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 = 184

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 = 184
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[0.0670896971539310007951875966415, 0.0670896971539310007951875966425]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 = 0.067089696

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 = 184 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": [
    [
      "1.322193816",
      "1.51609984"
    ],
    [
      "1.563677626",
      "0.256636585"
    ],
    [
      "0.721583677",
      "0.700345822"
    ],
    [
      "1.623107784",
      "1.327739673"
    ],
    [
      "1.274065652",
      "0.108256954"
    ],
    [
      "1.070933702",
      "1.930209677"
    ],
    [
      "0.943760915",
      "0.768533831"
    ],
    [
      "1.400930546",
      "1.259551664"
    ],
    [
      "1.6492591",
      "0.842388078"
    ],
    [
      "1.531703348",
      "1.229508624"
    ],
    [
      "0.62508938",
      "1.854262056"
    ],
    [
      "1.878610602",
      "0.686367982"
    ],
    [
      "0.890934626",
      "0.179468646"
    ],
    [
      "0.522359647",
      "1.3409608"
    ],
    [
      "0.965233446",
      "0.067737741"
    ],
    [
      "0.734251382",
      "1.055125047"
    ],
    [
      "1.112684281",
      "1.802691055"
    ],
    [
      "1.176970436",
      "0.206489705"
    ],
    [
      "1.557854663",
      "0.74415703"
    ],
    [
      "1.610442907",
      "0.972957816"
    ],
    [
      "1.257342589",
      "0.934952889"
    ],
    [
      "1.636591394",
      "0.487608853"
    ],
    [
      "0.590810875",
      "0.730388862"
    ],
    [
      "1.741016944",
      "0.940289089"
    ],
    [
      "0.979381816",
      "1.832116039"
    ],
    [
      "1.922674714",
      "1.137815116"
    ],
    [
      "0.574678755",
      "1.568672396"
    ],
    [
      "0.796572956",
      "1.635653946"
    ],
    [
      "0.95642862",
      "1.123313056"
    ],
    [
      "0.10810626",
      "0.726417857"
    ],
    [
      "0.865024184",
      "1.025082007"
    ],
    [
      "1.303570711",
      "0.250947329"
    ],
    [
      "0.272519825",
      "0.415967271"
    ],
    [
      "0.512074142",
      "0.986937037"
    ],
    [
      "1.336079319",
      "0.678404713"
    ],
    [
      "1.37489834",
      "0.547832343"
    ],
    [
      "0.564110213",
      "1.213442174"
    ],
    [
      "0.760952042",
      "0.572071734"
    ],
    [
      "0.883084128",
      "1.925555137"
    ],
    [
      "1.270157745",
      "1.289594703"
    ],
    [
      "0.075411241",
      "0.875672773"
    ],
    [
      "1.492334983",
      "1.357782712"
    ],
    [
      "0.840835959",
      "0.313740515"
    ],
    [
      "1.545039507",
      "0.389515215"
    ],
    [
      "1.139237493",
      "1.319775153"
    ],
    [
      "1.230789379",
      "1.417868791"
    ],
    [
      "0.800320407",
      "0.443797646"
    ],
    [
      "1.074111171",
      "0.373600939"
    ],
    [
      "0.329265272",
      "0.790474941"
    ],
    [
      "1.152052646",
      "1.674416966"
    ],
    [
      "1.440298912",
      "1.131277575"
    ],
    [
      "1.427631206",
      "0.776498351"
    ],
    [
      "0.642846946",
      "0.956893998"
    ],
    [
      "1.035165351",
      "0.86676488"
    ],
    [
      "0.352501517",
      "1.500484387"
    ],
    [
      "1.571071713",
      "1.101234536"
    ],
    [
      "0.904392549",
      "0.896807919"
    ],
    [
      "1.608751461",
      "1.706925238"
    ],
    [
      "0.401345356",
      "0.45349183"
    ],
    [
      "0.363827667",
      "0.317646429"
    ],
    [
      "1.931501744",
      "0.948754186"
    ],
    [
      "0.336888205",
      "1.143610984"
    ],
    [
      "1.374482265",
      "1.74236953"
    ],
    [
      "0.603478578",
      "1.085168086"
    ],
    [
      "0.746919085",
      "1.409904271"
    ],
    [
      "1.205306518",
      "0.708447752"
    ],
    [
      "1.715995017",
      "1.478653696"
    ],
    [
      "0.55144251",
      "0.85866295"
    ],
    [
      "0.237860836",
      "0.692243892"
    ],
    [
      "1.583739419",
      "1.456013761"
    ],
    [
      "0.757458403",
      "1.765161678"
    ],
    [
      "0.838323522",
      "1.508135319"
    ],
    [
      "1.399182844",
      "0.156806849"
    ],
    [
      "0.825655819",
      "1.153356095"
    ],
    [
      "1.66247615",
      "1.199465584"
    ],
    [
      "1.701844515",
      "1.071191496"
    ],
    [
      "0.468295834",
      "0.233441252"
    ],
    [
      "1.91451534",
      "0.81565433"
    ],
    [
      "1.770071699",
      "0.607479349"
    ],
    [
      "1.244674883",
      "0.580173664"
    ],
    [
      "1.28282545",
      "1.644373928"
    ],
    [
      "0.887948168",
      "1.73391218"
    ],
    [
      "1.178605858",
      "1.191501065"
    ],
    [
      "1.099869127",
      "1.448049241"
    ],
    [
      "0.613764084",
      "1.439191849"
    ],
    [
      "0.666083191",
      "1.666903444"
    ],
    [
      "1.361562181",
      "1.387825752"
    ],
    [
      "0.790028657",
      "0.091026035"
    ],
    [
      "0.773619748",
      "0.926850959"
