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 = 114

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 = 114
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[0.0847804410445982472677923831365, 0.0847804410445982472677923831375]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 = 0.084780440

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 = 114 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": [
    [
      "0.926922424",
      "0.08770262"
    ],
    [
      "1.09646934",
      "0.08987885"
    ],
    [
      "0.759883839",
      "0.116840292"
    ],
    [
      "1.262705021",
      "0.123294284"
    ],
    [
      "0.601087056",
      "0.176291736"
    ],
    [
      "1.419923555",
      "0.186801962"
    ],
    [
      "1.150648566",
      "0.250550889"
    ],
    [
      "0.853191907",
      "0.258418882"
    ],
    [
      "0.45598265",
      "0.264016329"
    ],
    [
      "1.562728539",
      "0.278222034"
    ],
    [
      "1.307042366",
      "0.316177522"
    ],
    [
      "0.69002847",
      "0.320676839"
    ],
    [
      "1.004070854",
      "0.335790944"
    ],
    [
      "0.329551219",
      "0.377002991"
    ],
    [
      "1.686218301",
      "0.394416575"
    ],
    [
      "0.537157399",
      "0.412883897"
    ],
    [
      "1.46106883",
      "0.413928321"
    ],
    [
      "1.151179763",
      "0.420110939"
    ],
    [
      "0.845687505",
      "0.427813619"
    ],
    [
      "1.30749667",
      "0.485806581"
    ],
    [
      "0.688039999",
      "0.490248836"
    ],
    [
      "0.999547668",
      "0.507868001"
    ],
    [
      "0.226132423",
      "0.511373544"
    ],
    [
      "0.410725967",
      "0.52587056"
    ],
    [
      "1.792188917",
      "0.54168407"
    ],
    [
      "1.61225462",
      "0.546995297"
    ],
    [
      "1.44653115",
      "0.582864843"
    ],
    [
      "1.149698687",
      "0.589665353"
    ],
    [
      "0.85529949",
      "0.597101842"
    ],
    [
      "0.561608567",
      "0.603235499"
    ],
    [
      "0.149276032",
      "0.662515821"
    ],
    [
      "1.298711944",
      "0.670573277"
    ],
    [
      "1.004622247",
      "0.677436186"
    ],
    [
      "0.710223049",
      "0.684872675"
    ],
    [
      "0.410252597",
      "0.695430781"
    ],
    [
      "1.69366027",
      "0.695736727"
    ],
    [
      "1.863221073",
      "0.695900969"
    ],
    [
      "1.447208926",
      "0.752424371"
    ],
    [
      "1.153172238",
      "0.759190652"
    ],
    [
      "0.858773041",
      "0.766627141"
    ],
    [
      "0.561135197",
      "0.77279572"
    ],
    [
      "0.266373046",
      "0.785150164"
    ],
    [
      "0.101620083",
      "0.825241981"
    ],
    [
      "1.595758918",
      "0.834178837"
    ],
    [
      "1.302132486",
      "0.840195204"
    ],
    [
      "1.008095798",
      "0.846961486"
    ],
    [
      "0.711933656",
      "0.854425584"
    ],
    [
      "1.904503941",
      "0.860359528"
    ],
    [
      "0.416012086",
      "0.864893818"
    ],
    [
      "1.744308909",
      "0.915933303"
    ],
    [
      "1.450682477",
      "0.92194967"
    ],
    [
      "1.156645789",
      "0.928715952"
    ],
    [
      "0.860556181",
      "0.936178647"
    ],
    [
      "0.566894686",
      "0.942258757"
