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 = 181

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 = 181
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[0.0676552009080910238327781695325, 0.0676552009080910238327781695335]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 = 0.067655200

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 = 181 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": [
    [
      "0.980166815",
      "0.06794792"
    ],
    [
      "1.148076003",
      "0.079489098"
    ],
    [
      "0.842936805",
      "0.080979882"
    ],
    [
      "0.71156589",
      "0.113392652"
    ],
    [
      "1.308515049",
      "0.120178831"
    ],
    [
      "0.58627011",
      "0.164479562"
    ],
    [
      "1.439155098",
      "0.177593533"
    ],
    [
      "1.057047213",
      "0.179602354"
    ],
    [
      "0.922188753",
      "0.190652274"
    ],
    [
      "1.201979067",
      "0.203599389"
    ],
    [
      "0.790817839",
      "0.223065044"
    ],
    [
      "0.469688503",
      "0.233164594"
    ],
    [
      "1.327491381",
      "0.254151977"
    ],
    [
      "1.576700534",
      "0.267413269"
    ],
    [
      "0.620606941",
      "0.295360756"
    ],
    [
      "1.110950277",
      "0.303712646"
    ],
    [
      "1.449315122",
      "0.313040692"
    ],
    [
      "0.364276564",
      "0.318001072"
    ],
    [
      "0.877625608",
      "0.32685963"
    ],
    [
      "0.744332975",
      "0.35014007"
    ],
    [
      "1.236469099",
      "0.354346755"
    ],
    [
      "0.504025334",
      "0.364045788"
    ],
    [
      "1.000957865",
      "0.382519842"
    ],
    [
      "1.627390334",
      "0.39287023"
    ],
    [
      "1.35841414",
      "0.41327001"
    ],
    [
      "0.27225453",
      "0.417202131"
    ],
    [
      "0.63502968",
      "0.429900302"
    ],
    [
      "1.125384811",
      "0.438250927"
    ],
    [
      "1.751116368",
      "0.447649545"
    ],
    [
      "0.402580785",
      "0.453588561"
    ],
    [
      "1.486961513",
      "0.462051642"
    ],
    [
      "0.892355482",
      "0.463231843"
    ],
    [
      "0.758755714",
      "0.484679616"
    ],
    [
      "1.249110845",
      "0.493030242"
    ],
    [
      "0.525726385",
      "0.509660533"
    ],
    [
      "1.016081516",
      "0.518011158"
    ],
    [
      "1.608714207",
      "0.526885554"
    ],
    [
      "1.372836879",
      "0.547809556"
    ],
    [
      "0.252787174",
      "0.551104808"
    ],
    [
      "0.649452419",
      "0.564439848"
    ],
    [
      "1.13980755",
      "0.572790473"
    ],
    [
      "1.732440241",
      "0.581664869"
    ],
    [
      "0.408595811",
      "0.58882087"
    ],
    [
      "0.90677822",
      "0.597771389"
    ],
    [
      "1.496562913",
      "0.602588871"
    ],
    [
      "0.773178453",
      "0.619219162"
    ],
    [
      "1.263533584",
      "0.627569788"
    ],
    [
      "0.143709397",
      "0.631173175"
    ],
    [
      "1.857359693",
      "0.633665207"
    ],
    [
      "0.540149124",
      "0.644200079"
    ],
    [
      "1.030504255",
      "0.652550704"
    ],
    [
      "1.623136946",
      "0.6614251"
    ],
    [
      "0.304222857",
      "0.676257816"
    ],
    [
      "1.387259618",
      "0.682349102"
    ],
    [
      "0.663875158",
      "0.698979394"
    ],
    [
      "1.154230289",
      "0.707330019"
    ],
    [
      "1.74686298",
      "0.716204415"
    ],
    [
      "0.430845828",
      "0.72396031"
    ],
    [
      "0.921200959",
      "0.732310935"
    ],
    [
      "1.510985652",
      "0.737128417"
    ],
    [
      "0.787601192",
      "0.753758708"
    ],
    [
      "0.194704523",
      "0.756506339"
    ],
    [
      "1.901356332",
      "0.761623015"
    ],
    [
      "1.277956323",
      "0.762109334"
    ],
    [
      "0.554571863",
      "0.778739625"
    ],
    [
      "1.044926993",
      "0.78709025"
    ],
    [
      "1.637559684",
      "0.795964646"
    ],
    [
      "0.318645596",
      "0.810797362"
    ],
    [
      "1.401682357",
      "0.816888648"
    ],
    [
      "0.082808808",
      "0.832586914"
    ],
    [
      "0.678297897",
      "0.83351894"
    ],
    [
      "1.168653028",
      "0.841869565"
    ],
    [
      "1.793437429",
      "0.843246634"
    ],
    [
      "0.445268567",
      "0.858499856"
    ],
    [
      "0.935623698",
      "0.866850481"
    ],
    [
      "1.525408391",
      "0.871667963"
    ],
    [
      "0.802023931",
      "0.888298254"
    ],
    [
      "0.209127261",
      "0.891045885"
    ],
    [
      "1.926385447",
      "0.89643419"
    ],
    [
      "1.292379062",
      "0.89664888"
    ],
    [
      "0.568994601",
      "0.913279171"
    ],
    [
      "1.059349732",
      "0.921629796"
    ],
    [
      "1.684134134",
      "0.923006865"
    ],
    [
      "0.333068335",
      "0.945336908"
    ],
    [
      "1.416105096",
      "0.951428194"
    ],
    [
      "0.068235498",
      "0.967110233"
    ],
    [
      "0.692720636",
      "0.968058486"
    ],
    [
      "1.183075766",
