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 = 197

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 = 197
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[0.0650729109795, 0.0650729109805]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 = 0.065072910

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 = 197 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": [
    [
      "1.033026522",
      "0.065656428"
    ],
    [
      "0.902957365",
      "0.070122917"
    ],
    [
      "1.162455696",
      "0.079295482"
    ],
    [
      "0.774768684",
      "0.0926084"
    ],
    [
      "1.291823964",
      "0.113573373"
    ],
    [
      "0.650944495",
      "0.132677156"
    ],
    [
      "1.414808663",
      "0.162132209"
    ],
    [
      "1.061234575",
      "0.192708546"
    ],
    [
      "0.859802872",
      "0.192905753"
    ],
    [
      "1.190663749",
      "0.206347601"
    ],
    [
      "0.490496625",
      "0.216102702"
    ],
    [
      "1.314805205",
      "0.245422324"
    ],
    [
      "0.743942781",
      "0.252187911"
    ],
    [
      "1.56142015",
      "0.252406476"
    ],
    [
      "0.613902992",
      "0.25744038"
    ],
    [
      "0.960599422",
      "0.275234305"
    ],
    [
      "1.436938289",
      "0.290382802"
    ],
    [
      "0.388229262",
      "0.296673885"
    ],
    [
      "1.113260587",
      "0.312003261"
    ],
    [
      "0.84473933",
      "0.334516463"
    ],
    [
      "1.660815054",
      "0.338641007"
    ],
    [
      "0.51640033",
      "0.34364459"
    ],
    [
      "1.237662436",
      "0.350240873"
    ],
    [
      "0.663666997",
      "0.377895804"
    ],
    [
      "1.532067683",
      "0.379199093"
    ],
    [
      "0.294212556",
      "0.386850145"
    ],
    [
      "1.35979552",
      "0.395201351"
    ],
    [
      "0.996445414",
      "0.400346254"
    ],
    [
      "0.41757246",
      "0.42832627"
    ],
    [
      "1.119749778",
      "0.441987204"
    ],
    [
      "0.775426747",
      "0.444669542"
    ],
    [
      "1.759601896",
      "0.454937049"
    ],
    [
      "0.565917056",
      "0.46400251"
    ],
    [
      "1.629852194",
      "0.465083459"
    ],
    [
      "1.244151627",
      "0.480224816"
    ],
    [
      "0.898731111",
      "0.486310491"
    ],
    [
      "0.215904799",
      "0.49080124"
    ],
    [
      "1.49467088",
      "0.503856279"
    ],
    [
      "1.366284712",
      "0.525185294"
    ],
    [
      "1.022035476",
      "0.527951441"
    ],
    [
      "0.677712444",
      "0.530633779"
    ],
    [
      "0.339264703",
      "0.532277366"
    ],
    [
      "0.468202753",
      "0.549966747"
    ],
    [
      "1.827933249",
      "0.565701489"
    ],
    [
      "1.14533984",
      "0.569592391"
    ],
    [
      "0.801016809",
      "0.572274729"
    ],
    [
      "1.59245539",
      "0.589740646"
    ],
    [
      "1.268644205",
      "0.61123334"
    ],
    [
      "0.924321173",
      "0.613915678"
    ],
    [
      "0.579998142",
      "0.616598016"
    ],
    [
      "0.235619571",
      "0.619445178"
    ],
    [
      "1.715692594",
      "0.63157994"
    ],
    [
      "1.391983601",
      "0.652770411"
    ],
    [
      "1.047625538",
      "0.655556628"
    ],
    [
      "0.703302506",
      "0.658238966"
    ],
    [
      "0.358979475",
      "0.660921303"
    ],
    [
      "0.121086645",
      "0.681252489"
    ],
    [
      "1.880221058",
      "0.684881688"
    ],
    [
      "1.515252933",
      "0.69451524"
    ],
    [
      "1.170929902",
      "0.697197578"
    ],
    [
      "0.826606871",
      "0.699879915"
    ],
    [
      "0.48228384",
      "0.702562253"
    ],
    [
      "1.639156891",
      "0.736842569"
    ],
    [
      "1.294234267",
      "0.738838527"
    ],
    [
      "0.949911235",
      "0.741520865"
    ],
    [
      "0.605588204",
      "0.744203203"
    ],
    [
      "0.261209633",
      "0.747050364"
    ],
    [
      "1.768465611",
      "0.751579927"
    ],
    [
      "1.417538631",
      "0.780479477"
    ],
    [
      "1.0732156",
      "0.783161815"
    ],
    [
      "0.728892568",
      "0.785844152"
    ],
    [
      "0.384569537",
      "0.78852649"
    ],
    [
      "0.146676707",
      "0.808857676"
    ],
    [
      "1.9154521",
      "0.810168195"
    ],
    [
      "1.540842995",
      "0.822120427"
    ],
    [
      "1.196519964",
      "0.824802764"
    ],
    [
      "0.852196933",
      "0.827485102"
    ],
    [
      "0.507873901",
      "0.83016744"
    ],
    [
      "1.674387933",
      "0.862129076"
    ],
    [
      "1.319824328",
      "0.866443714"
    ],
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      "0.975501297",
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    [
      "0.631178266",
      "0.87180839"
    ],
    [
      "0.280134012",
      "0.875899984"
    ],
    [
      "1.803696653",
      "0.876866434"
    ],
    [
      "1.443128693",
      "0.908084664"
    ],
    [
      "1.098805662",
      "0.910767002"
    ],
    [
      "0.069096695",
      "0.91335302"
    ],
    [
      "0.75448263",
      "0.913449339"
    ],
    [
      "0.410159599",
      "0.916131677"
    ],
    [
      "1.930944375",
      "0.939593777"
    ],
    [
      "1.566433057",
      "0.949725613"
    ],
    [
      "1.222110026",
      "0.952407951"
    ],
    [
      "0.877786995",
      "0.955090289"
    ],
    [
      "0.533463963",
      "0.957772627"
    ],
    [
      "0.186377171",
      "0.969773095"
    ],
    [
      "1.689737422",
      "0.991366563"
    ],
    [
      "1.34541439",
      "0.994048901"
    ],
    [