    ],
    [
      "0.250393048",
      "0.955682634"
    ],
    [
      "0.084800328",
      "0.993966575"
    ],
    [
      "1.599232469",
      "1.003704137"
    ],
    [
      "1.305606037",
      "1.009720503"
    ],
    [
      "1.009878938",
      "1.016512992"
    ],
    [
      "0.715479741",
      "1.02394948"
    ],
    [
      "1.914740412",
      "1.029611138"
    ],
    [
      "0.401923053",
      "1.033868349"
    ],
    [
      "1.75454538",
      "1.085184913"
    ],
    [
      "1.454156028",
      "1.09147497"
    ],
    [
      "1.15842893",
      "1.098267458"
    ],
    [
      "0.864029732",
      "1.105703947"
    ],
    [
      "0.566722289",
      "1.111819552"
    ],
    [
      "0.266870952",
      "1.136395529"
    ],
    [
      "0.099394094",
      "1.162898262"
    ],
    [
      "1.60946894",
      "1.172955746"
    ],
    [
      "1.307389178",
      "1.179272009"
    ],
    [
      "1.013352489",
      "1.186038291"
    ],
    [
      "0.718208583",
      "1.193488402"
    ],
    [
      "1.893579127",
      "1.197846368"
    ],
    [
      "0.423188145",
      "1.202090489"
    ],
    [
      "1.455939169",
      "1.261026476"
    ],
    [
      "1.161902481",
      "1.267792757"
    ],
    [
      "0.866758575",
      "1.275242869"
    ],
    [
      "0.573132143",
      "1.281259236"
    ],
    [
      "1.772194334",
      "1.316238196"
    ],
    [
      "0.14490046",
      "1.326238592"
    ],
    [
      "0.314359375",
      "1.332118146"
    ],
    [
      "1.604671501",
      "1.342448747"
    ],
    [
      "1.310862729",
      "1.348797309"
    ],
    [
      "1.016081332",
      "1.355577213"
    ],
    [
      "0.721682134",
      "1.363013702"
    ],
    [
      "1.45941272",
      "1.430551775"
    ],
    [
      "1.164631323",
      "1.43733168"
    ],
    [
      "0.870232126",
      "1.444768168"
    ],
    [
      "0.576605694",
      "1.450784535"
    ],
    [
      "0.408523495",
      "1.473128823"
    ],
    [
      "0.22117577",
      "1.480669828"
    ],
    [
      "1.775667885",
      "1.485763495"
    ],
    [
      "1.608145052",
      "1.511974047"
    ],
    [
      "1.313591571",
      "1.518336231"
    ],
    [
      "1.019554883",
      "1.525102513"
    ],
    [
      "0.725155685",
      "1.532539001"
    ],
    [
      "1.462141563",
      "1.600090697"
    ],
    [
      "1.168104874",
      "1.606856979"
    ],
    [
      "0.870305927",
      "1.620187732"
    ],
    [
      "0.580079245",
      "1.620309835"
    ],
    [
      "0.329876968",
      "1.623347386"
    ],
    [
      "1.610873894",
      "1.681512969"
    ],
    [
      "1.317065122",
      "1.68786153"
    ],
    [
      "1.022649202",
      "1.694635158"
    ],
    [
      "0.725229487",
      "1.707958565"
    ],
    [
      "0.456367444",
      "1.736267944"
    ],
    [
      "1.171199194",
      "1.776389625"
    ],
    [
      "1.474054744",
      "1.782868293"
    ],
    [
      "0.601517686",
      "1.823916674"
    ],
    [
      "0.921570758",
      "1.830774942"
    ],
    [
      "1.320159441",
      "1.857394176"
    ],
    [
      "0.760345525",
      "1.883285103"
    ],
    [
      "1.070120749",
      "1.912529409"
    ]
  ],
  "radius": "0.08478044"
}
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": 114
}

