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 = 177

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 = 177
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[0.0684251080473629769641274764615, 0.0684251080473629769641274764625]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 = 0.068425107

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 = 177 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": [
    [
      "0.486810165",
      "1.464198261"
    ],
    [
      "0.86310806",
      "0.69775881"
    ],
    [
      "0.719782502",
      "1.029241192"
    ],
    [
      "0.996931739",
      "0.726380562"
    ],
    [
      "0.628083498",
      "1.130825021"
    ],
    [
      "1.700631007",
      "1.362126596"
    ],
    [
      "1.929625885",
      "0.939771322"
    ],
    [
      "0.189558811",
      "0.540634287"
    ],
    [
      "1.157980783",
      "1.609197879"
    ],
    [
      "1.286820332",
      "1.886321542"
    ],
    [
      "1.273695472",
      "0.373763285"
    ],
    [
      "0.635533153",
      "0.768830029"
    ],
    [
      "1.433077795",
      "1.304446392"
    ],
    [
      "1.348828445",
      "1.044035229"
    ],
    [
      "1.415990675",
      "0.166463236"
    ],
    [
      "0.796582197",
      "1.651647347"
    ],
    [
      "0.662631612",
      "1.623612663"
    ],
    [
      "0.578509169",
      "1.362614432"
    ],
    [
      "1.172880092",
      "0.885207895"
    ],
    [
      "0.536384494",
      "1.23240885"
    ],
    [
      "0.501709474",
      "0.740208277"
    ],
    [
      "1.037322962",
      "1.930826931"
    ],
    [
      "0.52893484",
      "1.594403842"
    ],
    [
      "0.463253657",
      "0.238596597"
    ],
    [
      "1.249679787",
      "1.50761405"
    ],
    [
      "1.554022542",
      "1.588583681"
    ],
    [
      "0.551283803",
      "0.508418867"
    ],
    [
      "1.658600477",
      "1.231484315"
    ],
    [
      "0.971684642",
      "1.810745323"
    ],
    [
      "1.264579096",
      "0.783624066"
    ],
    [
      "0.257624462",
      "1.562770237"
    ],
    [
      "0.930405876",
      "1.680269099"
    ],
    [
      "1.088630743",
      "0.624796733"
    ],
    [
      "0.402560816",
      "1.203787098"
    ],
    [
      "1.79233001",
      "1.260542767"
    ],
    [
      "1.341378791",
      "1.406030221"
    ],
    [
      "1.238448445",
      "0.241530049"
    ],
    [
      "0.32561627",
      "1.090616791"
    ],
    [
      "1.081181088",
      "0.986791725"
    ],
    [
      "1.732040536",
      "0.423850338"
    ],
    [
      "1.440527449",
      "0.9424514"
    ],
    [
      "0.804031852",
      "1.289652355"
    ],
    [
      "0.316463903",
      "0.954072967"
    ],
    [
      "0.751041214",
      "0.102307791"
    ],
    [
      "1.604142139",
      "0.472734913"
    ],
    [
      "0.754457522",
      "1.521441765"
    ],
    [
      "0.226450368",
      "1.184925356"
    ],
    [
      "0.585958824",
      "1.00061944"
    ],
    [
      "1.450417532",
      "0.564486719"
    ],
    [
      "0.761907177",
      "1.159446774"
    ],
    [
      "1.257129442",
      "1.145619058"
    ],
    [
      "1.314153425",
      "0.551834656"
    ],
    [
      "0.729284382",
      "0.669137057"
    ],
    [
      "1.524776799",
      "1.202862563"
    ],
    [
      "0.897828748",
      "1.925955082"
    ],
    [
      "0.677657827",
      "0.899035611"
    ],
    [
      "1.306703771",
      "0.913829648"
    ],
    [
      "1.808550664",
      "0.537314791"
    ],
    [
      "1.782420437",
      "1.505618472"
    ],
    [
      "0.834982117",
      "1.804389122"
    ],
    [
      "0.387438399",
      "1.701854732"
    ],
    [
      "0.394453381",
      "1.565184429"
    ],
    [
      "1.033323313",
      "0.069021302"
    ],
    [
      "1.867612109",
      "0.660764082"
    ],
    [
      "0.310861812",
      "1.305370927"
    ],
    [
      "1.180329747",
