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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 = 44

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 = 44
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[0.1333682458860055851257759908285, 0.1333682458860055851257759908295]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 = 0.133368245

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 = 44 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": [
    [
      "1.034600707",
      "0.134059247"
    ],
    [
      "0.769612946",
      "0.164552573"
    ],
    [
      "1.32548171",
      "0.196811197"
    ],
    [
      "0.526450161",
      "0.274189317"
    ],
    [
      "1.554329799",
      "0.333840056"
    ],
    [
      "1.133355857",
      "0.381840958"
    ],
    [
      "0.868368096",
      "0.412334284"
    ],
    [
      "0.328147592",
      "0.452583396"
    ],
    [
      "0.625205311",
      "0.521971029"
    ],
    [
      "1.730665242",
      "0.533975429"
    ],
    [
      "1.396333883",
      "0.548748516"
    ],
    [
      "1.148802643",
      "0.64812981"
    ],
    [
      "0.883814883",
      "0.678623137"
    ],
    [
      "0.193490794",
      "0.68283522"
    ],
    [
      "1.572669326",
      "0.748883889"
    ],
    [
      "1.83778347",
      "0.778258137"
    ],
    [
      "0.640652097",
      "0.788259881"
    ],
    [
      "1.325138086",
      "0.848265183"
    ],
    [
      "0.385959912",
      "0.867507889"
    ],
    [
      "1.062284378",
      "0.900444982"
    ],
    [
      "0.134466386",
      "0.956386246"
    ],
    [
      "0.81200231",
      "0.992679695"
    ],
    [
      "1.600422839",
      "1.014172597"
    ],
    [
      "1.865536983",
      "1.043546844"
    ],
    [
      "0.557310125",
      "1.071927704"
    ],
    [
      "1.352891599",
      "1.113553891"
    ],
    [
      "1.017020949",
      "1.163312968"
    ],
    [
      "0.278388435",
      "1.180963181"
    ],
    [
      "0.753025027",
      "1.253156932"
    ],
    [
      "1.544621408",
      "1.298994006"
    ],
    [
      "1.811296642",
      "1.304710283"
    ],
    [
      "0.49821846",
      "1.332036395"
    ],
    [
      "1.215106809",
      "1.341947646"
    ],
    [
      "0.958283326",
      "1.423501837"
    ],
    [
      "0.257470206",
      "1.446878174"
    ],
    [
      "0.678254194",
      "1.528849734"
    ],
    [
      "1.413525496",
      "1.531291699"
    ],
    [
      "1.680200729",
      "1.537007975"
    ],
    [
      "1.156369185",
      "1.602136515"
    ],
    [
      "0.428544845",
      "1.651528667"
    ],
    [
      "0.896028424",
      "1.682871618"
    ],
    [
      "1.351892742",
      "1.791883535"
    ],
    [
      "0.653754445",
      "1.794458692"
    ],
    [
      "1.094114284",
      "1.861506296"
    ]
  ],
  "radius": "0.133368245"
}
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": 44
}

起步布局答案

{
  "centers": [
    [
      "1.034600707",
      "0.134059247"
    ],
    [
      "0.769612946",
      "0.164552573"
    ],
    [
      "1.32548171",
      "0.196811197"
    ],
    [
      "0.526450161",
      "0.274189317"
    ],
    [
      "1.554329799",
      "0.333840056"
    ],
    [
      "1.133355857",
      "0.381840958"
    ],
    [
      "0.868368096",
      "0.412334284"
    ],
    [
      "0.328147592",
      "0.452583396"
    ],
    [
      "0.625205311",
      "0.521971029"
    ],
    [
      "1.730665242",
      "0.533975429"
    ],
    [
      "1.396333883",
      "0.548748516"
    ],
    [
      "1.148802643",
      "0.64812981"
    ],
    [
      "0.883814883",
      "0.678623137"
    ],
    [
      "0.193490794",
      "0.68283522"
    ],
    [
      "1.572669326",
      "0.748883889"
    ],
    [
      "1.83778347",
      "0.778258137"
    ],
    [
      "0.640652097",
      "0.788259881"
    ],
    [
      "1.325138086",
      "0.848265183"
    ],
    [
      "0.385959912",
      "0.867507889"
    ],
    [
      "1.062284378",
      "0.900444982"
    ],
    [
      "0.134466386",
      "0.956386246"
    ],
    [
      "0.81200231",
      "0.992679695"
    ],
    [
      "1.600422839",
      "1.014172597"
    ],
    [
      "1.865536983",
      "1.043546844"
    ],
    [
      "0.557310125",
      "1.071927704"
    ],
    [
      "1.352891599",
      "1.113553891"
    ],
    [
      "1.017020949",
      "1.163312968"
    ],
    [
      "0.278388435",
      "1.180963181"
    ],
    [
      "0.753025027",
      "1.253156932"
    ],
    [
      "1.544621408",
      "1.298994006"
    ],
    [
      "1.811296642",
      "1.304710283"
    ],
    [
      "0.49821846",
      "1.332036395"
    ],
    [
      "1.215106809",
      "1.341947646"
    ],
    [
      "0.958283326",
      "1.423501837"
    ],
    [
      "0.257470206",
      "1.446878174"
    ],
    [
      "0.678254194",
      "1.528849734"
    ],
    [
      "1.413525496",
      "1.531291699"
    ],
    [
      "1.680200729",
      "1.537007975"
    ],
    [
      "1.156369185",
      "1.602136515"
    ],
    [
      "0.428544845",
      "1.651528667"
    ],
    [
      "0.896028424",
      "1.682871618"
    ],
    [
      "1.351892742",
      "1.791883535"
    ],
    [
      "0.653754445",
      "1.794458692"
    ],
    [
      "1.094114284",
      "1.861506296"
    ]
  ],
  "radius": "0.133368245"
}

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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