起步布局答案

{
  "centers": [
    [
      "0.926922424",
      "0.08770262"
    ],
    [
      "1.09646934",
      "0.08987885"
    ],
    [
      "0.759883839",
      "0.116840292"
    ],
    [
      "1.262705021",
      "0.123294284"
    ],
    [
      "0.601087056",
      "0.176291736"
    ],
    [
      "1.419923555",
      "0.186801962"
    ],
    [
      "1.150648566",
      "0.250550889"
    ],
    [
      "0.853191907",
      "0.258418882"
    ],
    [
      "0.45598265",
      "0.264016329"
    ],
    [
      "1.562728539",
      "0.278222034"
    ],
    [
      "1.307042366",
      "0.316177522"
    ],
    [
      "0.69002847",
      "0.320676839"
    ],
    [
      "1.004070854",
      "0.335790944"
    ],
    [
      "0.329551219",
      "0.377002991"
    ],
    [
      "1.686218301",
      "0.394416575"
    ],
    [
      "0.537157399",
      "0.412883897"
    ],
    [
      "1.46106883",
      "0.413928321"
    ],
    [
      "1.151179763",
      "0.420110939"
    ],
    [
      "0.845687505",
      "0.427813619"
    ],
    [
      "1.30749667",
      "0.485806581"
    ],
    [
      "0.688039999",
      "0.490248836"
    ],
    [
      "0.999547668",
      "0.507868001"
    ],
    [
      "0.226132423",
      "0.511373544"
    ],
    [
      "0.410725967",
      "0.52587056"
    ],
    [
      "1.792188917",
      "0.54168407"
    ],
    [
      "1.61225462",
      "0.546995297"
    ],
    [
      "1.44653115",
      "0.582864843"
    ],
    [
      "1.149698687",
      "0.589665353"
    ],
    [
      "0.85529949",
      "0.597101842"
    ],
    [
      "0.561608567",
      "0.603235499"
    ],
    [
      "0.149276032",
      "0.662515821"
    ],
    [
      "1.298711944",
      "0.670573277"
    ],
    [
      "1.004622247",
      "0.677436186"
    ],
    [
      "0.710223049",
      "0.684872675"
    ],
    [
      "0.410252597",
      "0.695430781"
    ],
    [
      "1.69366027",
      "0.695736727"
    ],
    [
      "1.863221073",
      "0.695900969"
    ],
    [
      "1.447208926",
      "0.752424371"
    ],
    [
      "1.153172238",
      "0.759190652"
    ],
    [
      "0.858773041",
      "0.766627141"
    ],
    [
      "0.561135197",
      "0.77279572"
    ],
    [
      "0.266373046",
      "0.785150164"
    ],
    [
      "0.101620083",
      "0.825241981"
    ],
    [
      "1.595758918",
      "0.834178837"
    ],
    [
      "1.302132486",
      "0.840195204"
    ],
    [
      "1.008095798",
      "0.846961486"
    ],
    [
      "0.711933656",
      "0.854425584"
    ],
    [
      "1.904503941",
      "0.860359528"
    ],
    [
      "0.416012086",
      "0.864893818"
    ],
    [
      "1.744308909",
      "0.915933303"
    ],
    [
      "1.450682477",
      "0.92194967"
    ],
    [
      "1.156645789",
      "0.928715952"
    ],
    [
      "0.860556181",
      "0.936178647"
    ],
    [
      "0.566894686",
      "0.942258757"
    ],
    [
      "0.250393048",
      "0.955682634"
    ],
    [
      "0.084800328",
      "0.993966575"
    ],
    [
      "1.599232469",
      "1.003704137"
    ],
    [
      "1.305606037",
      "1.009720503"
    ],
    [
      "1.009878938",
      "1.016512992"
    ],
    [
      "0.715479741",
      "1.02394948"
    ],
    [
      "1.914740412",
      "1.029611138"
    ],
    [
      "0.401923053",
      "1.033868349"
    ],
    [
      "1.75454538",
      "1.085184913"
    ],
    [
      "1.454156028",
      "1.09147497"
    ],
    [
      "1.15842893",
      "1.098267458"
    ],
    [
      "0.864029732",
      "1.105703947"
    ],
    [
      "0.566722289",
      "1.111819552"
    ],
    [
      "0.266870952",
      "1.136395529"
    ],
    [
      "0.099394094",
      "1.162898262"
    ],
    [
      "1.60946894",
      "1.172955746"
    ],
    [
      "1.307389178",
      "1.179272009"
    ],
    [
      "1.013352489",
      "1.186038291"
    ],
    [
      "0.718208583",
      "1.193488402"
    ],
    [
      "1.893579127",
      "1.197846368"
    ],
    [
      "0.423188145",
      "1.202090489"
    ],
    [
      "1.455939169",
      "1.261026476"
    ],
    [
      "1.161902481",
      "1.267792757"
    ],
    [
      "0.866758575",
      "1.275242869"
    ],
    [
      "0.573132143",
      "1.281259236"
    ],
    [
      "1.772194334",
      "1.316238196"
    ],
    [
      "0.14490046",
      "1.326238592"
    ],
    [
      "0.314359375",
      "1.332118146"
    ],
    [
      "1.604671501",
      "1.342448747"
    ],
    [
      "1.310862729",
      "1.348797309"
    ],
    [
      "1.016081332",
      "1.355577213"
    ],
    [
      "0.721682134",
      "1.363013702"
    ],
    [
      "1.45941272",
      "1.430551775"
    ],
    [
      "1.164631323",
      "1.43733168"
    ],
    [
      "0.870232126",
      "1.444768168"
    ],
    [
      "0.576605694",
      "1.450784535"
    ],
    [
      "0.408523495",
      "1.473128823"
    ],
    [
      "0.22117577",
      "1.480669828"
    ],
    [
      "1.775667885",
      "1.485763495"
    ],
    [
      "1.608145052",
      "1.511974047"
    ],
    [
      "1.313591571",
      "1.518336231"
    ],
    [
      "1.019554883",
      "1.525102513"
    ],
    [
      "0.725155685",
      "1.532539001"
    ],
    [
      "1.462141563",
      "1.600090697"
    ],
    [
      "1.168104874",
      "1.606856979"
    ],
    [
      "0.870305927",
      "1.620187732"
    ],
    [
      "0.580079245",
      "1.620309835"
    ],
    [
      "0.329876968",
      "1.623347386"
    ],
    [
      "1.610873894",
      "1.681512969"
    ],
    [
      "1.317065122",
      "1.68786153"
    ],
    [
      "1.022649202",
      "1.694635158"
    ],
    [
      "0.725229487",
      "1.707958565"
    ],
    [
      "0.456367444",
      "1.736267944"
    ],
    [
      "1.171199194",
      "1.776389625"
    ],
    [
      "1.474054744",
      "1.782868293"
    ],
    [
      "0.601517686",
      "1.823916674"
    ],
    [
      "0.921570758",
      "1.830774942"
    ],
    [
      "1.320159441",
      "1.857394176"
    ],
    [
      "0.760345525",
      "1.883285103"
    ],
    [
      "1.070120749",
      "1.912529409"
    ]
  ],
  "radius": "0.08478044"
}

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

DISCUSSION

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