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 = 126

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 = 126
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[0.0805338142529732300574007296545, 0.0805338142529732300574007296555]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 = 0.080533813

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 = 126 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": [
    [
      "0.50879152",
      "1.578752489"
    ],
    [
      "0.281363026",
      "1.57356688"
    ],
    [
      "0.250540298",
      "1.153241142"
    ],
    [
      "0.39767683",
      "1.462149287"
    ],
    [
      "1.0479894",
      "0.308069068"
    ],
    [
      "0.638548842",
      "1.6741769"
    ],
    [
      "1.081844518",
      "1.793342092"
    ],
    [
      "0.859935136",
      "1.43383635"
    ],
    [
      "1.145375751",
      "0.578864399"
    ],
    [
      "0.128023644",
      "1.291676708"
    ],
    [
      "1.386617498",
      "1.681836958"
    ],
    [
      "1.188314168",
      "1.158012433"
    ],
    [
      "0.809811938",
      "0.857735874"
    ],
    [
      "0.712831605",
      "0.596157668"
    ],
    [
      "0.659725484",
      "0.916189595"
    ],
    [
      "1.118354794",
      "0.737649313"
    ],
    [
      "0.560267662",
      "1.426132084"
    ],
    [
      "0.535724194",
      "0.813396303"
    ],
    [
      "0.836832895",
      "0.698950961"
    ],
    [
      "1.150272644",
      "0.092896809"
    ],
    [
      "1.574108354",
      "1.505808537"
    ],
    [
      "0.739852563",
      "0.437372754"
    ],
    [
      "0.797858385",
      "0.103476255"
    ],
    [
      "1.01177765",
      "1.376696246"
    ],
    [
      "1.856323944",
      "0.665145748"
    ],
    [
      "1.339336416",
      "1.102020812"
    ],
    [
      "1.767448873",
      "1.098756887"
    ],
    [
      "0.587165971",
      "0.495405847"
    ],
    [
      "0.715516959",
      "1.362518953"
    ],
    [
      "1.601129312",
      "1.347023623"
    ],
    [
      "0.192300669",
      "1.439363014"
    ],
    [
      "1.312315458",
      "1.260805725"
    ],
    [
      "1.23559525",
      "1.737828579"
    ],
    [
      "1.306258236",
      "0.13303751"
    ],
    [
      "1.784752512",
      "0.520853298"
    ],
    [
      "1.837127144",
      "1.380310678"
    ],
    [
      "1.532870922",
      "1.749310782"
    ],
    [
      "1.138607458",
      "1.476259372"
    ],
    [
      "1.199620887",
      "0.253748971"
    ],
    [
      "0.522235038",
      "1.785594493"
    ],
    [
      "0.821300582",
      "1.901933803"
    ],
    [
      "1.440993539",
      "1.364324352"
    ],
    [
      "1.517379622",
      "0.887244277"
    ],
    [
      "1.262616207",
      "1.579043666"
    ],
    [
      "0.886848126",
      "1.275033101"
    ],
    [
      "1.651720203",
      "0.430050293"
    ],
    [
      "1.544400579",
      "0.728459363"
    ],
    [
      "1.366357374",
      "0.943235898"
    ],
    [
      "1.641380912",
      "0.99003757"
    ],
    [
      "0.14571522",
      "0.659977088"
    ],
    [
      "0.409234372",
      "0.712661284"
    ],
    [
      "1.482453866",
      "0.415787078"
    ],
    [
      "1.205853268",
      "1.896126386"
    ],
    [
      "0.461400425",
      "0.394778772"
    ],
    [
      "0.288085708",
      "1.309871673"
    ],
    [
      "0.890874811",
      "0.381381133"
    ],
    [
      "0.782967019",
      "1.745494297"
    ],
    [
      "1.296397999",
      "0.522872778"
    ],
    [
      "0.989675735",
      "0.080591779"
    ],
    [
      "1.037291921",
      "1.214004054"
    ],
    [
      "1.614359954",
      "1.148822483"
    ],
    [
      "1.289971291",
      "1.420315973"
    ],
    [
      "1.901617933",
      "0.819713647"
    ],
    [
      "1.242356084",
      "0.840442606"
    ],
    [
      "0.488421383",
      "0.235993858"
    ],
    [
      "1.79446983",
      "0.939971973"
    ],
    [
      "0.623312301",
      "1.091004817"
    ],
    [
      "1.490358664",
      "1.046029191"
    ],
    [
      "1.166129517",
      "0.417548261"
    ],
    [
      "1.571421537",
      "0.56967445"
    ],
    [
      "0.473565002",
      "1.150321993"
    ],
    [
      "0.990838038",
      "0.639250847"
    ],
    [
      "1.91847112",
      "1.042765265"
    ],
    [
      "0.360531669",
      "1.035577727"
    ],
    [
      "0.748983497",
      "1.1917497"
    ],
    [
      "1.269377041",
      "0.681657692"
    ],
    [
      "0.863853853",
      "0.540166047"
    ],
    [
      "0.941145487",
      "1.714167283"
    ],
    [
      "1.317761004",
      "0.363228164"
    ],
    [
      "1.424398353",
      "0.242516703"
    ],
    [
      "1.017858996",
      "0.480465934"
    ],
    [
      "1.091333836",
      "0.896434227"
    ],
    [
      "0.614186929",
      "0.336620933"
    ],
    [
      "1.757918018",
      "1.520555804"
    ],
    [
      "0.080894195",
      "0.974259238"
    ],
    [
      "0.666653215",
      "1.85691189"
    ],
    [
      "0.449152972",
      "1.309528883"
    ],
    [
      "0.599236198",
      "1.251066876"
    ],
    [
      "0.273531234",
      "0.561967663"
    ],
    [
      "0.909775165",
      "1.115605588"
    ],
    [
      "0.629591159",
      "0.158444679"
    ],
    [
      "0.834443161",
      "1.592873893"
    ],
    [
      "0.984951091",
      "1.535514119"
    ],
    [
      "0.685810648",
      "0.754942582"
    ],
    [
      "0.090504377",
      "1.135039913"
    ],
    [
      "0.755356706",
      "0.259071755"
    ],
    [
      "1.363223053",
      "1.843454662"
    ],
    [
      "0.510278968",
      "0.976260551"
    ],
    [
      "0.434379467",
      "0.553563685"
    ],
    [
      "1.655451166",
      "1.644826981"
    ],
    [
      "0.300552192",
      "0.40318275"
    ],
    [
      "0.772758817",
      "1.030933861"
    ],
    [
      "0.099488002",
      "0.814268453"
    ],
    [
      "0.260033847",
      "0.773440881"
    ],
    [
      "1.579456354",
      "0.286103368"
    ],
    [
      "0.392477716",
      "1.690170081"
    ],
    [
      "1.734262492",
      "1.256368577"
    ],
    [
      "0.98143158",
      "1.919278674"
    ],
    [
      "0.386277678",
      "0.873467258"
    ],
    [
      "0.936796123",
      "0.956820675"
    ],
    [
      "1.413972581",
      "1.523109266"
    ],
    [
      "0.690024984",
      "1.521556496"
    ],
    [
      "1.695422827",
      "0.672467742"
    ],
    [
      "1.420399289",
      "0.625666071"
    ],
    [
      "1.215335126",
      "0.999227519"
    ],
    [
      "1.1115865",
      "1.635044286"
    ],
    [
      "0.96381708",
      "0.798035761"
    ],
    [
      "1.673616805",
      "0.832054129"
    ],
    [
      "1.064312878",
      "1.05521914"
    ],
    [
      "0.236530379",
      "0.932784434"
    ],
    [
      "1.161293211",
      "1.316797347"
    ],
    [
      "1.463337706",
      "1.204814104"
    ],
    [
      "0.560145014",
      "0.654190761"
    ],
    [
      "0.912030899",
      "0.221708966"
    ],
    [
      "1.892744903",
      "1.218223787"
    ],
    [
      "1.393378331",
      "0.784450985"
    ]
  ],
  "radius": "0.080533813"
}
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": 126
}

