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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 = 88

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 = 88
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[0.0958557927718008374134669219985, 0.0958557927718008374134669219995]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 = 0.095855792

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 = 88 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": [
    [
      "0.975984363",
      "0.096174798"
    ],
    [
      "1.167088103",
      "0.111429061"
    ],
    [
      "0.785960353",
      "0.121556048"
    ],
    [
      "1.350679648",
      "0.166633015"
    ],
    [
      "0.605559456",
      "0.186431684"
    ],
    [
      "1.518504818",
      "0.259304718"
    ],
    [
      "1.020348872",
      "0.282682498"
    ],
    [
      "0.442892401",
      "0.287884931"
    ],
    [
      "1.21141617",
      "0.298386632"
    ],
    [
      "0.830324863",
      "0.308063748"
    ],
    [
      "0.649923965",
      "0.372939385"
    ],
    [
      "1.663018279",
      "0.385277697"
    ],
    [
      "1.394088975",
      "0.405160914"
    ],
    [
      "0.305272616",
      "0.421354504"
    ],
    [
      "1.030638602",
      "0.47640037"
    ],
    [
      "0.488788796",
      "0.476808482"
    ],
    [
      "1.222073685",
      "0.489801755"
    ],
    [
      "0.839729641",
      "0.49954451"
    ],
    [
      "1.777722781",
      "0.538888274"
    ],
    [
      "1.58632827",
      "0.561569409"
    ],
    [
      "0.659328743",
      "0.564420147"
    ],
    [
      "0.198887416",
      "0.580839679"
    ],
    [
      "1.381382449",
      "0.596450946"
    ],
    [
      "0.382403596",
      "0.636293657"
    ],
    [
      "1.119733103",
      "0.651912018"
    ],
    [
      "0.928811883",
      "0.669302238"
    ],
    [
      "1.857461268",
      "0.713230196"
    ],
    [
      "0.552929687",
      "0.723896078"
    ],
    [
      "0.748410985",
      "0.734177874"
    ],
    [
      "1.6644551",
      "0.7368066"
    ],
    [
      "0.128519822",
      "0.759170086"
    ],
    [
      "1.466368687",
      "0.76829585"
    ],
    [
      "1.273634982",
      "0.768849694"
    ],
    [
      "0.312036003",
      "0.814624064"
    ],
    [
      "1.080268128",
      "0.839517583"
    ],
    [
      "0.890047107",
      "0.863377491"
    ],
    [
      "1.781156843",
      "0.889102214"
    ],
    [
      "0.482562094",
      "0.902226485"
    ],
    [
      "0.674077733",
      "0.910892049"
    ],
    [
      "1.591058184",
      "0.913918175"
    ],
    [
      "1.396846114",
      "0.946957385"
    ],
    [
      "0.097333528",
      "0.948328081"
    ],
    [
      "1.208388734",
      "0.982130494"
    ],
    [
      "0.280849708",
      "1.003782058"
    ],
    [
      "1.016500244",
      "1.020313017"
    ],
    [
      "1.903395485",
      "1.036787847"
    ],
    [
      "0.826279223",
      "1.044172925"
    ],
    [
      "1.71163427",
      "1.067763748"
    ],
    [
      "0.451375799",
      "1.09138448"
    ],
    [
      "1.521535611",
      "1.092579709"
    ],
    [
      "0.642891438",
      "1.100050044"
    ],
    [
      "1.33307823",
      "1.127752819"
    ],
    [
      "0.106730651",
      "1.139809218"
    ],
    [
      "1.144620849",
      "1.162925928"
    ],
    [
      "0.290246831",
      "1.195263196"
    ],
    [
      "0.95273236",
      "1.201108451"
    ],
    [
      "1.875330982",
      "1.226434139"
    ],
    [
      "0.766495154",
      "1.2465951"
    ],
    [
      "1.686199463",
      "1.257780596"
    ],
    [
      "0.532534546",
      "1.266212267"
    ],
    [
      "1.457767727",
      "1.273375144"
    ],
    [
      "1.269310346",
      "1.308548253"
    ],
    [
      "0.160921319",
      "1.336784375"
    ],
    [
      "1.080852965",
      "1.343721362"
    ],
    [
      "0.348145011",
      "1.378022964"
    ],
    [
      "0.89461576",
      "1.389208012"
    ],
    [
      "1.807912051",
      "1.405900068"
    ],
    [
      "1.582457224",
      "1.418997468"
    ],
    [
      "0.707049053",
      "1.428857255"
    ],
    [
      "1.393999843",
      "1.454170578"
    ],
    [
      "0.518834383",
      "1.465306813"
    ],
    [
      "1.205542462",
      "1.489343687"
    ],
    [
      "1.019305256",
      "1.534830336"
    ],
    [
      "0.290950612",
      "1.5610042"
    ],
    [
      "1.704169812",
      "1.56711694"
    ],
    [
      "0.831738549",
      "1.57447958"
    ],
    [
      "1.515712431",
      "1.602290049"
    ],
    [
      "0.641493144",
      "1.612643707"
    ],
    [
      "1.32725505",
      "1.637463159"
    ],
    [
      "0.455545319",
      "1.659299237"
    ],
    [
      "1.151988524",
      "1.715148272"
    ],
    [
      "0.960288262",
      "1.717231889"
    ],
    [
      "0.769138064",
      "1.755682536"
    ],
    [
      "1.451944547",
      "1.783085483"
    ],
    [
      "0.583190239",
      "1.802338065"
    ],
    [
      "1.276678021",
      "1.860770597"
    ],
    [
      "0.897266223",
      "1.898288661"
    ],
    [
      "1.088972194",
      "1.899755909"
    ]
  ],
  "radius": "0.095855792"
}
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": 88
}