      "0.976409111"
    ],
    [
      "1.809932599",
      "0.977547841"
    ],
    [
      "0.459691306",
      "0.993039402"
    ],
    [
      "0.950046437",
      "1.001390027"
    ],
    [
      "1.53983113",
      "1.006207509"
    ],
    [
      "0.81644667",
      "1.0228378"
    ],
    [
      "0.190504575",
      "1.025068646"
    ],
    [
      "1.306801801",
      "1.031188426"
    ],
    [
      "1.931621638",
      "1.036714407"
    ],
    [
      "0.58341734",
      "1.047818717"
    ],
    [
      "1.073772471",
      "1.056169342"
    ],
    [
      "1.665249404",
      "1.056992954"
    ],
    [
      "0.316323064",
      "1.07960716"
    ],
    [
      "1.430527835",
      "1.08596774"
    ],
    [
      "0.707143374",
      "1.102598032"
    ],
    [
      "0.073920479",
      "1.106089634"
    ],
    [
      "1.197498505",
      "1.110948657"
    ],
    [
      "1.789200727",
      "1.111260572"
    ],
    [
      "0.474114045",
      "1.127578948"
    ],
    [
      "0.964469176",
      "1.135929573"
    ],
    [
      "1.554253869",
      "1.140747055"
    ],
    [
      "0.830869408",
      "1.157377346"
    ],
    [
      "0.207019769",
      "1.159367392"
    ],
    [
      "1.321224539",
      "1.165727972"
    ],
    [
      "1.916496229",
      "1.171176769"
    ],
    [
      "0.597840079",
      "1.182358263"
    ],
    [
      "1.08819521",
      "1.190708888"
    ],
    [
      "1.679672143",
      "1.1915325"
    ],
    [
      "0.364013315",
      "1.206234743"
    ],
    [
      "1.444950574",
      "1.220507286"
    ],
    [
      "0.721566113",
      "1.237137578"
    ],
    [
      "0.09946871",
      "1.241475093"
    ],
    [
      "1.211921244",
      "1.245488203"
    ],
    [
      "1.803623466",
      "1.245800118"
    ],
    [
      "0.488536784",
      "1.262118494"
    ],
    [
      "0.978891915",
      "1.270469119"
    ],
    [
      "1.568676608",
      "1.275286601"
    ],
    [
      "0.25471002",
      "1.285994975"
    ],
    [
      "0.845292147",
      "1.291916892"
    ],
    [
      "1.335647278",
      "1.300267518"
    ],
    [
      "0.612262818",
      "1.316897809"
    ],
    [
      "1.102617949",
      "1.325248434"
    ],
    [
      "1.694094882",
      "1.326072046"
    ],
    [
      "0.378436054",
      "1.340774289"
    ],
    [
      "1.459373312",
      "1.355046832"
    ],
    [
      "0.143905101",
      "1.369280853"
    ],
    [
      "1.855439725",
      "1.370796038"
    ],
    [
      "0.735988852",
      "1.371677124"
    ],
    [
      "1.226343983",
      "1.380027749"
    ],
    [
      "0.502959523",
      "1.39665804"
    ],
    [
      "0.993314654",
      "1.405008666"
    ],
    [
      "1.588765157",
      "1.411033057"
    ],
    [
      "0.269132759",
      "1.420534521"
    ],
    [
      "0.859714886",
      "1.426456438"
    ],
    [
      "1.350070017",
      "1.434807064"
    ],
    [
      "1.746370778",
      "1.450876434"
    ],
    [
      "0.626685557",
      "1.451437355"
    ],
    [
      "1.117040688",
      "1.45978798"
    ],
    [
      "0.392858793",
      "1.475313836"
    ],
    [
      "1.473796051",
      "1.489586378"
    ],
    [
      "0.750411591",
      "1.50621667"
    ],
    [
      "1.240766722",
      "1.514567295"
    ],
    [
      "0.514667477",
      "1.536880422"
    ],
    [
      "1.007737393",
      "1.539548212"
    ],
    [
      "1.596202743",
      "1.547253584"
    ],
    [
      "0.250511638",
      "1.554557499"
    ],
    [
      "0.874137625",
      "1.560995984"
    ],
    [
      "1.365520113",
      "1.570735784"
    ],
    [
      "1.726234128",
      "1.584680097"
    ],
    [
      "0.641108296",
      "1.585976901"
    ],
    [
      "1.131463427",
      "1.594327526"
    ],
    [
      "0.374237672",
      "1.609336814"
    ],
    [
      "1.494995396",
      "1.637064371"
    ],
    [
      "0.765267246",
      "1.644754264"
    ],
    [
      "1.255214605",
      "1.649104143"
    ],
    [
      "1.028534295",
      "1.682159745"
    ],
    [
      "1.633955564",
      "1.683642572"
    ],
    [
      "0.553041408",
      "1.68870532"
    ],
    [
      "0.893821891",
      "1.694866947"
    ],
    [
      "1.378236349",
      "1.705447336"
    ],
    [
      "0.425707344",
      "1.734475847"
    ],
    [
      "0.674928657",
      "1.747462476"
    ],
    [
      "1.528324332",
      "1.768205848"
    ],
    [
      "1.208909112",
      "1.776244641"
    ],
    [
      "0.801918943",
      "1.794178341"
    ],
    [
      "1.077454719",
      "1.808317175"
    ],
    [
      "0.942742315",
      "1.821024376"
    ],
    [
      "0.563099842",
      "1.823641352"
    ],
    [
      "1.411565285",
      "1.836588812"
    ],
    [
      "0.695529506",
      "1.881200881"
    ],
    [
      "1.286137657",
      "1.887351151"
    ],
    [
      "1.154683264",
      "1.919423685"
    ],
    [
      "0.850839367",
      "1.92033577"
    ],
    [
      "1.01997086",
      "1.932130886"
    ]
  ],
  "radius": "0.0676552"
}
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": 181
}

