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 = 200

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 = 200
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[0.0646693541862641875059442629185, 0.0646693541862641875059442629195]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 = 0.064669353

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 = 200 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": [
    [
      "0.95721687",
      "0.065648342"
    ],
    [
      "1.086618695",
      "0.070606063"
    ],
    [
      "0.828731774",
      "0.08048349"
    ],
    [
      "1.215457824",
      "0.089823455"
    ],
    [
      "1.338957037",
      "0.128248003"
    ],
    [
      "0.633100265",
      "0.139634961"
    ],
    [
      "1.011118321",
      "0.183220171"
    ],
    [
      "0.753862309",
      "0.185949431"
    ],
    [
      "1.139935547",
      "0.194822918"
    ],
    [
      "0.882633225",
      "0.19805532"
    ],
    [
      "1.521565483",
      "0.223589758"
    ],
    [
      "1.26343476",
      "0.233247466"
    ],
    [
      "1.394055688",
      "0.245263558"
    ],
    [
      "0.55912078",
      "0.245727078"
    ],
    [
      "0.430377937",
      "0.258127961"
    ],
    [
      "0.679882824",
      "0.292041548"
    ],
    [
      "1.066080755",
      "0.301001876"
    ],
    [
      "0.808639348",
      "0.304299562"
    ],
    [
      "0.93752042",
      "0.315170209"
    ],
    [
      "1.189579968",
      "0.339426424"
    ],
    [
      "1.576591762",
      "0.340639364"
    ],
    [
      "0.333482503",
      "0.343800455"
    ],
    [
      "1.318533411",
      "0.350263021"
    ],
    [
      "1.449081967",
      "0.362313164"
    ],
    [
      "1.699933258",
      "0.379567206"
    ],
    [
      "0.469028594",
      "0.381556597"
    ],
    [
      "0.597936211",
      "0.392107796"
    ],
    [
      "0.729150904",
      "0.412109114"
    ],
    [
      "0.857963033",
      "0.423864087"
    ],
    [
      "0.986844104",
      "0.434734734"
    ],
    [
      "0.249332022",
      "0.442020606"
    ],
    [
      "1.115725175",
      "0.445605382"
    ],
    [
      "1.244678619",
      "0.456441979"
    ],
    [
      "1.37355969",
      "0.467312627"
    ],
    [
      "0.375425025",
      "0.470813941"
    ],
    [
      "1.504108245",
      "0.479362769"
    ],
    [
      "1.778830219",
      "0.482055112"
    ],
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      "1.632989316",
      "0.490233417"
    ],
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      "0.527254396",
      "0.500424842"
    ],
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      "0.655227169",
      "0.519172397"
    ],
    [
      "0.78410824",
      "0.530043045"
    ],
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      "0.912989312",
      "0.540913692"
    ],
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      "0.179535596",
      "0.550910277"
    ],
    [
      "1.041870383",
      "0.55178434"
    ],
    [
      "1.170823826",
      "0.562620937"
    ],
    [
      "1.299704897",
      "0.573491585"
    ],
    [
      "0.305628599",
      "0.579703612"
    ],
    [
      "1.428585969",
      "0.584362232"
    ],
    [
      "0.43450967",
      "0.590574259"
    ],
    [
      "1.711886278",
      "0.592721323"
    ],
    [
      "1.559134524",
      "0.596412375"
    ],
    [
      "1.846620749",
      "0.603069585"
    ],
    [
      "0.57841129",
      "0.623241975"
    ],
    [
      "0.710253448",
      "0.636222002"
    ],
    [
      "0.839134519",
      "0.64709265"
    ],
    [
      "0.96801559",
      "0.657963298"
    ],
    [
      "0.125427852",
      "0.668387311"
    ],
    [
      "1.096969034",
      "0.668799895"
    ],
    [
      "1.225850105",
      "0.679670542"
    ],
    [
      "1.354731176",
      "0.69054119"
    ],
    [
      "0.251520855",
      "0.697180646"
    ],
    [
      "1.638031485",
      "0.698900281"
    ],
    [
      "1.483612247",
      "0.701411838"
    ],
    [
      "0.380401926",
      "0.708051293"
    ],
    [
      "1.768125869",
      "0.709192821"
    ],
    [
      "1.895685323",
      "0.730572422"
    ],
    [
      "0.507583259",
      "0.731575964"
    ],
    [
      "0.63646433",
      "0.742446612"
    ],
    [
      "0.765345401",
      "0.753317259"
    ],
    [
      "0.894226472",
      "0.764187907"
    ],
    [
      "1.023114241",
      "0.774978853"
    ],
    [
      "1.151995312",
      "0.7858495"
    ],
    [
      "0.088043423",
      "0.792205347"
    ],
    [
      "1.280876384",
      "0.796720148"
    ],
    [
      "1.409757455",
      "0.807590796"
    ],
    [
      "0.306547134",
      "0.814230251"
    ],
    [
      "1.538638526",
      "0.818461443"
    ],
    [
      "1.66782373",
      "0.824761012"
    ],
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      "1.796729981",
      "0.835328884"
    ],
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      "0.433728466",
      "0.837754922"
    ],
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      "0.562609537",
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    ],
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      "1.924289434",
      "0.856708484"
    ],
    [
      "0.691490609",
      "0.859496217"
    ],
    [
      "0.82037168",
      "0.870366865"
    ],
    [
      "0.190440758",
      "0.871219821"
    ],
    [
      "0.949252751",
      "0.881237512"
    ],
    [
      "1.07814052",
      "0.892028458"
    ],
    [
      "1.207021591",
      "0.902899106"
    ],
    [
      "1.335902662",
      "0.913769753"
    ],
    [
      "1.464783733",
      "0.924640401"
    ],
    [
      "0.069112383",
      "0.925101624"
    ],
    [
      "1.59559267",
      "0.934613869"
    ],
    [
      "0.348794184",
      "0.940899573"
    ],
    [
      "1.724656314",
      "0.943074679"
    ],
    [
      "0.488754745",
      "0.954804527"
    ],
    [
      "0.617635816",
      "0.965675175"
    ],
    [
      "1.875612319",
      "0.976537696"
    ],
    [
      "0.746516887",
      "0.976545823"
    ],
    [
      "0.875397958",
      "0.98741647"
    ],
    [
      "1.00427903",
      "0.998287118"
    ],
    [
      "0.182260474",
      "1.000299581"
    ],
    [
      "1.133166798",
