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 = 190

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 = 190
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[0.0662093739499438218537726064585, 0.0662093739499438218537726064595]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 = 0.066209373

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 = 190 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": [
    [
      "1.096540516",
      "0.071213231"
    ],
    [
      "0.836950375",
      "0.080554649"
    ],
    [
      "1.226947499",
      "0.094207658"
    ],
    [
      "0.967673619",
      "0.101677107"
    ],
    [
      "0.708533474",
      "0.112862921"
    ],
    [
      "1.377762025",
      "0.146032328"
    ],
    [
      "0.560525241",
      "0.176148336"
    ],
    [
      "0.807694385",
      "0.209701124"
    ],
    [
      "1.038739086",
      "0.213410827"
    ],
    [
      "1.505541043",
      "0.214969166"
    ],
    [
      "1.267997433",
      "0.220102963"
    ],
    [
      "0.679277484",
      "0.242009395"
    ],
    [
      "0.447852375",
      "0.24694095"
    ],
    [
      "0.922178059",
      "0.276245944"
    ],
    [
      "1.387026781",
      "0.278126571"
    ],
    [
      "1.151436406",
      "0.282938079"
    ],
    [
      "1.611600786",
      "0.294429919"
    ],
    [
      "0.564935608",
      "0.308797569"
    ],
    [
      "0.346883227",
      "0.332614493"
    ],
    [
      "0.805147392",
      "0.342095374"
    ],
    [
      "1.034875379",
      "0.345773196"
    ],
    [
      "1.264133727",
      "0.352465331"
    ],
    [
      "1.49271131",
      "0.357910687"
    ],
    [
      "0.677734422",
      "0.378160038"
    ],
    [
      "0.46396646",
      "0.394471112"
    ],
    [
      "0.91895713",
      "0.409786352"
    ],
    [
      "1.1475727",
      "0.415300448"
    ],
    [
      "1.376831047",
      "0.421992583"
    ],
    [
      "1.60540863",
      "0.427437939"
    ],
    [
      "1.737827196",
      "0.427657479"
    ],
    [
      "0.313387382",
      "0.460726761"
    ],
    [
      "0.576765275",
      "0.463833582"
    ],
    [
      "0.803430155",
      "0.474502987"
    ],
    [
      "1.03165445",
      "0.479313604"
    ],
    [
      "1.26027002",
      "0.4848277"
    ],
    [
      "1.489528367",
      "0.491519835"
    ],
    [
      "0.196860825",
      "0.523625778"
    ],
    [
      "0.460094547",
      "0.526833241"
    ],
    [
      "0.686887289",
      "0.537371781"
    ],
    [
      "1.811366883",
      "0.5377785"
    ],
    [
      "0.916127475",
      "0.544030239"
    ],
    [
      "1.14435177",
      "0.548840855"
    ],
    [
      "1.372967341",
      "0.554354952"
    ],
    [
      "1.605042886",
      "0.559856181"
    ],
    [
      "0.34353352",
      "0.589668357"
    ],
    [
      "0.570326263",
      "0.600206898"
    ],
    [
      "0.79958461",
      "0.606899033"
    ],
    [
      "1.028824796",
      "0.613557491"
    ],
    [
      "1.257049091",
      "0.618368107"
    ],
    [
      "1.485664661",
      "0.623882204"
    ],
    [
      "1.716959577",
      "0.630633147"
    ],
    [
      "0.227006963",
      "0.652567374"
    ],
    [
      "1.868590437",
      "0.657194536"
    ],
    [
      "0.453765236",
      "0.663042014"
    ],
    [
      "0.683023583",
      "0.66973415"
    ],
    [
      "0.91228193",
      "0.676426285"
    ],
    [
      "1.141522116",
      "0.683084743"
    ],
    [
      "1.369746411",
      "0.687895359"
    ],
    [
      "1.599133544",
      "0.692143008"
    ],
    [
      "0.110565016",
      "0.715622887"
    ],
    [
      "0.337238679",
      "0.725941031"
    ],
    [
      "0.566462556",
      "0.732569266"
    ],
    [
      "0.795720904",
      "0.739261402"
    ],
    [
      "1.024979251",
      "0.745953537"
    ],
    [
      "1.254219436",
      "0.752611995"
    ],
    [
      "1.483142555",
      "0.756276931"
    ],
    [
      "1.711050236",
      "0.762919974"
    ],
    [
      "1.84130322",
      "0.78677128"
    ],
    [
      "0.220796732",
      "0.788996544"
    ],
    [
      "0.449935999",
      "0.795468283"
    ],
    [
      "0.679159877",
      "0.802096518"
    ],
    [
      "0.908418224",
      "0.808788654"
    ],
    [
      "1.137676571",
      "0.815480789"
    ],
    [
      "1.367615581",
      "0.820993566"
    ],
    [
      "1.59513364",
      "0.826936125"
    ],
    [
      "0.079282653",
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    ],
    [
      "0.330622454",
      "0.862976511"
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      "0.56263332",
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    ],
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    ],
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      "1.021115544",
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      "1.250393977",
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      "1.927014327",
      "0.887708543"
    ],
    [
      "1.479606665",
      "0.89165276"
    ],
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      "1.708531551",
      "0.895314766"
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    [
      "0.189514369",
      "0.917667204"
    ],
    [
      "0.446072293",
      "0.927830651"
    ],
    [
      "0.67533064",
      "0.934522787"
    ],
    [
      "0.904554518",
      "0.941151022"
    ],
    [
      "1.133812865",
      "0.947843158"
    ],
    [
      "1.363071206",
      "0.954535282"
    ],
    [
      "1.593004576",
      "0.960031401"
    ],
    [
      "1.820373469",
      "0.966209831"
    ],
    [
      "0.066549952",