      "0.523212904"
    ],
    [
      "0.454571965",
      "0.605835457"
    ],
    [
      "1.299254116",
      "1.27582464"
    ],
    [
      "1.574351128",
      "0.971073152"
    ],
    [
      "0.897783081",
      "1.189959383"
    ],
    [
      "0.896570537",
      "0.074184616"
    ],
    [
      "0.98203243",
      "1.450370546"
    ],
    [
      "1.848053775",
      "1.385534141"
    ],
    [
      "0.236506282",
      "0.751487045"
    ],
    [
      "1.389981567",
      "0.301068391"
    ],
    [
      "1.207555112",
      "1.377408469"
    ],
    [
      "1.438301461",
      "1.661635808"
    ],
    [
      "0.982387129",
      "0.196038965"
    ],
    [
      "0.855658406",
      "1.059753801"
    ],
    [
      "1.532226453",
      "0.840867571"
    ],
    [
      "0.729009477",
      "1.891171963"
    ],
    [
      "1.511836586",
      "0.371507428"
    ],
    [
      "0.771409056",
      "0.799342639"
    ],
    [
      "1.490101778",
      "0.710661989"
    ],
    [
      "0.174143802",
      "1.311384855"
    ],
    [
      "1.075022999",
      "1.718129422"
    ],
    [
      "1.102958045",
      "0.260774086"
    ],
    [
      "1.00538514",
      "0.359317201"
    ],
    [
      "0.35749565",
      "0.325448342"
    ],
    [
      "1.773147557",
      "0.759781404"
    ],
    [
      "1.580262434",
      "0.607710361"
    ],
    [
      "1.046506068",
      "0.494591151"
    ],
    [
      "0.70723654",
      "1.755307056"
    ],
    [
      "0.912682389",
      "0.465969399"
    ],
    [
      "0.870557715",
      "0.335763818"
    ],
    [
      "1.53361956",
      "0.236401974"
    ],
    [
      "0.888281201",
      "1.550063518"
    ],
    [
      "0.592984726",
      "0.282160705"
    ],
    [
      "1.393404898",
      "0.440077952"
    ],
    [
      "0.395213256",
      "0.470779327"
    ],
    [
      "0.090779947",
      "1.20285629"
    ],
    [
      "0.905232735",
      "0.827964391"
    ],
    [
      "1.073731434",
      "1.348786717"
    ],
    [
      "1.289384629",
      "0.114512385"
    ],
    [
      "1.710730316",
      "0.99887598"
    ],
    [
      "1.511897867",
      "1.4583781"
    ],
    [
      "0.813533731",
      "0.92954822"
    ],
    [
      "0.521142058",
      "1.731032003"
    ],
    [
      "1.031606759",
      "1.218581135"
    ],
    [
      "0.645035032",
      "0.408725895"
    ],
    [
      "0.846156526",
      "1.419857936"
    ],
    [
      "0.622210933",
      "0.148467734"
    ],
    [
      "0.325168255",
      "1.443187661"
    ],
    [
      "1.616475803",
      "1.101278734"
    ],
    [
      "0.601304384",
      "1.841946308"
    ],
    [
      "0.620633844",
      "1.492820013"
    ],
    [
      "1.39095312",
      "1.174240811"
    ],
    [
      "1.800719124",
      "0.89382539"
    ],
    [
      "1.666050132",
      "0.869489323"
    ],
    [
      "1.906380712",
      "0.793617204"
    ],
    [
      "1.215004767",
      "1.015413477"
    ],
    [
      "0.202634776",
      "1.030479093"
    ],
    [
      "1.222454421",
      "0.653418485"
    ],
    [
      "0.399822787",
      "0.845540585"
    ],
    [
      "0.452135145",
      "0.971997688"
    ],
    [
      "0.265603014",
      "0.426857045"
    ],
    [
      "1.138205072",
      "0.393007322"
    ],
    [
      "0.820983386",
      "0.567553228"
    ],
    [
      "0.736683308",
      "0.286482085"
    ],
    [
      "0.49425982",
      "1.102203269"
    ],
    [
      "1.024157105",
      "1.580576127"
    ],
    [
      "1.911359798",
      "1.193015799"
    ],
    [
      "1.212778779",
      "1.734597885"
    ],
    [
      "0.670208173",
      "1.261030603"
    ],
    [
      "0.954807064",
      "0.59617498"
    ],
    [
      "1.039056414",
      "0.856586143"
    ],
    [
      "1.398402775",
      "0.812245819"
    ],
    [
      "0.778858711",
      "0.437347647"
    ],
    [
      "1.482652124",
      "1.072656981"
    ],
    [
      "0.845634352",
