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 = 175

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 = 175
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[0.0687926980848295389432156154135, 0.0687926980848295389432156154145]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 = 0.068792697

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 = 175 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": [
    [
      "1.591754551",
      "0.280991244"
    ],
    [
      "1.727552498",
      "1.348603926"
    ],
    [
      "1.845335937",
      "0.611134105"
    ],
    [
      "1.424088062",
      "1.330234962"
    ],
    [
      "1.143034976",
      "0.857070062"
    ],
    [
      "1.078675107",
      "1.215352068"
    ],
    [
      "1.210346546",
      "0.737074672"
    ],
    [
      "1.491399632",
      "1.210239572"
    ],
    [
      "1.488447931",
      "0.971952956"
    ],
    [
      "1.356486594",
      "0.139730073"
    ],
    [
      "0.586352585",
      "1.614800907"
    ],
    [
      "1.213298247",
      "0.975361287"
    ],
    [
      "1.225838432",
      "0.096593092"
    ],
    [
      "1.515869689",
      "1.775258346"
    ],
    [
      "0.108945351",
      "1.270497046"
    ],
    [
      "1.347921388",
      "0.735370506"
    ],
    [
      "0.311317002",
      "1.626787657"
    ],
    [
      "1.494351333",
      "1.448526187"
    ],
    [
      "0.660047179",
      "0.743891334"
    ],
    [
      "0.800573722",
      "0.980473784"
    ],
    [
      "0.961062564",
      "0.120481759"
    ],
    [
      "0.588307909",
      "0.364571013"
    ],
    [
      "1.396007964",
      "1.842807648"
    ],
    [
      "1.148938378",
      "1.333643293"
    ],
    [
      "1.356776491",
      "1.450230353"
    ],
    [
      "0.589783908",
      "0.625600109"
    ],
    [
      "0.451068477",
      "1.589741949"
    ],
    [
      "0.797622021",
      "0.742187168"
    ],
    [
      "0.29231654",
      "1.366739344"
    ],
    [
      "1.283561519",
      "1.093652512"
    ],
    [
      "1.261285791",
      "1.715907643"
    ],
    [
      "1.15189008",
      "1.571929909"
    ],
    [
      "1.479352685",
      "0.201646668"
    ],
    [
      "1.26201643",
      "0.36911614"
    ],
    [
      "0.197860091",
      "1.067273872"
    ],
    [
      "0.999556732",
      "0.382200996"
    ],
    [
      "0.525311926",
      "0.983819156"
    ],
    [
      "1.861651015",
      "0.880837995"
    ],
    [
      "1.931207191",
      "0.999546375"
    ],
    [
      "0.662886768",
      "0.982114991"
    ],
    [
      "0.804937036",
      "1.456171785"
    ],
    [
      "0.448302063",
      "0.249815739"
    ],
    [
      "1.865080997",
      "1.344647511"
    ],
    [
      "1.76639426",
      "1.216615067"
    ],
    [
      "1.626022772",
      "0.970248791"
    ],
    [
      "0.671187316",
      "1.723119002"
    ],
    [
      "1.476799489",
      "0.356652348"
    ],
    [
      "1.128754406",
      "0.334896639"
    ],
    [
      "1.702469229",
      "0.601213083"
    ],
    [
      "1.289164421",
      "1.581176343"
    ],
    [
      "1.01572685",
      "1.811045461"
    ],
    [
      "1.072028475",
      "0.209549481"
    ],
    [
      "1.090024067",
      "1.926845568"
    ],
    [
      "0.734673764",
      "1.33788056"
    ],
    [
      "0.226479442",
      "1.518471778"
    ],
    [
      "0.158527765",
      "1.398837707"
    ],
    [
      "0.522360225",
      "0.74553254"
    ],
    [
      "0.695211532",
      "0.120084647"
    ],
    [
      "0.360268217",
      "1.486373415"
    ],
    [
      "0.667362194",
      "1.457875951"
    ],
    [
      "0.676979769",
      "0.256456726"
    ],
    [
      "0.315900366",
      "0.368213066"
    ],
    [
      "1.005460134",
      "0.858774228"
    ],
    [
      "0.642654624",
      "1.859913555"
    ],
    [
      "1.628810012",
      "1.217177065"
    ],
    [
      "0.078814579",
      "1.136251458"
    ],
    [
      "0.297122932",
      "0.617933877"
    ],
    [
      "1.83370583",
      "1.096619676"
    ],
    [
      "1.485496229",
      "0.733666341"
    ],
    [
      "1.277658116",
      "0.617079281"
    ],
    [
      "0.161454948",
      "0.595044157"
    ],
    [
      "1.28651322",
      "1.331939128"
    ],
    [
      "1.084466397",
      "1.691862341"
    ],
    [
      "1.5528078",
      "0.61367095"
    ],
    [
      "0.582679705",
      "1.349438592"
    ],
    [
      "0.873676582",
      "1.336988666"
    ],
    [
      "1.090260238",
      "0.073177402"
    ],
    [
      "0.455048655",
      "0.865527931"
    ],
    [
      "1.563785685",
      "1.567305866"
    ],
    [
      "1.216249949",
      "1.213647903"
    ],
    [
      "1.2280827",
      "0.234252759"
    ],
    [
      "0.875711447",
      "1.81134758"
    ],
    [
      "1.618074985",
      "0.49255144"
    ],
    [
      "0.932245162",
      "0.502196387"
    ],
    [
      "1.280609818",