      "0.976320265"
    ],
    [
      "0.299340091",
      "0.99164717"
    ],
    [
      "0.558769614",
      "0.997357903"
    ],
    [
      "0.788027961",
      "1.004050039"
    ],
    [
      "1.017251838",
      "1.010678274"
    ],
    [
      "1.246510185",
      "1.01737041"
    ],
    [
      "1.475796797",
      "1.024016689"
    ],
    [
      "1.704846494",
      "1.030926466"
    ],
    [
      "1.933108894",
      "1.035675281"
    ],
    [
      "0.177082393",
      "1.049501079"
    ],
    [
      "0.442208587",
      "1.06019302"
    ],
    [
      "0.671466934",
      "1.066885155"
    ],
    [
      "0.900725281",
      "1.073577291"
    ],
    [
      "1.129949159",
      "1.080205526"
    ],
    [
      "1.359207512",
      "1.086897651"
    ],
    [
      "1.588319937",
      "1.093825483"
    ],
    [
      "1.817581919",
      "1.100391917"
    ],
    [
      "0.324878398",
      "1.121579921"
    ],
    [
      "0.554905907",
      "1.129720272"
    ],
    [
      "0.076030828",
      "1.135077396"
    ],
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      "0.784164255",
      "1.136412407"
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      "1.013422602",
      "1.143104543"
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      "1.914019679",
      "1.191135969"
    ],
    [
      "0.436706109",
      "1.192497394"
    ],
    [
      "0.667603228",
      "1.199247524"
    ],
    [
      "0.896861575",
      "1.205939659"
    ],
    [
      "1.126119922",
      "1.212631795"
    ],
    [
      "1.355197963",
      "1.219495864"
    ],
    [
      "1.584528805",
      "1.22618995"
    ],
    [
      "1.797493122",
      "1.254034986"
    ],
    [
      "0.316035356",
      "1.254867945"
    ],
    [
      "0.549403429",
      "1.262024646"
    ],
    [
      "0.10442789",
      "1.264415448"
    ],
    [
      "0.780300548",
      "1.268774776"
    ],
    [
      "1.009558895",
      "1.275466911"
    ],
    [
      "1.238671406",
      "1.282394881"
    ],
    [
      "1.467967858",
      "1.289025214"
    ],
    [
      "0.429649969",
      "1.32472801"
    ],
    [
      "0.662100749",
      "1.331551898"
    ],
    [
      "0.892997869",
      "1.338302028"
    ],
    [
      "1.12211038",
      "1.345229997"
    ],
    [
      "1.351406831",
      "1.351860331"
    ],
    [
      "1.59227511",
      "1.35838193"
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    [
      "0.181583968",
      "1.37203359"
    ],
    [
      "1.856170022",
      "1.372743647"
    ],
    [
      "1.723758178",
      "1.374095855"
    ],
    [
      "0.313088942",
      "1.387563126"
    ],
    [
      "0.542347289",
      "1.394255262"
    ],
    [
      "0.77479807",
      "1.40107915"
    ],
    [
      "1.005100166",
      "1.408784648"
    ],
    [
      "1.234803345",
      "1.414764308"
    ],
    [
      "1.475714163",
      "1.421217194"
    ],
    [
      "0.425786262",
      "1.457090378"
    ],
    [
      "0.65504461",
      "1.463782514"
    ],
    [
      "0.88749539",
      "1.470606402"
    ],
    [
      "1.118238383",
      "1.477592123"
    ],
    [
      "1.644407986",
      "1.48010657"
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    [
      "1.347543119",
      "1.484222699"
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    [
      "1.79483657",
      "1.490101785"
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    [
      "0.211940232",
      "1.500925878"
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    [
      "0.538483583",
      "1.52661763"
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    [
      "0.76774193",
      "1.533309766"
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    [
      "0.999597688",
      "1.541089022"
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    [
      "1.230066093",
      "1.548509597"
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    [
      "1.460823404",
      "1.55279603"
    ],
    [
      "0.421922556",
      "1.589452747"
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    [
      "0.651180903",
      "1.596144882"
    ],
    [
      "1.717519402",
      "1.597604251"
    ],
    [
      "1.585103704",
      "1.598503052"
    ],
    [
      "0.88043925",
      "1.602837018"
    ],
    [
      "0.290720339",
      "1.607360927"
    ],
    [
      "1.112735904",
      "1.609896498"
    ],
    [
      "1.343646836",
      "1.616584112"
    ],
    [
      "0.763878224",
      "1.665672134"
    ],
    [
      "0.528153418",
      "1.668507974"
    ],
    [
      "0.992541548",
      "1.673319638"
    ],
    [
      "1.223722907",
      "1.682122602"
    ],
    [
      "1.456927121",
      "1.685157444"
    ],
    [
      "0.406574152",
      "1.720978985"
    ],
    [
      "0.646217847",
      "1.72847059"
    ],
    [
      "1.581207421",
      "1.730864466"
    ],
    [
      "0.876575798",
      "1.737246703"
    ],
    [
      "1.105679765",
      "1.742127113"
    ],
    [
      "1.338079386",
      "1.748885769"
    ],
    [
      "0.758915167",
      "1.797997842"
    ],
    [
      "0.523911006",
      "1.803257227"
    ],
    [
      "0.989714015",
      "1.806054178"
    ],
    [
      "1.451359671",
      "1.8174591"
    ],
    [
      "1.186053595",
      "1.847363857"
    ],
    [
      "0.643004683",
      "1.862786217"
    ],
    [
      "1.314587357",
      "1.879204031"
    ],
    [
      "0.818885834",
      "1.916058182"
    ],
    [
      "1.082682099",
      "1.930122897"
    ],
    [
      "0.950284087",
      "1.932466225"
    ]
  ],
  "radius": "0.066209373"
}
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": 190
}