      "0.201202279"
    ],
    [
      "0.939907755",
      "1.320164964"
    ],
    [
      "0.94735741",
      "0.958169972"
    ],
    [
      "0.712332848",
      "1.391236184"
    ],
    [
      "1.304477783",
      "1.633014055"
    ],
    [
      "0.367690332",
      "0.7125162"
    ],
    [
      "0.989482084",
      "1.088375554"
    ],
    [
      "0.444685491",
      "1.33399268"
    ],
    [
      "1.346602458",
      "1.763219637"
    ],
    [
      "1.566901473",
      "1.333068144"
    ],
    [
      "0.070871021",
      "1.067461991"
    ],
    [
      "0.091202471",
      "0.795259115"
    ],
    [
      "1.165430438",
      "1.247202887"
    ],
    [
      "1.590558896",
      "1.720466494"
    ],
    [
      "1.152996654",
      "1.85769979"
    ],
    [
      "1.639732829",
      "0.32281931"
    ],
    [
      "1.623925457",
      "0.739283742"
    ],
    [
      "1.478634624",
      "1.799212535"
    ],
    [
      "1.115856109",
      "1.478992298"
    ],
    [
      "0.543834149",
      "0.870413859"
    ],
    [
      "1.714086112",
      "0.636332113"
    ],
    [
      "1.130755417",
      "0.755002314"
    ],
    [
      "0.071012883",
      "0.930611848"
    ],
    [
      "0.131004089",
      "0.664324732"
    ],
    [
      "0.183186005",
      "1.447936018"
    ],
    [
      "1.153894229",
      "0.133756423"
    ],
    [
      "0.499233496",
      "0.381853677"
    ],
    [
      "0.319169053",
      "0.584556569"
    ],
    [
      "1.693424958",
      "1.621941282"
    ],
    [
      "1.3562781",
      "0.682040237"
    ],
    [
      "1.396176787",
      "1.531430226"
    ],
    [
      "1.123305763",
      "1.116997306"
    ],
    [
      "0.593408478",
      "0.638624448"
    ],
    [
      "1.750299481",
      "1.129900486"
    ],
    [
      "1.86952361",
      "1.062717238"
    ],
    [
      "1.646720698",
      "1.487910777"
    ],
    [
      "0.199390677",
      "0.883208013"
    ],
    [
      "0.687159707",
      "0.538931476"
    ]
  ],
  "radius": "0.068425107"
}
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": 177
}

起步布局答案

{
  "centers": [
    [
      "0.486810165",
      "1.464198261"
    ],
    [
      "0.86310806",
      "0.69775881"
    ],
    [
      "0.719782502",
      "1.029241192"
    ],
    [
      "0.996931739",
      "0.726380562"
    ],
    [
      "0.628083498",
      "1.130825021"
    ],
    [
      "1.700631007",
      "1.362126596"
    ],
    [
      "1.929625885",
      "0.939771322"
    ],
    [
      "0.189558811",
      "0.540634287"
    ],
    [
      "1.157980783",
      "1.609197879"
    ],
    [
      "1.286820332",
      "1.886321542"
    ],
    [
      "1.273695472",
      "0.373763285"
    ],
    [
      "0.635533153",
      "0.768830029"
    ],
    [
      "1.433077795",
      "1.304446392"
    ],
    [
      "1.348828445",
      "1.044035229"
    ],
    [
      "1.415990675",
      "0.166463236"
    ],
    [
      "0.796582197",
      "1.651647347"
    ],
    [
      "0.662631612",
      "1.623612663"
    ],
    [
      "0.578509169",
      "1.362614432"
    ],
    [
      "1.172880092",
      "0.885207895"
    ],
    [
      "0.536384494",
      "1.23240885"
    ],
    [
      "0.501709474",
      "0.740208277"
    ],
    [
      "1.037322962",
      "1.930826931"
    ],
    [
      "0.52893484",
      "1.594403842"
    ],
    [
      "0.463253657",
      "0.238596597"
    ],
    [
      "1.249679787",
      "1.50761405"
    ],
    [
      "1.554022542",
      "1.588583681"
    ],
    [
      "0.551283803",
      "0.508418867"
    ],
    [
      "1.658600477",
      "1.231484315"
    ],
    [
      "0.971684642",
      "1.810745323"
    ],
    [
      "1.264579096",
      "0.783624066"
    ],
    [
      "0.257624462",
      "1.562770237"
    ],
    [
      "0.930405876",
      "1.680269099"
    ],
    [
      "1.088630743",
      "0.624796733"
    ],
    [
      "0.402560816",
      "1.203787098"