      "0.855365897"
    ],
    [
      "1.21920165",
      "1.451934518"
    ],
    [
      "1.896379157",
      "0.747707618"
    ],
    [
      "1.140083275",
      "0.618783447"
    ],
    [
      "0.564860291",
      "0.176713614"
    ],
    [
      "1.696121582",
      "1.097181674"
    ],
    [
      "0.733150039",
      "1.100406216"
    ],
    [
      "0.862442864",
      "0.368511894"
    ],
    [
      "1.775632713",
      "0.48469336"
    ],
    [
      "1.008411835",
      "1.097060843"
    ],
    [
      "1.42113636",
      "1.091948347"
    ],
    [
      "1.80025528",
      "1.468913928"
    ],
    [
      "0.79467032",
      "0.503900553"
    ],
    [
      "1.558711202",
      "1.090244181"
    ],
    [
      "1.691238469",
      "0.376031717"
    ],
    [
      "0.515499614",
      "1.228808853"
    ],
    [
      "0.072412556",
      "0.997221522"
    ],
    [
      "0.773259649",
      "1.903180963"
    ],
    [
      "0.440446358",
      "1.374564619"
    ],
    [
      "0.592623496",
      "0.863823765"
    ],
    [
      "1.002508433",
      "0.620487612"
    ],
    [
      "0.110939103",
      "0.723020289"
    ],
    [
      "1.555759501",
      "0.851957566"
    ],
    [
      "0.72735875",
      "0.623895943"
    ],
    [
      "1.398332215",
      "1.703741331"
    ],
    [
      "1.343670832",
      "0.479851409"
    ],
    [
      "1.075723406",
      "0.977065453"
    ],
    [
      "0.935196863",
      "0.740483003"
    ],
    [
      "0.079831376",
      "0.857042864"
    ],
    [
      "1.760525039",
      "0.725949863"
    ],
    [
      "0.317473603",
      "0.863840829"
    ],
    [
      "1.267501421",
      "1.891958535"
    ],
    [
      "1.426210844",
      "1.569010031"
    ],
    [
      "0.411188449",
      "1.721420818"
    ],
    [
      "1.014203125",
      "1.573571116"
    ],
    [
      "1.072771704",
      "0.738778837"
    ],
    [
      "1.415232958",
      "0.615375116"
    ],
    [
      "1.699824991",
      "0.853981723"
    ],
    [
      "1.207394845",
      "0.498788056"
    ],
    [
      "0.366354959",
      "0.49621336"
    ],
    [
      "0.945974719",
      "1.929638805"
    ],
    [
      "0.875200307",
      "1.57446301"
    ],
    [
      "0.434689028",
      "0.615629424"
    ],
    [
      "0.377368627",
      "0.991436455"
    ],
    [
      "0.864933591",
      "0.622191778"
    ],
    [
      "0.458000356",
      "1.103814547"
    ],
    [
      "1.364298295",
      "0.277093527"
    ],
    [
      "1.145986677",
      "1.095356678"
    ],
    [
      "1.658753898",
      "1.467752966"
    ],
    [
      "0.870836994",
      "1.098765009"
    ],
    [
      "0.230276156",
      "0.475908173"
    ],
    [
      "0.730310451",
      "0.862182559"
    ],
    [
      "1.589319545",
      "1.348973287"
    ],
    [
      "1.719438237",
      "1.591232325"
    ],
    [
      "0.540893669",
      "1.767315232"
    ],
    [
      "1.069820003",
      "0.500492222"
    ],
    [
      "1.158763613",
      "1.807662448"
    ],
    [
      "0.724407048",
      "0.385609327"
    ],
    [
      "0.595575198",
      "1.102110381"
    ],
    [
      "1.35382479",
      "1.211943737"
    ],
    [
      "0.384785173",
      "0.743845439"
    ],
    [
      "0.322232741",
      "1.126105711"
    ],
    [
      "0.242734126",
      "1.238398682"
    ],
    [
      "0.196671482",
      "0.929693611"
    ],
    [
      "0.531246619",
      "1.477933153"
    ],
    [
      "1.906421027",
      "1.213419682"
    ],
    [
      "0.809958318",
      "0.221151674"
    ],
    [
      "0.720613445",
      "1.58473826"
    ],
    [
      "0.943939854",
      "1.455279891"
    ],
    [
      "0.519520637",
      "0.507308883"
    ],
    [
      "0.803413311",
      "1.218697441"
    ],
    [
      "1.081626808",
      "1.453638684"
    ],
    [
      "0.867885293",
      "0.860478393"
    ],
    [
      "0.805448176",
      "1.693056355"
    ],
    [
      "0.941100265",
      "1.217056234"
    ],
    [
      "0.942830801",
      "0.256853839"
    ],
    [
      "1.623071071",
      "0.731962175"
    ],
    [
      "1.763433152",
      "0.977186284"
    ],
    [
      "1.480500143",
      "0.494255605"
    ],
    [
      "0.653074456",
      "1.227104687"
    ],
    [
      "0.452148743",
      "0.387347351"
    ],
    [
      "1.350873089",
      "0.973657122"
    ],
    [
      "0.938148564",
      "0.978769618"
    ],
    [
      "1.011363537",
      "1.335347459"
    ],
    [
      "0.657095478",
      "0.505604718"
    ],
    [
      "0.828190081",
      "0.084779594"
    ],
    [
      "1.418184659",
      "0.853661731"
    ],
    [
      "1.624470025",
      "1.690785225"
    ],
    [
      "0.945463579",
      "1.692754235"
    ],
    [
      "0.379731999",
      "1.251100017"
    ],
    [
      "0.246607087",
      "0.74591001"
    ]
  ],
  "radius": "0.068792697"
}
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": 175
}