起步布局答案

{
  "centers": [
    [
      "1.096540516",
      "0.071213231"
    ],
    [
      "0.836950375",
      "0.080554649"
    ],
    [
      "1.226947499",
      "0.094207658"
    ],
    [
      "0.967673619",
      "0.101677107"
    ],
    [
      "0.708533474",
      "0.112862921"
    ],
    [
      "1.377762025",
      "0.146032328"
    ],
    [
      "0.560525241",
      "0.176148336"
    ],
    [
      "0.807694385",
      "0.209701124"
    ],
    [
      "1.038739086",
      "0.213410827"
    ],
    [
      "1.505541043",
      "0.214969166"
    ],
    [
      "1.267997433",
      "0.220102963"
    ],
    [
      "0.679277484",
      "0.242009395"
    ],
    [
      "0.447852375",
      "0.24694095"
    ],
    [
      "0.922178059",
      "0.276245944"
    ],
    [
      "1.387026781",
      "0.278126571"
    ],
    [
      "1.151436406",
      "0.282938079"
    ],
    [
      "1.611600786",
      "0.294429919"
    ],
    [
      "0.564935608",
      "0.308797569"
    ],
    [
      "0.346883227",
      "0.332614493"
    ],
    [
      "0.805147392",
      "0.342095374"
    ],
    [
      "1.034875379",
      "0.345773196"
    ],
    [
      "1.264133727",
      "0.352465331"
    ],
    [
      "1.49271131",
      "0.357910687"
    ],
    [
      "0.677734422",
      "0.378160038"
    ],
    [
      "0.46396646",
      "0.394471112"
    ],
    [
      "0.91895713",
      "0.409786352"
    ],
    [
      "1.1475727",
      "0.415300448"
    ],
    [
      "1.376831047",
      "0.421992583"
    ],
    [
      "1.60540863",
      "0.427437939"
    ],
    [
      "1.737827196",
      "0.427657479"
    ],
    [
      "0.313387382",
      "0.460726761"
    ],
    [
      "0.576765275",
      "0.463833582"
    ],
    [
      "0.803430155",
      "0.474502987"
    ],
    [
      "1.03165445",
      "0.479313604"
    ],
    [
      "1.26027002",
      "0.4848277"
    ],
    [
      "1.489528367",
      "0.491519835"
    ],
    [
      "0.196860825",
      "0.523625778"
    ],
    [
      "0.460094547",
      "0.526833241"
    ],
    [
      "0.686887289",
      "0.537371781"
    ],
    [
      "1.811366883",
      "0.5377785"
    ],
    [
      "0.916127475",
      "0.544030239"
    ],
    [
      "1.14435177",
      "0.548840855"
    ],
    [
      "1.372967341",
      "0.554354952"
    ],
    [
      "1.605042886",
      "0.559856181"
    ],
    [
      "0.34353352",
      "0.589668357"
    ],
    [
      "0.570326263",
      "0.600206898"
    ],
    [
      "0.79958461",
      "0.606899033"
    ],
    [
      "1.028824796",
      "0.613557491"
    ],
    [
      "1.257049091",
      "0.618368107"
    ],
    [
      "1.485664661",
      "0.623882204"
    ],
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  "radius": "0.066209373"
}

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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