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 = 153

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 = 153
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[0.0735751977883484093484798432685, 0.0735751977883484093484798432695]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 = 0.073575197

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 = 153 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": [
    [
      "1.033665147",
      "0.074187075"
    ],
    [
      "0.850473233",
      "0.085721782"
    ],
    [
      "1.179829195",
      "0.091196185"
    ],
    [
      "0.707597079",
      "0.120930352"
    ],
    [
      "1.321456307",
      "0.131133637"
    ],
    [
      "0.572098",
      "0.178317097"
    ],
    [
      "1.454973352",
      "0.192991844"
    ],
    [
      "0.94929728",
      "0.194749515"
    ],
    [
      "1.095461329",
      "0.211758625"
    ],
    [
      "0.779255791",
      "0.249453766"
    ],
    [
      "0.447394522",
      "0.256434199"
    ],
    [
      "1.290937988",
      "0.27508457"
    ],
    [
      "1.577011808",
      "0.275210177"
    ],
    [
      "0.643756712",
      "0.306840511"
    ],
    [
      "0.900091871",
      "0.333429242"
    ],
    [
      "1.424455032",
      "0.336942777"
    ],
    [
      "1.160733077",
      "0.343640533"
    ],
    [
      "0.336632808",
      "0.35331083"
    ],
    [
      "1.684492749",
      "0.375714336"
    ],
    [
      "0.49706853",
      "0.394946769"
    ],
    [
      "0.769883499",
      "0.401978631"
    ],
    [
      "1.027779978",
      "0.40670874"
    ],
    [
      "1.544722452",
      "0.42173067"
    ],
    [
      "1.285899159",
      "0.422148669"
    ],
    [
      "0.632056771",
      "0.453525038"
    ],
    [
      "0.900106948",
      "0.480579637"
    ],
    [
      "0.226858798",
      "0.489592716"
    ],
    [
      "1.155498959",
      "0.490697811"
    ],
    [
      "1.415423923",
      "0.491981144"
    ],
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      "0.373883721",
      "0.495668156"
    ],
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      "1.711522287",
      "0.52036095"
    ],
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      "0.513832382",
      "0.541139149"
    ],
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      "0.376704702",
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    ],
    [
      "0.633587621",
      "0.764717159"
    ],
    [
      "0.888454401",
      "0.774649589"
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    [
      "1.143032523",
      "0.784733772"
    ],
    [
      "1.397899303",
      "0.794666202"
    ],
    [
      "1.652577947",
      "0.80459118"
    ],
    [
      "0.092440898",
      "0.813990884"
    ],
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      "1.90793028",
      "0.815811181"
    ],
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      "0.49999827",
      "0.835107811"
    ],
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      "0.758061849",
      "0.84319904"
    ],
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      "1.012828107",
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      "0.214934382",
      "0.895529742"
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      "0.901195262"
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      "0.625928814",
      "0.911684181"
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      "1.088736317"
    ],
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      "1.640929433",
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    ],
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      "0.080179325",
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    ],
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      "1.919885201",
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    ],
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    ],
    [
      "0.746601471",
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    ],
    [
      "1.00136773",
      "1.147368242"
    ],
    [
      "1.256046373",
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    ],
    [
      "1.510725017",
      "1.167218198"
    ],
    [
      "1.766852768",
      "1.17479639"
    ],
    [
      "0.203423226",
      "1.189079338"
    ],
    [
      "0.359273547",
      "1.195353282"
    ],
    [
      "0.616397055",
      "1.20583351"
    ],
    [
      "0.871075699",
      "1.215758488"
    ],
    [
      "1.125841957",
      "1.225850123"
    ],
    [
      "1.380520601",
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    ],
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      "1.635199245",
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      "1.890969792",
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      "0.740871283",
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      "0.995637541",
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    [
      "1.250316185",
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      "1.745423984"
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      "1.767349424"
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      "1.782324457"
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    [
      "1.030694892",
      "1.795312328"
    ],
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      "1.80093659"
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    [
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  ],
  "radius": "0.073575197"
}
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": 153
}

起步布局答案

{
  "centers": [
    [
      "1.033665147",
      "0.074187075"
    ],
    [
      "0.850473233",
      "0.085721782"
    ],
    [
      "1.179829195",
      "0.091196185"
    ],
    [
      "0.707597079",
      "0.120930352"
    ],
    [
      "1.321456307",
      "0.131133637"
    ],
    [
      "0.572098",
      "0.178317097"
    ],
    [
      "1.454973352",
      "0.192991844"
    ],
    [
      "0.94929728",
      "0.194749515"
    ],
    [
      "1.095461329",
      "0.211758625"
    ],
    [
      "0.779255791",
      "0.249453766"
    ],
    [
      "0.447394522",
      "0.256434199"
    ],
    [
      "1.290937988",
      "0.27508457"
    ],
    [
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      "0.275210177"
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      "0.306840511"
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      "0.336942777"
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      "1.088736317"
    ],
    [
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  "radius": "0.073575197"
}

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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还没有帖子。第一个聊聊这道题的思路?