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 = 290

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 = 290
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[0.0539123520405, 0.0539123520415]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 = 0.053912351

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 = 290 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": [
    [
      "1.06414506",
      "0.056089385"
    ],
    [
      "0.923294923",
      "0.057026952"
    ],
    [
      "1.174633402",
      "0.071885748"
    ],
    [
      "0.814435094",
      "0.072543051"
    ],
    [
      "1.284211981",
      "0.097611288"
    ],
    [
      "0.70494574",
      "0.101097991"
    ],
    [
      "0.994263463",
      "0.138203508"
    ],
    [
      "0.60458143",
      "0.140508294"
    ],
    [
      "1.102649811",
      "0.156804582"
    ],
    [
      "0.888478233",
      "0.159075788"
    ],
    [
      "1.455172679",
      "0.170602827"
    ],
    [
      "1.208957014",
      "0.174830824"
    ],
    [
      "0.782063677",
      "0.17645707"
    ],
    [
      "1.34823098",
      "0.184373806"
    ],
    [
      "0.471513449",
      "0.215281068"
    ],
    [
      "0.679855405",
      "0.217709301"
    ],
    [
      "1.546588577",
      "0.227780624"
    ],
    [
      "1.032768213",
      "0.238918705"
    ],
    [
      "0.575360384",
      "0.244297973"
    ],
    [
      "1.140192229",
      "0.257882442"
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    [
      "0.872060868",
      "0.265643311"
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    [
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    [
      "0.765646311",
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      "1.360080669",
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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": 290
}

起步布局答案

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  "radius": "0.053912351"
}

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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