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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 = 75

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 = 75
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[0.1033909156664448825435827300995, 0.1033909156664448825435827301005]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 = 0.103390915

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 = 75 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": [
    [
      "1.103388776",
      "0.109371789"
    ],
    [
      "0.896606945",
      "0.109372285"
    ],
    [
      "1.304671504",
      "0.156742551"
    ],
    [
      "0.695324444",
      "0.156744015"
    ],
    [
      "1.489749184",
      "0.24896499"
    ],
    [
      "0.510247207",
      "0.248967343"
    ],
    [
      "1.114036946",
      "0.315879276"
    ],
    [
      "0.8859561",
      "0.315879634"
    ],
    [
      "1.315319674",
      "0.363250038"
    ],
    [
      "0.6846736",
      "0.363251364"
    ],
    [
      "1.648777795",
      "0.381133922"
    ],
    [
      "0.351219231",
      "0.381137039"
    ],
    [
      "0.999996795",
      "0.488371506"
    ],
    [
      "1.475515909",
      "0.494001299"
    ],
    [
      "0.5244801",
      "0.494005976"
    ],
    [
      "1.201279523",
      "0.535742269"
    ],
    [
      "0.798714294",
      "0.535743236"
    ],
    [
      "1.773298829",
      "0.546219469"
    ],
    [
      "0.22669899",
      "0.546223184"
    ],
    [
      "1.600036944",
      "0.659086845"
    ],
    [
      "0.399959859",
      "0.659092122"
    ],
    [
      "1.361475758",
      "0.666493529"
    ],
    [
      "0.638520795",
      "0.666497848"
    ],
    [
      "1.103382186",
      "0.717881878"
    ],
    [
      "0.896600355",
      "0.717888905"
    ],
    [
      "1.856689189",
      "0.735440963"
    ],
    [
      "0.14330954",
      "0.735445079"
    ],
    [
      "1.47654062",
      "0.840041718"
    ],
    [
      "0.529681995",
      "0.843697836"
    ],
    [
      "1.683427303",
      "0.848308339"
    ],
    [
      "0.316570409",
      "0.848314016"
    ],
    [
      "1.263578422",
      "0.848633139"
    ],
    [
      "0.736406856",
      "0.848643517"
    ],
    [
      "0.999997356",
      "0.896963711"
    ],
    [
      "1.894513457",
      "0.938733979"
    ],
    [
      "0.105486249",
      "0.938738277"
    ],
    [
      "1.573716163",
      "1.02358581"
    ],
    [
      "0.426281519",
      "1.023591505"
    ],
    [
      "1.366975423",
      "1.027707944"
    ],
    [
      "1.160193591",
      "1.027714971"
    ],
    [
      "0.839803857",
      "1.027718323"
    ],
    [
      "0.633022026",
      "1.02772535"
    ],
    [
      "1.884759815",
      "1.145285649"
    ],
    [
      "0.115240883",
      "1.1452899"
    ],
    [
      "1.000000092",
      "1.158469583"
    ],
    [
      "1.677113164",
      "1.202660615"
    ],
    [
      "0.322896689",
      "1.202673338"
    ],
    [
      "1.470372424",
      "1.206782749"
    ],
    [
      "1.263590593",
      "1.206789776"
    ],
    [
      "0.736419027",
      "1.206800155"
    ],
    [
      "0.529637196",
      "1.206807182"
    ],
    [
      "1.103397093",
      "1.337544388"
    ],
    [
      "0.896615262",
      "1.337551415"
    ],
    [
      "1.827947045",
      "1.344109779"
    ],
    [
      "0.172054608",
      "1.344113757"
    ],
    [
      "1.366987594",
      "1.385864582"
    ],
    [
      "0.426223012",
      "1.385879072"
    ],
    [
      "0.633034197",
      "1.385881987"
    ],
    [
      "1.57377069",
      "1.385885601"
    ],
    [
      "1.206773692",
      "1.516630972"
    ],
    [
      "0.793230432",
      "1.516633248"
    ],
    [
      "1.000002073",
      "1.518686066"
    ],
    [
      "1.727096939",
      "1.524631196"
    ],
    [
      "0.272905582",
      "1.524634689"
    ],
    [
      "1.400110681",
      "1.589976284"
    ],
    [
      "0.599898402",
      "1.589991627"
    ],
    [
      "1.587573571",
      "1.677248218"
    ],
    [
      "0.412429683",
      "1.677251041"
    ],
    [
      "1.103457784",
      "1.69772696"
    ],
    [
      "0.896675966",
      "1.697801786"
    ],
    [
      "1.411843532",
      "1.796424984"
    ],
    [
      "0.58818868",
      "1.79644164"
    ],
    [
      "1.218439397",
      "1.869593054"
    ],
    [
      "0.781595773",
      "1.869601888"
    ],
    [
      "1.002235595",
      "1.880026339"
    ]
  ],
  "radius": "0.103390915"
}
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": 75
}