      "1.001091359",
      "0.996731239"
    ],
    [
      "0.656768328",
      "0.999413576"
    ],
    [
      "0.312445291",
      "1.002095908"
    ],
    [
      "1.819046142",
      "1.006103921"
    ],
    [
      "1.468718755",
      "1.035689851"
    ],
    [
      "1.124395724",
      "1.038372188"
    ],
    [
      "0.780072692",
      "1.041054526"
    ],
    [
      "0.066083759",
      "1.043463962"
    ],
    [
      "0.435749653",
      "1.043736866"
    ],
    [
      "1.932283815",
      "1.070253469"
    ],
    [
      "1.592023119",
      "1.0773308"
    ],
    [
      "1.247700088",
      "1.080013138"
    ],
    [
      "0.903377057",
      "1.082695476"
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    [
      "0.559054025",
      "1.085377814"
    ],
    [
      "0.183364235",
      "1.099884037"
    ],
    [
      "1.715327671",
      "1.118971712"
    ],
    [
      "1.371004452",
      "1.121654088"
    ],
    [
      "1.026681421",
      "1.124336425"
    ],
    [
      "0.68235839",
      "1.127018763"
    ],
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      "0.336468685",
      "1.130005296"
    ],
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      "1.838628581",
      "1.160622889"
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      "1.494308817",
      "1.163295037"
    ],
    [
      "1.149985785",
      "1.165977375"
    ],
    [
      "0.805662754",
      "1.168659713"
    ],
    [
      "0.461339723",
      "1.171342051"
    ],
    [
      "0.083936967",
      "1.186809151"
    ],
    [
      "1.617613369",
      "1.204935949"
    ],
    [
      "1.27329015",
      "1.207618325"
    ],
    [
      "0.928967119",
      "1.210300663"
    ],
    [
      "0.584644087",
      "1.212983"
    ],
    [
      "0.239808344",
      "1.217152948"
    ],
    [
      "1.740914279",
      "1.246587126"
    ],
    [
      "1.396594528",
      "1.24925929"
    ],
    [
      "1.052271483",
      "1.251941612"
    ],
    [
      "0.707948452",
      "1.25462395"
    ],
    [
      "0.363610595",
      "1.257289433"
    ],
    [
      "1.892263645",
      "1.279202883"
    ],
    [
      "1.519899066",
      "1.290900187"
    ],
    [
      "1.175575847",
      "1.293582562"
    ],
    [
      "0.831252816",
      "1.2962649"
    ],
    [
      "0.486929785",
      "1.298947238"
    ],
    [
      "0.118872736",
      "1.312206489"
    ],
    [
      "1.298880212",
      "1.335223512"
    ],
    [
      "0.954557181",
      "1.337905849"
    ],
    [
      "1.648676131",
      "1.338998803"
    ],
    [
      "0.610234149",
      "1.340588187"
    ],
    [
      "0.265398405",
      "1.344758135"
    ],
    [
      "1.794549342",
      "1.36516712"
    ],
    [
      "1.42225391",
      "1.376942958"
    ],
    [
      "1.077861545",
      "1.379546799"
    ],
    [
      "0.733538514",
      "1.382229137"
    ],
    [
      "0.389200657",
      "1.38489462"
    ],
    [
      "1.545591714",
      "1.418484758"
    ],
    [
      "1.201165909",
      "1.421187749"
    ],
    [
      "0.856842878",
      "1.423870087"
    ],
    [
      "0.512519847",
      "1.426552424"
    ],
    [
      "0.173380913",
      "1.436794628"
    ],
    [
      "1.324470557",
      "1.462828642"
    ],
    [
      "0.980147243",
      "1.465511036"
    ],
    [
      "0.635824211",
      "1.468193374"
    ],
    [
      "1.665440506",
      "1.469221353"
    ],
    [
      "1.79294611",
      "1.495303067"
    ],
    [
      "0.284933934",
      "1.503830876"
    ],
    [
      "1.448934211",
      "1.505635559"
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    [
      "1.103451607",
      "1.507151986"
    ],
    [
      "0.759128576",
      "1.509834324"
    ],
    [
      "0.414790719",
      "1.512499807"
    ],
    [
      "1.226756254",
      "1.548792879"
    ],
    [
      "0.88243294",
      "1.551475273"
    ],
    [
      "0.538109909",
      "1.554157611"
    ],
    [
      "1.588976642",
      "1.574536178"
    ],
    [
      "1.350184904",
      "1.590408841"
    ],
    [
      "1.005737304",
      "1.593116223"
    ],
    [
      "0.661414273",
      "1.595798561"
    ],
    [
      "1.716482246",
      "1.600617893"
    ],
    [
      "0.310523996",
      "1.631436063"
    ],
    [
      "1.472975374",
      "1.633541608"
    ],
    [
      "1.129041952",
      "1.634757116"
    ],
    [
      "0.440380781",
      "1.640104994"
    ],
    [
      "0.773184704",
      "1.662471688"
    ],
    [
      "1.252470602",
      "1.676373078"
    ],
    [
      "0.902128078",
      "1.680122219"
    ],
    [
      "0.568037984",
      "1.686456178"
    ],
    [
      "1.616475692",
      "1.702489649"
    ],
    [
      "1.375261071",
      "1.719505845"
    ],
    [
      "1.031327649",
      "1.720721353"
    ],
    [
      "0.679808414",
      "1.753129305"
    ],
    [
      "1.154756299",
      "1.762337316"
    ],
    [
      "0.466317342",
      "1.767640204"
    ],
    [
      "1.501125527",
      "1.771664383"
    ],
    [
      "0.804099102",
      "1.791726713"
    ],
    [
      "0.933409099",
      "1.806452867"
    ],
    [
      "1.263271067",
      "1.834187718"
    ],
    [
      "0.578087772",
      "1.834313331"
    ],
    [
      "1.057041997",
      "1.848301553"
    ],
    [
      "1.392628897",
      "1.848487602"
    ],
    [
      "0.700217339",
      "1.885552451"
    ],
    [
      "0.829813449",
      "1.919306913"
    ],
    [
      "1.165556764",
      "1.920151955"
    ],
    [
      "0.959123446",
      "1.934033066"
    ]
  ],
  "radius": "0.06507291"
}
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": 197
}