    ],
    [
      "1.79233001",
      "1.260542767"
    ],
    [
      "1.341378791",
      "1.406030221"
    ],
    [
      "1.238448445",
      "0.241530049"
    ],
    [
      "0.32561627",
      "1.090616791"
    ],
    [
      "1.081181088",
      "0.986791725"
    ],
    [
      "1.732040536",
      "0.423850338"
    ],
    [
      "1.440527449",
      "0.9424514"
    ],
    [
      "0.804031852",
      "1.289652355"
    ],
    [
      "0.316463903",
      "0.954072967"
    ],
    [
      "0.751041214",
      "0.102307791"
    ],
    [
      "1.604142139",
      "0.472734913"
    ],
    [
      "0.754457522",
      "1.521441765"
    ],
    [
      "0.226450368",
      "1.184925356"
    ],
    [
      "0.585958824",
      "1.00061944"
    ],
    [
      "1.450417532",
      "0.564486719"
    ],
    [
      "0.761907177",
      "1.159446774"
    ],
    [
      "1.257129442",
      "1.145619058"
    ],
    [
      "1.314153425",
      "0.551834656"
    ],
    [
      "0.729284382",
      "0.669137057"
    ],
    [
      "1.524776799",
      "1.202862563"
    ],
    [
      "0.897828748",
      "1.925955082"
    ],
    [
      "0.677657827",
      "0.899035611"
    ],
    [
      "1.306703771",
      "0.913829648"
    ],
    [
      "1.808550664",
      "0.537314791"
    ],
    [
      "1.782420437",
      "1.505618472"
    ],
    [
      "0.834982117",
      "1.804389122"
    ],
    [
      "0.387438399",
      "1.701854732"
    ],
    [
      "0.394453381",
      "1.565184429"
    ],
    [
      "1.033323313",
      "0.069021302"
    ],
    [
      "1.867612109",
      "0.660764082"
    ],
    [
      "0.310861812",
      "1.305370927"
    ],
    [
      "1.180329747",
      "0.523212904"
    ],
    [
      "0.454571965",
      "0.605835457"
    ],
    [
      "1.299254116",
      "1.27582464"
    ],
    [
      "1.574351128",
      "0.971073152"
    ],
    [
      "0.897783081",
      "1.189959383"
    ],
    [
      "0.896570537",
      "0.074184616"
    ],
    [
      "0.98203243",
      "1.450370546"
    ],
    [
      "1.848053775",
      "1.385534141"
    ],
    [
      "0.236506282",
      "0.751487045"
    ],
    [
      "1.389981567",
      "0.301068391"
    ],
    [
      "1.207555112",
      "1.377408469"
    ],
    [
      "1.438301461",
      "1.661635808"
    ],
    [
      "0.982387129",
      "0.196038965"
    ],
    [
      "0.855658406",
      "1.059753801"
    ],
    [
      "1.532226453",
      "0.840867571"
    ],
    [
      "0.729009477",
      "1.891171963"
    ],
    [
      "1.511836586",
      "0.371507428"
    ],
    [
      "0.771409056",
      "0.799342639"
    ],
    [
      "1.490101778",
      "0.710661989"
    ],
    [
      "0.174143802",
      "1.311384855"
    ],
    [
      "1.075022999",
      "1.718129422"
    ],
    [
      "1.102958045",
      "0.260774086"
    ],
    [
      "1.00538514",
      "0.359317201"
    ],
    [
      "0.35749565",
      "0.325448342"
    ],
    [
      "1.773147557",
      "0.759781404"
    ],
    [
      "1.580262434",
      "0.607710361"
    ],
    [
      "1.046506068",
      "0.494591151"
    ],
    [
      "0.70723654",
      "1.755307056"
    ],
    [
      "0.912682389",
      "0.465969399"
    ],
    [
      "0.870557715",
      "0.335763818"
    ],
    [
      "1.53361956",
      "0.236401974"
    ],
    [
      "0.888281201",
      "1.550063518"
    ],
    [
      "0.592984726",
      "0.282160705"
    ],
    [
      "1.393404898",
      "0.440077952"
    ],
    [
      "0.395213256",
      "0.470779327"
    ],
    [
      "0.090779947",
      "1.20285629"
    ],
    [
      "0.905232735",
      "0.827964391"
    ],
    [
      "1.073731434",
      "1.348786717"
    ],
    [
      "1.289384629",
      "0.114512385"
    ],
    [
      "1.710730316",
      "0.99887598"
    ],
    [
      "1.511897867",
      "1.4583781"
    ],
    [
      "0.813533731",
      "0.92954822"
    ],
    [
      "0.521142058",
