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 = 252

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 = 252
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[0.0577194050654640124473660883935, 0.0577194050654640124473660883945]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 = 0.057719404

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 = 252 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": [
    [
      "1.111418732",
      "0.506842895"
    ],
    [
      "0.697117599",
      "0.998233022"
    ],
    [
      "0.346133629",
      "0.777459539"
    ],
    [
      "0.28076315",
      "0.966417357"
    ],
    [
      "0.850835192",
      "0.204411544"
    ],
    [
      "1.59173856",
      "1.413516075"
    ],
    [
      "1.569660304",
      "0.771159292"
    ],
    [
      "1.65710904",
      "1.224558256"
    ],
    [
      "0.087761782",
      "1.236038455"
    ],
    [
      "0.915091221",
      "0.544709308"
    ],
    [
      "1.447662425",
      "1.70361988"
    ],
    [
      "0.631531047",
      "0.658183798"
    ],
    [
      "1.863371762",
      "0.622532227"
    ],
    [
      "0.297885077",
      "1.194541143"
    ],
    [
      "0.626425588",
      "1.865063511"
    ],
    [
      "1.1551431",
      "0.733542377"
    ],
    [
      "1.376491899",
      "0.137054475"
    ],
    [
      "1.111850876",
      "1.564856978"
    ],
    [
      "0.746890118",
      "1.906828158"
    ],
    [
      "0.25960084",
      "1.374696675"
    ],
    [
      "0.813615679",
      "0.076336856"
    ],
    [
      "1.767531059",
      "0.728495588"
    ],
    [
      "0.5224362",
      "0.620442134"
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    [
      "0.258916158",
      "0.418031393"
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    [
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      "0.675255415",
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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": 252
}

起步布局答案

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  "radius": "0.057719404"
}

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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