      "1.009078064"
    ],
    [
      "1.26204787",
      "1.019948711"
    ],
    [
      "1.390928941",
      "1.030819359"
    ],
    [
      "1.519810012",
      "1.041690007"
    ],
    [
      "1.648691083",
      "1.052560654"
    ],
    [
      "0.066464206",
      "1.057916642"
    ],
    [
      "0.414899952",
      "1.060983485"
    ],
    [
      "1.777747509",
      "1.061101166"
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      "0.543781023",
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      "0.286534481",
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      "1.9306367",
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      "0.801543166",
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    ],
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      "1.57483629",
      "1.158739612"
    ],
    [
      "1.703709126",
      "1.169707459"
    ],
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      "0.469926231",
      "1.178033091"
    ],
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      "1.83277189",
      "1.178151663"
    ],
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      "0.212679689",
      "1.182999339"
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    ],
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      "0.34156076",
      "1.193869987"
    ],
    [
      "0.59810852",
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    ],
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      "0.727364107",
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    ],
    [
      "0.856245178",
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    ],
    [
      "0.985126249",
      "1.221667575"
    ],
    [
      "1.11400732",
      "1.232538222"
    ],
    [
      "1.242895089",
      "1.243329168"
    ],
    [
      "1.37177616",
      "1.254199816"
    ],
    [
      "1.500657231",
      "1.265070463"
    ],
    [
      "1.629538303",
      "1.275941111"
    ],
    [
      "1.758411138",
      "1.286908958"
    ],
    [
      "1.887473902",
      "1.295353162"
    ],
    [
      "0.520527768",
      "1.299752447"
    ],
    [
      "0.247122066",
      "1.307667796"
    ],
    [
      "0.117821604",
      "1.310812957"
    ],
    [
      "0.653134798",
      "1.312339646"
    ],
    [
      "0.782390385",
      "1.316975885"
    ],
    [
      "0.376003137",
      "1.318538444"
    ],
    [
      "0.911271457",
      "1.327846532"
    ],
    [
      "1.040152528",
      "1.33871718"
    ],
    [
      "1.169033599",
      "1.349587828"
    ],
    [
      "1.297921368",
      "1.360378774"
    ],
    [
      "1.426802439",
      "1.371249421"
    ],
    [
      "1.580309221",
      "1.395544619"
    ],
    [
      "1.709182057",
      "1.406512465"
    ],
    [
      "1.83824482",
      "1.41495667"
    ],
    [
      "0.454786529",
      "1.421113677"
    ],
    [
      "0.584123775",
      "1.421728768"
    ],
    [
      "0.298433201",
      "1.426392978"
    ],
    [
      "0.713363046",
      "1.426799506"
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    [
      "0.16913274",
      "1.429538139"
    ],
    [
      "0.842279076",
      "1.437247414"
    ],
    [
      "0.971160147",
      "1.448118061"
    ],
    [
      "1.100041218",
      "1.458988709"
    ],
    [
      "1.228928987",
      "1.469779655"
    ],
    [
      "1.357810058",
      "1.480650303"
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    [
      "1.487393406",
      "1.485517685"
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    [
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      "1.513207102"
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    [
      "0.380085854",
      "1.526699238"
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    [
      "1.772593945",
      "1.527113615"
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    [
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      "1.53859914"
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    [
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      "1.539109241"
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    [
      "0.236331478",
      "1.540049817"
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    [
      "0.768391273",
      "1.543848196"
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    [
      "0.897307128",
      "1.554298253"
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    [
      "1.026145962",
      "1.565658552"
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    [
      "1.155067496",
      "1.576038315"
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    [
      "1.283955265",
      "1.586829261"
    ],
    [
      "1.413281687",
      "1.597489508"
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    [
      "1.542522574",
      "1.602518866"
    ],
    [
      "1.691100499",
      "1.630256708"
    ],
    [
      "0.318132863",
      "1.64023482"
    ],
    [
      "0.570535623",
      "1.652294296"
    ],
    [
      "0.699868297",
      "1.653543698"
    ],
    [
      "0.828784152",
      "1.663993756"
    ],
    [
      "0.443013949",
      "1.673898093"
    ],
    [
      "1.081617591",
      "1.682497757"
    ],
    [
      "1.21050536",
      "1.693288703"
    ],
    [
      "0.953323256",
      "1.698900962"
    ],
    [
      "1.339426894",
      "1.703668466"
    ],
    [
      "1.468307965",
      "1.714539113"
    ],
    [
      "1.597548853",
      "1.719568472"
    ],
    [
      "0.627107779",
      "1.768604626"
    ],
    [
      "0.756420554",
      "1.771194554"
    ],
    [
      "0.499586105",
      "1.790208422"
    ],
    [
      "1.137636041",
      "1.800146383"
    ],
    [
      "0.880959658",
      "1.80610176"
    ],
    [
      "1.009663829",
      "1.81889777"
    ],
    [
      "1.292465643",
      "1.824180467"
    ],
    [
      "1.421346714",
      "1.835051114"
    ],
    [
      "0.689067796",
      "1.882136374"
    ],
    [
      "1.199264826",
      "1.913858275"
    ],
    [
      "0.813731596",
      "1.916595603"
    ],
    [
      "1.071292614",
      "1.932609661"
    ],
    [
      "0.941957166",
      "1.933527957"
    ]
  ],
  "radius": "0.064669353"
}
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": 200
}

