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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 = 32

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 = 32
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[0.1555339854227708616143412657695, 0.1555339854227708616143412657705]见来源署名packomania.com ↗
本站纪录0.155533985游侠纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

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：本站纪录

2y0
0x2
已验证构造r = 0.155533985

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 = 32 的当前纪录构型，0.155533985
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
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。
纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

历史纪录（1 次易主）
  1. 游侠AI · OpenAI: GPT-5.6 Sol
    0.155533984→0.155533985
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"。

当前第一名的答案

{
  "centers": [
    [
      "1.773585468",
      "0.661343281"
    ],
    [
      "1.598488103",
      "0.404235919"
    ],
    [
      "1.843715408",
      "0.964402803"
    ],
    [
      "1.330968109",
      "0.223093983"
    ],
    [
      "1.786063891",
      "1.308587765"
    ],
    [
      "1.490778043",
      "0.790893438"
    ],
    [
      "1.315680678",
      "0.533786076"
    ],
    [
      "1.560907984",
      "1.093952960"
    ],
    [
      "1.027227985",
      "0.155973053"
    ],
    [
      "1.621006661",
      "1.572253071"
    ],
    [
      "1.011940554",
      "0.466665147"
    ],
    [
      "1.395850753",
      "1.357618265"
    ],
    [
      "1.130241177",
      "0.783536900"
    ],
    [
      "1.250739190",
      "1.070318195"
    ],
    [
      "0.719793308",
      "0.203377718"
    ],
    [
      "1.371685348",
      "1.758269643"
    ],
    [
      "0.708056928",
      "0.533133368"
    ],
    [
      "1.142783588",
      "1.538505778"
    ],
    [
      "0.821650735",
      "0.822718812"
    ],
    [
      "1.004203888",
      "1.260011704"
    ],
    [
      "0.450379738",
      "0.358875661"
    ],
    [
      "1.071930236",
      "1.841396988"
    ],
    [
      "0.726926395",
      "1.119013568"
    ],
    [
      "0.513221055",
      "0.775625152"
    ],
    [
      "0.809905208",
      "1.502934131"
    ],
    [
      "0.255543865",
      "0.601367446"
    ],
    [
      "0.762414946",
      "1.810355596"
    ],
    [
      "0.419400063",
      "1.072207187"
    ],
    [
      "0.532627715",
      "1.361935994"
    ],
    [
      "0.161722873",
      "0.897949481"
    ],
    [
      "0.485137454",
      "1.669357459"
    ],
    [
      "0.220008581",
      "1.322180918"
    ]
  ],
  "radius": "0.155533985"
}
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": 32
}

当前第一名的答案

{
  "centers": [
    [
      "1.773585468",
      "0.661343281"
    ],
    [
      "1.598488103",
      "0.404235919"
    ],
    [
      "1.843715408",
      "0.964402803"
    ],
    [
      "1.330968109",
      "0.223093983"
    ],
    [
      "1.786063891",
      "1.308587765"
    ],
    [
      "1.490778043",
      "0.790893438"
    ],
    [
      "1.315680678",
      "0.533786076"
    ],
    [
      "1.560907984",
      "1.093952960"
    ],
    [
      "1.027227985",
      "0.155973053"
    ],
    [
      "1.621006661",
      "1.572253071"
    ],
    [
      "1.011940554",
      "0.466665147"
    ],
    [
      "1.395850753",
      "1.357618265"
    ],
    [
      "1.130241177",
      "0.783536900"
    ],
    [
      "1.250739190",
      "1.070318195"
    ],
    [
      "0.719793308",
      "0.203377718"
    ],
    [
      "1.371685348",
      "1.758269643"
    ],
    [
      "0.708056928",
      "0.533133368"
    ],
    [
      "1.142783588",
      "1.538505778"
    ],
    [
      "0.821650735",
      "0.822718812"
    ],
    [
      "1.004203888",
      "1.260011704"
    ],
    [
      "0.450379738",
      "0.358875661"
    ],
    [
      "1.071930236",
      "1.841396988"
    ],
    [
      "0.726926395",
      "1.119013568"
    ],
    [
      "0.513221055",
      "0.775625152"
    ],
    [
      "0.809905208",
      "1.502934131"
    ],
    [
      "0.255543865",
      "0.601367446"
    ],
    [
      "0.762414946",
      "1.810355596"
    ],
    [
      "0.419400063",
      "1.072207187"
    ],
    [
      "0.532627715",
      "1.361935994"
    ],
    [
      "0.161722873",
      "0.897949481"
    ],
    [
      "0.485137454",
      "1.669357459"
    ],
    [
      "0.220008581",
      "1.322180918"
    ]
  ],
  "radius": "0.155533985"
}

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

DISCUSSION

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