      "1.731032003"
    ],
    [
      "1.031606759",
      "1.218581135"
    ],
    [
      "0.645035032",
      "0.408725895"
    ],
    [
      "0.846156526",
      "1.419857936"
    ],
    [
      "0.622210933",
      "0.148467734"
    ],
    [
      "0.325168255",
      "1.443187661"
    ],
    [
      "1.616475803",
      "1.101278734"
    ],
    [
      "0.601304384",
      "1.841946308"
    ],
    [
      "0.620633844",
      "1.492820013"
    ],
    [
      "1.39095312",
      "1.174240811"
    ],
    [
      "1.800719124",
      "0.89382539"
    ],
    [
      "1.666050132",
      "0.869489323"
    ],
    [
      "1.906380712",
      "0.793617204"
    ],
    [
      "1.215004767",
      "1.015413477"
    ],
    [
      "0.202634776",
      "1.030479093"
    ],
    [
      "1.222454421",
      "0.653418485"
    ],
    [
      "0.399822787",
      "0.845540585"
    ],
    [
      "0.452135145",
      "0.971997688"
    ],
    [
      "0.265603014",
      "0.426857045"
    ],
    [
      "1.138205072",
      "0.393007322"
    ],
    [
      "0.820983386",
      "0.567553228"
    ],
    [
      "0.736683308",
      "0.286482085"
    ],
    [
      "0.49425982",
      "1.102203269"
    ],
    [
      "1.024157105",
      "1.580576127"
    ],
    [
      "1.911359798",
      "1.193015799"
    ],
    [
      "1.212778779",
      "1.734597885"
    ],
    [
      "0.670208173",
      "1.261030603"
    ],
    [
      "0.954807064",
      "0.59617498"
    ],
    [
      "1.039056414",
      "0.856586143"
    ],
    [
      "1.398402775",
      "0.812245819"
    ],
    [
      "0.778858711",
      "0.437347647"
    ],
    [
      "1.482652124",
      "1.072656981"
    ],
    [
      "0.845634352",
      "0.201202279"
    ],
    [
      "0.939907755",
      "1.320164964"
    ],
    [
      "0.94735741",
      "0.958169972"
    ],
    [
      "0.712332848",
      "1.391236184"
    ],
    [
      "1.304477783",
      "1.633014055"
    ],
    [
      "0.367690332",
      "0.7125162"
    ],
    [
      "0.989482084",
      "1.088375554"
    ],
    [
      "0.444685491",
      "1.33399268"
    ],
    [
      "1.346602458",
      "1.763219637"
    ],
    [
      "1.566901473",
      "1.333068144"
    ],
    [
      "0.070871021",
      "1.067461991"
    ],
    [
      "0.091202471",
      "0.795259115"
    ],
    [
      "1.165430438",
      "1.247202887"
    ],
    [
      "1.590558896",
      "1.720466494"
    ],
    [
      "1.152996654",
      "1.85769979"
    ],
    [
      "1.639732829",
      "0.32281931"
    ],
    [
      "1.623925457",
      "0.739283742"
    ],
    [
      "1.478634624",
      "1.799212535"
    ],
    [
      "1.115856109",
      "1.478992298"
    ],
    [
      "0.543834149",
      "0.870413859"
    ],
    [
      "1.714086112",
      "0.636332113"
    ],
    [
      "1.130755417",
      "0.755002314"
    ],
    [
      "0.071012883",
      "0.930611848"
    ],
    [
      "0.131004089",
      "0.664324732"
    ],
    [
      "0.183186005",
      "1.447936018"
    ],
    [
      "1.153894229",
      "0.133756423"
    ],
    [
      "0.499233496",
      "0.381853677"
    ],
    [
      "0.319169053",
      "0.584556569"
    ],
    [
      "1.693424958",
      "1.621941282"
    ],
    [
      "1.3562781",
      "0.682040237"
    ],
    [
      "1.396176787",
      "1.531430226"
    ],
    [
      "1.123305763",
      "1.116997306"
    ],
    [
      "0.593408478",
      "0.638624448"
    ],
    [
      "1.750299481",
      "1.129900486"
    ],
    [
      "1.86952361",
      "1.062717238"
    ],
    [
      "1.646720698",
      "1.487910777"
    ],
    [
      "0.199390677",
      "0.883208013"
    ],
    [
      "0.687159707",
      "0.538931476"
    ]
  ],
  "radius": "0.068425107"
}

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

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖;发言公开署名,与纪录使用同一个名字,署名后的 #编号是账号注册序号,冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路?