起步布局答案

{
  "centers": [
    [
      "1.591754551",
      "0.280991244"
    ],
    [
      "1.727552498",
      "1.348603926"
    ],
    [
      "1.845335937",
      "0.611134105"
    ],
    [
      "1.424088062",
      "1.330234962"
    ],
    [
      "1.143034976",
      "0.857070062"
    ],
    [
      "1.078675107",
      "1.215352068"
    ],
    [
      "1.210346546",
      "0.737074672"
    ],
    [
      "1.491399632",
      "1.210239572"
    ],
    [
      "1.488447931",
      "0.971952956"
    ],
    [
      "1.356486594",
      "0.139730073"
    ],
    [
      "0.586352585",
      "1.614800907"
    ],
    [
      "1.213298247",
      "0.975361287"
    ],
    [
      "1.225838432",
      "0.096593092"
    ],
    [
      "1.515869689",
      "1.775258346"
    ],
    [
      "0.108945351",
      "1.270497046"
    ],
    [
      "1.347921388",
      "0.735370506"
    ],
    [
      "0.311317002",
      "1.626787657"
    ],
    [
      "1.494351333",
      "1.448526187"
    ],
    [
      "0.660047179",
      "0.743891334"
    ],
    [
      "0.800573722",
      "0.980473784"
    ],
    [
      "0.961062564",
      "0.120481759"
    ],
    [
      "0.588307909",
      "0.364571013"
    ],
    [
      "1.396007964",
      "1.842807648"
    ],
    [
      "1.148938378",
      "1.333643293"
    ],
    [
      "1.356776491",
      "1.450230353"
    ],
    [
      "0.589783908",
      "0.625600109"
    ],
    [
      "0.451068477",
      "1.589741949"
    ],
    [
      "0.797622021",
      "0.742187168"
    ],
    [
      "0.29231654",
      "1.366739344"
    ],
    [
      "1.283561519",
      "1.093652512"
    ],
    [
      "1.261285791",
      "1.715907643"
    ],
    [
      "1.15189008",
      "1.571929909"
    ],
    [
      "1.479352685",
      "0.201646668"
    ],
    [
      "1.26201643",
      "0.36911614"
    ],
    [
      "0.197860091",
      "1.067273872"
    ],
    [
      "0.999556732",
      "0.382200996"
    ],
    [
      "0.525311926",
      "0.983819156"
    ],
    [
      "1.861651015",
      "0.880837995"
    ],
    [
      "1.931207191",
      "0.999546375"
    ],
    [
      "0.662886768",
      "0.982114991"
    ],
    [
      "0.804937036",
      "1.456171785"
    ],
    [
      "0.448302063",
      "0.249815739"
    ],
    [
      "1.865080997",
      "1.344647511"
    ],
    [
      "1.76639426",
      "1.216615067"
    ],
    [
      "1.626022772",
      "0.970248791"
    ],
    [
      "0.671187316",
      "1.723119002"
    ],
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      "1.476799489",
      "0.356652348"
    ],
    [
      "1.128754406",
      "0.334896639"
    ],
    [
      "1.702469229",
      "0.601213083"
    ],
    [
      "1.289164421",
      "1.581176343"
    ],
    [
      "1.01572685",
      "1.811045461"
    ],
    [
      "1.072028475",
      "0.209549481"
    ],
    [
      "1.090024067",
      "1.926845568"
    ],
    [
      "0.734673764",
      "1.33788056"
    ],
    [
      "0.226479442",
      "1.518471778"
    ],
    [
      "0.158527765",
      "1.398837707"
    ],
    [
      "0.522360225",
      "0.74553254"
    ],
    [
      "0.695211532",
      "0.120084647"
    ],
    [
      "0.360268217",
      "1.486373415"
    ],
    [
      "0.667362194",
      "1.457875951"
    ],
    [
      "0.676979769",
      "0.256456726"
    ],
    [
      "0.315900366",
      "0.368213066"
    ],
    [
      "1.005460134",
      "0.858774228"
    ],
    [
      "0.642654624",
      "1.859913555"
    ],
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  "radius": "0.068792697"
}

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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