起步布局答案

{
  "centers": [
    [
      "1.103388776",
      "0.109371789"
    ],
    [
      "0.896606945",
      "0.109372285"
    ],
    [
      "1.304671504",
      "0.156742551"
    ],
    [
      "0.695324444",
      "0.156744015"
    ],
    [
      "1.489749184",
      "0.24896499"
    ],
    [
      "0.510247207",
      "0.248967343"
    ],
    [
      "1.114036946",
      "0.315879276"
    ],
    [
      "0.8859561",
      "0.315879634"
    ],
    [
      "1.315319674",
      "0.363250038"
    ],
    [
      "0.6846736",
      "0.363251364"
    ],
    [
      "1.648777795",
      "0.381133922"
    ],
    [
      "0.351219231",
      "0.381137039"
    ],
    [
      "0.999996795",
      "0.488371506"
    ],
    [
      "1.475515909",
      "0.494001299"
    ],
    [
      "0.5244801",
      "0.494005976"
    ],
    [
      "1.201279523",
      "0.535742269"
    ],
    [
      "0.798714294",
      "0.535743236"
    ],
    [
      "1.773298829",
      "0.546219469"
    ],
    [
      "0.22669899",
      "0.546223184"
    ],
    [
      "1.600036944",
      "0.659086845"
    ],
    [
      "0.399959859",
      "0.659092122"
    ],
    [
      "1.361475758",
      "0.666493529"
    ],
    [
      "0.638520795",
      "0.666497848"
    ],
    [
      "1.103382186",
      "0.717881878"
    ],
    [
      "0.896600355",
      "0.717888905"
    ],
    [
      "1.856689189",
      "0.735440963"
    ],
    [
      "0.14330954",
      "0.735445079"
    ],
    [
      "1.47654062",
      "0.840041718"
    ],
    [
      "0.529681995",
      "0.843697836"
    ],
    [
      "1.683427303",
      "0.848308339"
    ],
    [
      "0.316570409",
      "0.848314016"
    ],
    [
      "1.263578422",
      "0.848633139"
    ],
    [
      "0.736406856",
      "0.848643517"
    ],
    [
      "0.999997356",
      "0.896963711"
    ],
    [
      "1.894513457",
      "0.938733979"
    ],
    [
      "0.105486249",
      "0.938738277"
    ],
    [
      "1.573716163",
      "1.02358581"
    ],
    [
      "0.426281519",
      "1.023591505"
    ],
    [
      "1.366975423",
      "1.027707944"
    ],
    [
      "1.160193591",
      "1.027714971"
    ],
    [
      "0.839803857",
      "1.027718323"
    ],
    [
      "0.633022026",
      "1.02772535"
    ],
    [
      "1.884759815",
      "1.145285649"
    ],
    [
      "0.115240883",
      "1.1452899"
    ],
    [
      "1.000000092",
      "1.158469583"
    ],
    [
      "1.677113164",
      "1.202660615"
    ],
    [
      "0.322896689",
      "1.202673338"
    ],
    [
      "1.470372424",
      "1.206782749"
    ],
    [
      "1.263590593",
      "1.206789776"
    ],
    [
      "0.736419027",
      "1.206800155"
    ],
    [
      "0.529637196",
      "1.206807182"
    ],
    [
      "1.103397093",
      "1.337544388"
    ],
    [
      "0.896615262",
      "1.337551415"
    ],
    [
      "1.827947045",
      "1.344109779"
    ],
    [
      "0.172054608",
      "1.344113757"
    ],
    [
      "1.366987594",
      "1.385864582"
    ],
    [
      "0.426223012",
      "1.385879072"
    ],
    [
      "0.633034197",
      "1.385881987"
    ],
    [
      "1.57377069",
      "1.385885601"
    ],
    [
      "1.206773692",
      "1.516630972"
    ],
    [
      "0.793230432",
      "1.516633248"
    ],
    [
      "1.000002073",
      "1.518686066"
    ],
    [
      "1.727096939",
      "1.524631196"
    ],
    [
      "0.272905582",
      "1.524634689"
    ],
    [
      "1.400110681",
      "1.589976284"
    ],
    [
      "0.599898402",
      "1.589991627"
    ],
    [
      "1.587573571",
      "1.677248218"
    ],
    [
      "0.412429683",
      "1.677251041"
    ],
    [
      "1.103457784",
      "1.69772696"
    ],
    [
      "0.896675966",
      "1.697801786"
    ],
    [
      "1.411843532",
      "1.796424984"
    ],
    [
      "0.58818868",
      "1.79644164"
    ],
    [
      "1.218439397",
      "1.869593054"
    ],
    [
      "0.781595773",
      "1.869601888"
    ],
    [
      "1.002235595",
      "1.880026339"
    ]
  ],
  "radius": "0.103390915"
}

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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