    ],
    [
      "1.479667277",
      "1.003003487"
    ],
    [
      "0.218639154",
      "1.509702719"
    ],
    [
      "1.126569787",
      "0.964995928"
    ],
    [
      "0.630179241",
      "0.602114774"
    ],
    [
      "1.419779967",
      "1.6160674"
    ],
    [
      "1.834841427",
      "1.416366935"
    ],
    [
      "0.471685959",
      "1.116170044"
    ],
    [
      "0.193196992",
      "1.377957487"
    ],
    [
      "1.243457083",
      "1.772648015"
    ],
    [
      "0.420657435",
      "0.888717408"
    ],
    [
      "1.204334628",
      "0.341259618"
    ],
    [
      "1.597223029",
      "0.615882941"
    ],
    [
      "0.597168423",
      "0.270803879"
    ],
    [
      "0.655514649",
      "1.311673222"
    ],
    [
      "0.381305095",
      "1.016996413"
    ],
    [
      "0.70802071",
      "0.346407319"
    ],
    [
      "0.19646108",
      "1.045372977"
    ],
    [
      "0.682215311",
      "0.82861991"
    ],
    [
      "1.217974223",
      "1.063226977"
    ],
    [
      "0.891724844",
      "0.542028695"
    ],
    [
      "0.108260147",
      "1.274083322"
    ],
    [
      "1.75581641",
      "1.307927291"
    ],
    [
      "0.98312928",
      "0.640259743"
    ],
    [
      "0.199510696",
      "0.824648905"
    ],
    [
      "1.432904825",
      "0.286679624"
    ],
    [
      "1.139658331",
      "0.077602481"
    ],
    [
      "1.452966617",
      "1.4860568"
    ],
    [
      "1.046694773",
      "0.174359324"
    ],
    [
      "0.274280675",
      "0.936065062"
    ],
    [
      "1.296710954",
      "0.806678801"
    ],
    [
      "1.780385309",
      "0.812015001"
    ],
    [
      "1.022497645",
      "0.511985655"
    ],
    [
      "1.688627465",
      "0.71411399"
    ],
    [
      "1.30952611",
      "1.161320615"
    ],
    [
      "1.204309447",
      "1.909462429"
    ],
    [
      "1.06050076",
      "1.576323328"
    ],
    [
      "0.067139061",
      "1.009596934"
    ],
    [
      "0.812988113",
      "0.798576871"
    ],
    [
      "0.634145957",
      "0.141820268"
    ],
    [
      "0.929698743",
      "1.606393549"
    ],
    [
      "0.877691887",
      "1.379861231"
    ],
    [
      "0.995796985",
      "0.995038968"
    ],
    [
      "0.786287451",
      "1.281630183"
    ],
    [
      "1.466302776",
      "0.646063391"
    ],
    [
      "1.770005658",
      "0.473299971"
    ],
    [
      "1.501041954",
      "1.78694256"
    ],
    [
      "1.550552769",
      "1.586024361"
    ],
    [
      "0.694883015",
      "1.183399134"
    ],
    [
      "0.42726907",
      "1.242784616"
    ],
    [
      "1.348894475",
      "1.033046527"
    ],
    [
      "0.289926502",
      "1.284965488"
    ],
    [
      "1.335113898",
      "1.870643618"
    ],
    [
      "0.298991734",
      "0.547509462"
    ],
    [
      "1.165938152",
      "0.83672184"
    ],
    [
      "1.008464691",
      "1.349818192"
    ],
    [
      "1.113902082",
      "0.610216704"
    ],
    [
      "1.832067493",
      "1.038848245"
    ],
    [
      "0.391869883",
      "1.372210299"
    ],
    [
      "1.892293214",
      "1.272276176"
    ],
    [
      "0.917060255",
      "1.251587144"
    ],
    [
      "0.483274319",
      "1.470441347"
    ],
    [
      "0.931093209",
      "0.413754606"
    ],
    [
      "0.972395953",
      "0.286090229"
    ],
    [
      "1.795184775",
      "1.179653203"
    ],
    [
      "0.526898813",
      "0.500825109"
    ],
    [
      "1.047833056",
      "1.221544104"
    ],
    [
      "1.153270447",
      "0.481942615"
    ],
    [
      "0.852356478",
      "0.670302783"
    ],
    [
      "1.283493904",
      "0.449601294"
    ],
    [
      "0.168218933",
      "0.577552501"
    ],
    [
      "1.677305671",
      "0.35859386"
    ],
    [
      "0.460038074",
      "0.760431901"
    ],
    [
      "0.443905954",
      "1.598715435"
    ],
    [
      "1.519035642",
      "0.874729399"
    ],
    [
      "0.553903319",
      "1.740522641"
    ],
    [
      "0.420079376",
      "1.730762412"
    ],
    [
      "1.70386789",
      "1.612283942"
    ],
    [
      "1.505671142",
      "0.517789303"
    ],
    [
      "0.078164724",
      "1.143322566"
    ],
    [
      "0.657671615",
      "0.470782069"
    ],
    [
      "0.751516599",
      "1.899209449"
    ],
    [
      "0.744998244",
      "0.217423709"
    ],
    [
      "0.70516852",
      "1.537422897"
    ],
    [
      "1.414266706",
      "0.419558255"
    ],
    [
      "1.388262841",
      "0.904772439"
    ],
    [
      "0.368633637",
      "0.662200853"
    ],
    [
      "1.087201422",
      "1.093270016"
    ],
    [
      "1.074533716",
      "0.738490792"
    ],
    [
      "0.505113906",
      "0.368425986"
    ],
    [
      "0.207486743",
      "1.179098609"
    ],
    [
      "1.021132394",
      "1.704597416"
    ],
    [
      "0.969096323",
      "1.47809228"
    ],
    [
      "0.499406439",
      "0.632157813"
    ],
    [
      "1.191421014",
      "1.546142879"
    ],
    [
      "0.312936484",
      "1.629528455"
    ]
  ],
  "radius": "0.067089696"
}
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": 184
}

