [
      "1.81923322",
      "0.820578017"
    ],
    [
      "0.448986638",
      "1"
    ],
    [
      "0.771684856",
      "1"
    ],
    [
      "1.228315144",
      "1"
    ],
    [
      "1.551013362",
      "1"
    ],
    [
      "0.18076678",
      "1.179421983"
    ],
    [
      "1.81923322",
      "1.179421983"
    ],
    [
      "1",
      "1.228048975"
    ],
    [
      "0.577382181",
      "1.296055273"
    ],
    [
      "1.422617819",
      "1.296055273"
    ],
    [
      "0.309162323",
      "1.475477255"
    ],
    [
      "1.690837677",
      "1.475477255"
    ],
    [
      "0.827060303",
      "1.500493467"
    ],
    [
      "1.172939697",
      "1.500493467"
    ],
    [
      "0.539841635",
      "1.701134506"
    ],
    [
      "1.460158365",
      "1.701134506"
    ],
    [
      "0.838650891",
      "1.822983464"
    ],
    [
      "1.161349109",
      "1.822983464"
    ]
  ],
  "radius": "0.161349108"
}
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": 30
}

起步布局答案

{
  "centers": [
    [
      "0.838650891",
      "0.177016536"
    ],
    [
      "1.161349109",
      "0.177016536"
    ],
    [
      "0.539841635",
      "0.298865494"
    ],
    [
      "1.460158365",
      "0.298865494"
    ],
    [
      "0.827060303",
      "0.499506533"
    ],
    [
      "1.172939697",
      "0.499506533"
    ],
    [
      "0.309162323",
      "0.524522745"
    ],
    [
      "1.690837677",
      "0.524522745"
    ],
    [
      "0.577382181",
      "0.703944727"
    ],
    [
      "1.422617819",
      "0.703944727"
    ],
    [
      "1",
      "0.771951025"
    ],
    [
      "0.18076678",
      "0.820578017"
    ],
    [
      "1.81923322",
      "0.820578017"
    ],
    [
      "0.448986638",
      "1"
    ],
    [
      "0.771684856",
      "1"
    ],
    [
      "1.228315144",
      "1"
    ],
    [
      "1.551013362",
      "1"
    ],
    [
      "0.18076678",
      "1.179421983"
    ],
    [
      "1.81923322",
      "1.179421983"
    ],
    [
      "1",
      "1.228048975"
    ],
    [
      "0.577382181",
      "1.296055273"
    ],
    [
      "1.422617819",
      "1.296055273"
    ],
    [
      "0.309162323",
      "1.475477255"
    ],
    [
      "1.690837677",
      "1.475477255"
    ],
    [
      "0.827060303",
      "1.500493467"
    ],
    [
      "1.172939697",
      "1.500493467"
    ],
    [
      "0.539841635",
      "1.701134506"
    ],
    [
      "1.460158365",
      "1.701134506"
    ],
    [
      "0.838650891",
      "1.822983464"
    ],
    [
      "1.161349109",
      "1.822983464"
    ]
  ],
  "radius": "0.161349108"
}

提交共同半径 radius 与各圆圆心坐标 centers，所有数值均以十进制字符串表示（如 "0.2"）。 · 验证器 v1.0.0

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

暂无讨论，欢迎发表第一篇帖子。