起步布局答案

{
  "centers": [
    [
      "0.975984363",
      "0.096174798"
    ],
    [
      "1.167088103",
      "0.111429061"
    ],
    [
      "0.785960353",
      "0.121556048"
    ],
    [
      "1.350679648",
      "0.166633015"
    ],
    [
      "0.605559456",
      "0.186431684"
    ],
    [
      "1.518504818",
      "0.259304718"
    ],
    [
      "1.020348872",
      "0.282682498"
    ],
    [
      "0.442892401",
      "0.287884931"
    ],
    [
      "1.21141617",
      "0.298386632"
    ],
    [
      "0.830324863",
      "0.308063748"
    ],
    [
      "0.649923965",
      "0.372939385"
    ],
    [
      "1.663018279",
      "0.385277697"
    ],
    [
      "1.394088975",
      "0.405160914"
    ],
    [
      "0.305272616",
      "0.421354504"
    ],
    [
      "1.030638602",
      "0.47640037"
    ],
    [
      "0.488788796",
      "0.476808482"
    ],
    [
      "1.222073685",
      "0.489801755"
    ],
    [
      "0.839729641",
      "0.49954451"
    ],
    [
      "1.777722781",
      "0.538888274"
    ],
    [
      "1.58632827",
      "0.561569409"
    ],
    [
      "0.659328743",
      "0.564420147"
    ],
    [
      "0.198887416",
      "0.580839679"
    ],
    [
      "1.381382449",
      "0.596450946"
    ],
    [
      "0.382403596",
      "0.636293657"
    ],
    [
      "1.119733103",
      "0.651912018"
    ],
    [
      "0.928811883",
      "0.669302238"
    ],
    [
      "1.857461268",
      "0.713230196"
    ],
    [
      "0.552929687",
      "0.723896078"
    ],
    [
      "0.748410985",
      "0.734177874"
    ],
    [
      "1.6644551",
      "0.7368066"
    ],
    [
      "0.128519822",
      "0.759170086"
    ],
    [
      "1.466368687",
      "0.76829585"
    ],
    [
      "1.273634982",
      "0.768849694"
    ],
    [
      "0.312036003",
      "0.814624064"
    ],
    [
      "1.080268128",
      "0.839517583"
    ],
    [
      "0.890047107",
      "0.863377491"
    ],
    [
      "1.781156843",
      "0.889102214"
    ],
    [
      "0.482562094",
      "0.902226485"
    ],
    [
      "0.674077733",
      "0.910892049"
    ],
    [
      "1.591058184",
      "0.913918175"
    ],
    [
      "1.396846114",
      "0.946957385"
    ],
    [
      "0.097333528",
      "0.948328081"
    ],
    [
      "1.208388734",
      "0.982130494"
    ],
    [
      "0.280849708",
      "1.003782058"
    ],
    [
      "1.016500244",
      "1.020313017"
    ],
    [
      "1.903395485",
      "1.036787847"
    ],
    [
      "0.826279223",
      "1.044172925"
    ],
    [
      "1.71163427",
      "1.067763748"
    ],
    [
      "0.451375799",
      "1.09138448"
    ],
    [
      "1.521535611",
      "1.092579709"
    ],
    [
      "0.642891438",
      "1.100050044"
    ],
    [
      "1.33307823",
      "1.127752819"
    ],
    [
      "0.106730651",
      "1.139809218"
    ],
    [
      "1.144620849",
      "1.162925928"
    ],
    [
      "0.290246831",
      "1.195263196"
    ],
    [
      "0.95273236",
      "1.201108451"
    ],
    [
      "1.875330982",
      "1.226434139"
    ],
    [
      "0.766495154",
      "1.2465951"
    ],
    [
      "1.686199463",
      "1.257780596"
    ],
    [
      "0.532534546",
      "1.266212267"
    ],
    [
      "1.457767727",
      "1.273375144"
    ],
    [
      "1.269310346",
      "1.308548253"
    ],
    [
      "0.160921319",
      "1.336784375"
    ],
    [
      "1.080852965",
      "1.343721362"
    ],
    [
      "0.348145011",
      "1.378022964"
    ],
    [
      "0.89461576",
      "1.389208012"
    ],
    [
      "1.807912051",
      "1.405900068"
    ],
    [
      "1.582457224",
      "1.418997468"
    ],
    [
      "0.707049053",
      "1.428857255"
    ],
    [
      "1.393999843",
      "1.454170578"
    ],
    [
      "0.518834383",
      "1.465306813"
    ],
    [
      "1.205542462",
      "1.489343687"
    ],
    [
      "1.019305256",
      "1.534830336"
    ],
    [
      "0.290950612",
      "1.5610042"
    ],
    [
      "1.704169812",
      "1.56711694"
    ],
    [
      "0.831738549",
      "1.57447958"
    ],
    [
      "1.515712431",
      "1.602290049"
    ],
    [
      "0.641493144",
      "1.612643707"
    ],
    [
      "1.32725505",
      "1.637463159"
    ],
    [
      "0.455545319",
      "1.659299237"
    ],
    [
      "1.151988524",
      "1.715148272"
    ],
    [
      "0.960288262",
      "1.717231889"
    ],
    [
      "0.769138064",
      "1.755682536"
    ],
    [
      "1.451944547",
      "1.783085483"
    ],
    [
      "0.583190239",
      "1.802338065"
    ],
    [
      "1.276678021",
      "1.860770597"
    ],
    [
      "0.897266223",
      "1.898288661"
    ],
    [
      "1.088972194",
      "1.899755909"
    ]
  ],
  "radius": "0.095855792"
}

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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