起步布局答案

{
  "centers": [
    [
      "1.322193816",
      "1.51609984"
    ],
    [
      "1.563677626",
      "0.256636585"
    ],
    [
      "0.721583677",
      "0.700345822"
    ],
    [
      "1.623107784",
      "1.327739673"
    ],
    [
      "1.274065652",
      "0.108256954"
    ],
    [
      "1.070933702",
      "1.930209677"
    ],
    [
      "0.943760915",
      "0.768533831"
    ],
    [
      "1.400930546",
      "1.259551664"
    ],
    [
      "1.6492591",
      "0.842388078"
    ],
    [
      "1.531703348",
      "1.229508624"
    ],
    [
      "0.62508938",
      "1.854262056"
    ],
    [
      "1.878610602",
      "0.686367982"
    ],
    [
      "0.890934626",
      "0.179468646"
    ],
    [
      "0.522359647",
      "1.3409608"
    ],
    [
      "0.965233446",
      "0.067737741"
    ],
    [
      "0.734251382",
      "1.055125047"
    ],
    [
      "1.112684281",
      "1.802691055"
    ],
    [
      "1.176970436",
      "0.206489705"
    ],
    [
      "1.557854663",
      "0.74415703"
    ],
    [
      "1.610442907",
      "0.972957816"
    ],
    [
      "1.257342589",
      "0.934952889"
    ],
    [
      "1.636591394",
      "0.487608853"
    ],
    [
      "0.590810875",
      "0.730388862"
    ],
    [
      "1.741016944",
      "0.940289089"
    ],
    [
      "0.979381816",
      "1.832116039"
    ],
    [
      "1.922674714",
      "1.137815116"
    ],
    [
      "0.574678755",
      "1.568672396"
    ],
    [
      "0.796572956",
      "1.635653946"
    ],
    [
      "0.95642862",
      "1.123313056"
    ],
    [
      "0.10810626",
      "0.726417857"
    ],
    [
      "0.865024184",
      "1.025082007"
    ],
    [
      "1.303570711",
      "0.250947329"
    ],
    [
      "0.272519825",
      "0.415967271"
    ],
    [
      "0.512074142",
      "0.986937037"
    ],
    [
      "1.336079319",
      "0.678404713"
    ],
    [
      "1.37489834",
      "0.547832343"
    ],
    [
      "0.564110213",
      "1.213442174"
    ],
    [
      "0.760952042",
      "0.572071734"
    ],
    [
      "0.883084128",
      "1.925555137"
    ],
    [
      "1.270157745",
      "1.289594703"
    ],
    [
      "0.075411241",
      "0.875672773"
    ],
    [
      "1.492334983",
      "1.357782712"
    ],
    [
      "0.840835959",
      "0.313740515"
    ],
    [
      "1.545039507",
      "0.389515215"
    ],
    [
      "1.139237493",
      "1.319775153"
    ],
    [
      "1.230789379",
      "1.417868791"
    ],
    [
      "0.800320407",
      "0.443797646"
    ],
    [
      "1.074111171",
      "0.373600939"
    ],
    [
      "0.329265272",
      "0.790474941"
    ],
    [
      "1.152052646",
      "1.674416966"
    ],
    [
      "1.440298912",
      "1.131277575"
    ],
    [
      "1.427631206",
      "0.776498351"
    ],
    [
      "0.642846946",
      "0.956893998"
    ],
    [
      "1.035165351",
      "0.86676488"
    ],
    [
      "0.352501517",
      "1.500484387"
    ],
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  "radius": "0.067089696"
}

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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