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 = 300

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 = 300
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[0.0531542991468511875009930257855, 0.0531542991468511875009930257865]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 = 0.053154298

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 = 300 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": [
    [
      "1.729110383",
      "1.165261157"
    ],
    [
      "1.186850199",
      "1.810863904"
    ],
    [
      "0.081781788",
      "1.231067299"
    ],
    [
      "0.658859039",
      "0.869813175"
    ],
    [
      "0.524721443",
      "0.704849004"
    ],
    [
      "0.38185828",
      "0.82101557"
    ],
    [
      "0.787659304",
      "1.922728676"
    ],
    [
      "0.982482547",
      "0.538155718"
    ],
    [
      "1.422792517",
      "0.933723905"
    ],
    [
      "1.431518083",
      "0.652593169"
    ],
    [
      "0.842614299",
      "1.679520904"
    ],
    [
      "1.275063563",
      "0.300693147"
    ],
    [
      "1.405341383",
      "1.495985378"
    ],
    [
      "0.579440727",
      "1.736491519"
    ],
    [
      "1.317826536",
      "0.950566145"
    ],
    [
      "0.160642611",
      "0.561826571"
    ],
    [
      "1.481491871",
      "0.27348232"
    ],
    [
      "0.839619384",
      "0.654322284"
    ],
    [
      "1.087448527",
      "0.521313478"
    ],
    [
      "0.688030654",
      "1.051619586"
    ],
    [
      "1.07872296",
      "0.802444215"
    ],
    [
      "1.456104273",
      "0.170249632"
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    [
      "1.402346468",
      "0.470786758"
    ],
    [
      "1.291816179",
      "1.794021664"
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      "1.160986858"
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      "1.548241214"
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      "0.066107056"
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      "1.804537444",
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  "radius": "0.053154298"
}
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": 300
}

起步布局答案

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  "radius": "0.053154298"
}

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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