P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 173

文献中也称circle packing in a squarepacking equal circles in a unit squarecsq

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆,使共同半径尽可能大。

子题n = 173
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[0.0392515334535, 0.0392515334545]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器单位正方形:左下角是原点 (0, 0),右上角是 (1, 1)
  • 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
放大来摆,然后提交 ↗

当前展示:起步布局

1y0
0x1
已验证构造r = 0.039251533

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 = 1–30 已证明最优,适合观察经典接触结构;n = 31–300 接入 Packomania 的公开前沿,除 n = 36 外最优性仍未知。大规模部分集中成一张表,供算法批量挑战。

查看来源
单位正方形内的等圆装箱 n = 173 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

容器是边长 1 的正方形,左下角是原点 (0, 0),右上角是 (1, 1)。坐标和长度用同一个单位,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 "0.25"。

起步布局答案

{
  "centers": [
    [
      "0.144793265",
      "0.039251533"
    ],
    [
      "0.223296331",
      "0.039251533"
    ],
    [
      "0.301799399",
      "0.039251533"
    ],
    [
      "0.380302466",
      "0.039251533"
    ],
    [
      "0.45972774",
      "0.039251533"
    ],
    [
      "0.54027226",
      "0.039251533"
    ],
    [
      "0.619697534",
      "0.039251533"
    ],
    [
      "0.698200601",
      "0.039251533"
    ],
    [
      "0.776703669",
      "0.039251533"
    ],
    [
      "0.855206735",
      "0.039251533"
    ],
    [
      "0.047100106",
      "0.047100106"
    ],
    [
      "0.952899894",
      "0.047100106"
    ],
    [
      "0.5",
      "0.10663756"
    ],
    [
      "0.420015103",
      "0.106968872"
    ],
    [
      "0.579984897",
      "0.106968872"
    ],
    [
      "0.262547865",
      "0.107237184"
    ],
    [
      "0.737452135",
      "0.107237184"
    ],
    [
      "0.105541731",
      "0.107237184"
    ],
    [
      "0.184044798",
      "0.107237184"
    ],
    [
      "0.341050932",
      "0.107237184"
    ],
    [
      "0.658949068",
      "0.107237184"
    ],
    [
      "0.815955202",
      "0.107237184"
    ],
    [
      "0.894458269",
      "0.107237184"
    ],
    [
      "0.039251533",
      "0.149288833"
    ],
    [
      "0.960748467",
      "0.149288833"
    ],
    [
      "0.460287363",
      "0.174354899"
    ],
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      "0.539712637",
      "0.174354899"
    ],
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      "0.380763569",
      "0.174954523"
    ],
    [
      "0.619236431",
      "0.174954523"
    ],
    [
      "0.223296332",
      "0.175222834"
    ],
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      "0.301799399",
      "0.175222834"
    ],
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      "0.698200601",
      "0.175222834"
    ],
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      "0.776703668",
      "0.175222834"
    ],
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      "0.185987732"
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      "0.872889568",
      "0.185987732"
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      "0.039251533",
      "0.2277919"
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      "0.960748467",
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    ],
    [
      "0.5",
      "0.242072238"
    ],
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      "0.42103583",
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      "0.460748467",
      "0.310057888"
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      "0.539251533",
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    ],
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      "0.381784296",
      "0.310326199"
    ],
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      "0.618215704",
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      "0.815955202",
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    ],
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      "0.696257667",
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    ],
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      "0.067875129",
      "0.614691812"
    ],
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      "0.932124871",
      "0.614691812"
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  ],
  "radius": "0.039251533"
}
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": 173
}

起步布局答案

{
  "centers": [
    [
      "0.144793265",
      "0.039251533"
    ],
    [
      "0.223296331",
      "0.039251533"
    ],
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      "0.039251533"
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      "0.54027226",
      "0.039251533"
    ],
    [
      "0.619697534",
      "0.039251533"
    ],
    [
      "0.698200601",
      "0.039251533"
    ],
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      "0.039251533"
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      "0.047100106"
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    [
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    [
      "0.5",
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}

提交 radius 与 centers。每个数写成十进制字符串,例如 "0.25"。 · 验证器 v1.0.0

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