起步布局答案

{
  "centers": [
    [
      "0.980166815",
      "0.06794792"
    ],
    [
      "1.148076003",
      "0.079489098"
    ],
    [
      "0.842936805",
      "0.080979882"
    ],
    [
      "0.71156589",
      "0.113392652"
    ],
    [
      "1.308515049",
      "0.120178831"
    ],
    [
      "0.58627011",
      "0.164479562"
    ],
    [
      "1.439155098",
      "0.177593533"
    ],
    [
      "1.057047213",
      "0.179602354"
    ],
    [
      "0.922188753",
      "0.190652274"
    ],
    [
      "1.201979067",
      "0.203599389"
    ],
    [
      "0.790817839",
      "0.223065044"
    ],
    [
      "0.469688503",
      "0.233164594"
    ],
    [
      "1.327491381",
      "0.254151977"
    ],
    [
      "1.576700534",
      "0.267413269"
    ],
    [
      "0.620606941",
      "0.295360756"
    ],
    [
      "1.110950277",
      "0.303712646"
    ],
    [
      "1.449315122",
      "0.313040692"
    ],
    [
      "0.364276564",
      "0.318001072"
    ],
    [
      "0.877625608",
      "0.32685963"
    ],
    [
      "0.744332975",
      "0.35014007"
    ],
    [
      "1.236469099",
      "0.354346755"
    ],
    [
      "0.504025334",
      "0.364045788"
    ],
    [
      "1.000957865",
      "0.382519842"
    ],
    [
      "1.627390334",
      "0.39287023"
    ],
    [
      "1.35841414",
      "0.41327001"
    ],
    [
      "0.27225453",
      "0.417202131"
    ],
    [
      "0.63502968",
      "0.429900302"
    ],
    [
      "1.125384811",
      "0.438250927"
    ],
    [
      "1.751116368",
      "0.447649545"
    ],
    [
      "0.402580785",
      "0.453588561"
    ],
    [
      "1.486961513",
      "0.462051642"
    ],
    [
      "0.892355482",
      "0.463231843"
    ],
    [
      "0.758755714",
      "0.484679616"
    ],
    [
      "1.249110845",
      "0.493030242"
    ],
    [
      "0.525726385",
      "0.509660533"
    ],
    [
      "1.016081516",
      "0.518011158"
    ],
    [
      "1.608714207",
      "0.526885554"
    ],
    [
      "1.372836879",
      "0.547809556"
    ],
    [
      "0.252787174",
      "0.551104808"
    ],
    [
      "0.649452419",
      "0.564439848"
    ],
    [
      "1.13980755",
      "0.572790473"
    ],
    [
      "1.732440241",
      "0.581664869"
    ],
    [
      "0.408595811",
      "0.58882087"
    ],
    [
      "0.90677822",
      "0.597771389"
    ],
    [
      "1.496562913",
      "0.602588871"
    ],
    [
      "0.773178453",
      "0.619219162"
    ],
    [
      "1.263533584",
      "0.627569788"
    ],
    [
      "0.143709397",
      "0.631173175"
    ],
    [
      "1.857359693",
      "0.633665207"
    ],
    [
      "0.540149124",
      "0.644200079"
    ],
    [
      "1.030504255",
      "0.652550704"
    ],
    [
      "1.623136946",
      "0.6614251"
    ],
    [
      "0.304222857",
      "0.676257816"
    ],
    [
      "1.387259618",
      "0.682349102"
    ],
    [
      "0.663875158",
      "0.698979394"
    ],
    [
      "1.154230289",
      "0.707330019"
    ],
    [
      "1.74686298",
      "0.716204415"
    ],
    [
      "0.430845828",
      "0.72396031"
    ],
    [
      "0.921200959",
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  "radius": "0.0676552"
}

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?