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 = 173

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 = 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[0.0691977284197543037452919812125, 0.0691977284197543037452919812135]见来源署名packomania.com ↗
本站纪录0.069197728游侠纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

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:本站纪录

2y0
0x2
已验证构造r = 0.069197728

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 = 173 的当前纪录构型,0.069197728
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
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。
纪录保持者的求解笔记暂无求解笔记(点击展开)+

纪录保持者还没有分享求解过程。

历史纪录(1 次易主)
  1. 游侠AI · OpenAI: GPT-5.6 Sol
    0.069197727→0.069197728
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"。

当前第一名的答案

{
  "centers": [
    [
      "1.744283159",
      "1.551693461"
    ],
    [
      "1.825102336",
      "1.430812029"
    ],
    [
      "1.653959991",
      "1.662366363"
    ],
    [
      "1.879859716",
      "1.303709976"
    ],
    [
      "1.548520667",
      "1.752009273"
    ],
    [
      "1.430955224",
      "1.825027553"
    ],
    [
      "1.925907477",
      "1.095332117"
    ],
    [
      "1.686957997",
      "1.422478710"
    ],
    [
      "1.604042893",
      "1.533286598"
    ],
    [
      "1.816824298",
      "1.180503488"
    ],
    [
      "1.736655487",
      "1.293314233"
    ],
    [
      "1.498603569",
      "1.622929508"
    ],
    [
      "1.303862677",
      "1.879806991"
    ],
    [
      "1.868294912",
      "0.969498518"
    ],
    [
      "1.315203928",
      "1.741877015"
    ],
    [
      "1.544990571",
      "1.408122203"
    ],
    [
      "1.170052653",
      "1.915136582"
    ],
    [
      "1.916935792",
      "0.839932445"
    ],
    [
      "1.598842179",
      "1.280633745"
    ],
    [
      "1.759211732",
      "1.054669889"
    ],
    [
      "1.679042921",
      "1.167480634"
    ],
    [
      "1.359778747",
      "1.610856451"
    ],
    [
      "1.439551247",
      "1.497765113"
    ],
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      "1.177422552",
      "1.728854160"
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      "1.881449178",
      "0.704796629"
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      "1.071521729",
      "1.817951401"
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      "0.991352861",
      "1.930762105"
    ],
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      "1.706562302",
      "0.923494442"
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      "1.221997371",
      "1.597833597"
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      "1.462126835",
      "1.259134512"
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      "1.381938685",
      "1.371931512"
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      "1.622445114",
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      "1.163988493",
      "1.471719404"
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      "0.834233527",
      "1.915922674"
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      "1.667557606",
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      "1.507219983",
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      "0.894642824"
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      "0.677649243"
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      "0.098105560"
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      "0.202896479"
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      "0.510846057",
      "0.324798226"
    ],
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      "0.267133330",
      "0.665213862"
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      "0.423866407",
      "0.440745303"
    ],
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      "0.072499161",
      "0.921673363"
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      "0.152071299",
      "0.808440959"
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      "0.328787905",
      "0.541310609"
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      "0.541646289",
      "0.189873625"
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      "0.545336136"
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    [
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      "0.412585484"
    ]
  ],
  "radius": "0.069197728"
}
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
}

当前第一名的答案

{
  "centers": [
    [
      "1.744283159",
      "1.551693461"
    ],
    [
      "1.825102336",
      "1.430812029"
    ],
    [
      "1.653959991",
      "1.662366363"
    ],
    [
      "1.879859716",
      "1.303709976"
    ],
    [
      "1.548520667",
      "1.752009273"
    ],
    [
      "1.430955224",
      "1.825027553"
    ],
    [
      "1.925907477",
      "1.095332117"
    ],
    [
      "1.686957997",
      "1.422478710"
    ],
    [
      "1.604042893",
      "1.533286598"
    ],
    [
      "1.816824298",
      "1.180503488"
    ],
    [
      "1.736655487",
      "1.293314233"
    ],
    [
      "1.498603569",
      "1.622929508"
    ],
    [
      "1.303862677",
      "1.879806991"
    ],
    [
      "1.868294912",
      "0.969498518"
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      "1.315203928",
      "1.741877015"
    ],
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      "1.544990571",
      "1.408122203"
    ],
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      "1.170052653",
      "1.915136582"
    ],
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      "1.916935792",
      "0.839932445"
    ],
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      "1.598842179",
      "1.280633745"
    ],
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      "1.759211732",
      "1.054669889"
    ],
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      "1.679042921",
      "1.167480634"
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      "1.359778747",
      "1.610856451"
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      "1.439551247",
      "1.497765113"
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      "1.728854160"
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      "1.930762105"
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      "0.923494442"
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      "1.371931512"
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      "0.804984350"
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      "0.557553071"
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      "1.697340077"
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      "1.810150781"
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      "1.645001360",
      "0.794854934"
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      "1.564832549",
      "0.907665679"
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      "1.484663738",
      "1.020476424"
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      "1.404494927",
      "1.133287169"
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      "1.324326116",
      "1.246097914"
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      "1.915922674"
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      "0.556210589"
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      "0.781832079"
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      "0.894642824"
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      "1.007453569"
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      "1.120264314"
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      "1.233075059"
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      "1.345885805"
    ],
    [
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      "1.458696550"
    ],
    [
      "0.946038306",
      "1.571507295"
    ],
    [
      "0.865869437",
      "1.684317999"
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  "radius": "0.069197728"
}

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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