起步布局答案

{
  "centers": [
    [
      "1.033026522",
      "0.065656428"
    ],
    [
      "0.902957365",
      "0.070122917"
    ],
    [
      "1.162455696",
      "0.079295482"
    ],
    [
      "0.774768684",
      "0.0926084"
    ],
    [
      "1.291823964",
      "0.113573373"
    ],
    [
      "0.650944495",
      "0.132677156"
    ],
    [
      "1.414808663",
      "0.162132209"
    ],
    [
      "1.061234575",
      "0.192708546"
    ],
    [
      "0.859802872",
      "0.192905753"
    ],
    [
      "1.190663749",
      "0.206347601"
    ],
    [
      "0.490496625",
      "0.216102702"
    ],
    [
      "1.314805205",
      "0.245422324"
    ],
    [
      "0.743942781",
      "0.252187911"
    ],
    [
      "1.56142015",
      "0.252406476"
    ],
    [
      "0.613902992",
      "0.25744038"
    ],
    [
      "0.960599422",
      "0.275234305"
    ],
    [
      "1.436938289",
      "0.290382802"
    ],
    [
      "0.388229262",
      "0.296673885"
    ],
    [
      "1.113260587",
      "0.312003261"
    ],
    [
      "0.84473933",
      "0.334516463"
    ],
    [
      "1.660815054",
      "0.338641007"
    ],
    [
      "0.51640033",
      "0.34364459"
    ],
    [
      "1.237662436",
      "0.350240873"
    ],
    [
      "0.663666997",
      "0.377895804"
    ],
    [
      "1.532067683",
      "0.379199093"
    ],
    [
      "0.294212556",
      "0.386850145"
    ],
    [
      "1.35979552",
      "0.395201351"
    ],
    [
      "0.996445414",
      "0.400346254"
    ],
    [
      "0.41757246",
      "0.42832627"
    ],
    [
      "1.119749778",
      "0.441987204"
    ],
    [
      "0.775426747",
      "0.444669542"
    ],
    [
      "1.759601896",
      "0.454937049"
    ],
    [
      "0.565917056",
      "0.46400251"
    ],
    [
      "1.629852194",
      "0.465083459"
    ],
    [
      "1.244151627",
      "0.480224816"
    ],
    [
      "0.898731111",
      "0.486310491"
    ],
    [
      "0.215904799",
      "0.49080124"
    ],
    [
      "1.49467088",
      "0.503856279"
    ],
    [
      "1.366284712",
      "0.525185294"
    ],
    [
      "1.022035476",
      "0.527951441"
    ],
    [
      "0.677712444",
      "0.530633779"
    ],
    [
      "0.339264703",
      "0.532277366"
    ],
    [
      "0.468202753",
      "0.549966747"
    ],
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  "radius": "0.06507291"
}

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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