起步布局答案

{
  "centers": [
    [
      "0.95721687",
      "0.065648342"
    ],
    [
      "1.086618695",
      "0.070606063"
    ],
    [
      "0.828731774",
      "0.08048349"
    ],
    [
      "1.215457824",
      "0.089823455"
    ],
    [
      "1.338957037",
      "0.128248003"
    ],
    [
      "0.633100265",
      "0.139634961"
    ],
    [
      "1.011118321",
      "0.183220171"
    ],
    [
      "0.753862309",
      "0.185949431"
    ],
    [
      "1.139935547",
      "0.194822918"
    ],
    [
      "0.882633225",
      "0.19805532"
    ],
    [
      "1.521565483",
      "0.223589758"
    ],
    [
      "1.26343476",
      "0.233247466"
    ],
    [
      "1.394055688",
      "0.245263558"
    ],
    [
      "0.55912078",
      "0.245727078"
    ],
    [
      "0.430377937",
      "0.258127961"
    ],
    [
      "0.679882824",
      "0.292041548"
    ],
    [
      "1.066080755",
      "0.301001876"
    ],
    [
      "0.808639348",
      "0.304299562"
    ],
    [
      "0.93752042",
      "0.315170209"
    ],
    [
      "1.189579968",
      "0.339426424"
    ],
    [
      "1.576591762",
      "0.340639364"
    ],
    [
      "0.333482503",
      "0.343800455"
    ],
    [
      "1.318533411",
      "0.350263021"
    ],
    [
      "1.449081967",
      "0.362313164"
    ],
    [
      "1.699933258",
      "0.379567206"
    ],
    [
      "0.469028594",
      "0.381556597"
    ],
    [
      "0.597936211",
      "0.392107796"
    ],
    [
      "0.729150904",
      "0.412109114"
    ],
    [
      "0.857963033",
      "0.423864087"
    ],
    [
      "0.986844104",
      "0.434734734"
    ],
    [
      "0.249332022",
      "0.442020606"
    ],
    [
      "1.115725175",
      "0.445605382"
    ],
    [
      "1.244678619",
      "0.456441979"
    ],
    [
      "1.37355969",
      "0.467312627"
    ],
    [
      "0.375425025",
      "0.470813941"
    ],
    [
      "1.504108245",
      "0.479362769"
    ],
    [
      "1.778830219",
      "0.482055112"
    ],
    [
      "1.632989316",
      "0.490233417"
    ],
    [
      "0.527254396",
      "0.500424842"
    ],
    [
      "0.655227169",
      "0.519172397"
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  "radius": "0.064669353"
}

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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