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 = 245

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 = 245
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[0.0585662197400308459403117443975, 0.0585662197400308459403117443985]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 = 0.058566219

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 = 245 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": [
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      "1.452869135",
      "0.3470519"
    ],
    [
      "1.563700189",
      "0.3849529"
    ],
    [
      "1.015628458",
      "1.632254953"
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    [
      "1.884210997",
      "0.676995201"
    ],
    [
      "1.255359587",
      "0.093969079"
    ],
    [
      "1.128864261",
      "0.724456421"
    ],
    [
      "0.86017576",
      "0.570033146"
    ],
    [
      "1.720286425",
      "0.543083578"
    ],
    [
      "0.248364588",
      "0.965564862"
    ],
    [
      "0.680687193",
      "0.114372034"
    ],
    [
      "0.933075089",
      "0.173603697"
    ],
    [
      "0.631874864",
      "1.455922085"
    ],
    [
      "1.322265476",
      "1.587379698"
    ],
    [
      "1.320436178",
      "0.6576715"
    ],
    [
      "0.82283701",
      "1.079234458"
    ],
    [
      "1.628292828",
      "1.23260155"
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    [
      "1.918077716",
      "0.791599256"
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    [
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      "1.167928968"
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}
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": 245
}

起步布局答案

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}

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