起步布局答案

{
  "centers": [
    [
      "0.50879152",
      "1.578752489"
    ],
    [
      "0.281363026",
      "1.57356688"
    ],
    [
      "0.250540298",
      "1.153241142"
    ],
    [
      "0.39767683",
      "1.462149287"
    ],
    [
      "1.0479894",
      "0.308069068"
    ],
    [
      "0.638548842",
      "1.6741769"
    ],
    [
      "1.081844518",
      "1.793342092"
    ],
    [
      "0.859935136",
      "1.43383635"
    ],
    [
      "1.145375751",
      "0.578864399"
    ],
    [
      "0.128023644",
      "1.291676708"
    ],
    [
      "1.386617498",
      "1.681836958"
    ],
    [
      "1.188314168",
      "1.158012433"
    ],
    [
      "0.809811938",
      "0.857735874"
    ],
    [
      "0.712831605",
      "0.596157668"
    ],
    [
      "0.659725484",
      "0.916189595"
    ],
    [
      "1.118354794",
      "0.737649313"
    ],
    [
      "0.560267662",
      "1.426132084"
    ],
    [
      "0.535724194",
      "0.813396303"
    ],
    [
      "0.836832895",
      "0.698950961"
    ],
    [
      "1.150272644",
      "0.092896809"
    ],
    [
      "1.574108354",
      "1.505808537"
    ],
    [
      "0.739852563",
      "0.437372754"
    ],
    [
      "0.797858385",
      "0.103476255"
    ],
    [
      "1.01177765",
      "1.376696246"
    ],
    [
      "1.856323944",
      "0.665145748"
    ],
    [
      "1.339336416",
      "1.102020812"
    ],
    [
      "1.767448873",
      "1.098756887"
    ],
    [
      "0.587165971",
      "0.495405847"
    ],
    [
      "0.715516959",
      "1.362518953"
    ],
    [
      "1.601129312",
      "1.347023623"
    ],
    [
      "0.192300669",
      "1.439363014"
    ],
    [
      "1.312315458",
      "1.260805725"
    ],
    [
      "1.23559525",
      "1.737828579"
    ],
    [
      "1.306258236",
      "0.13303751"
    ],
    [
      "1.784752512",
      "0.520853298"
    ],
    [
      "1.837127144",
      "1.380310678"
    ],
    [
      "1.532870922",
      "1.749310782"
    ],
    [
      "1.138607458",
      "1.476259372"
    ],
    [
      "1.199620887",
      "0.253748971"
    ],
    [
      "0.522235038",
      "1.785594493"
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      "1.901933803"
    ],
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      "1.364324352"
    ],
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      "1.517379622",
      "0.887244277"
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      "1.579043666"
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      "1.275033101"
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      "1.651720203",
      "0.430050293"
    ],
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      "1.544400579",
      "0.728459363"
    ],
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      "0.943235898"
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      "0.14571522",
      "0.659977088"
    ],
    [
      "0.409234372",
      "0.712661284"
    ],
    [
      "1.482453866",
      "0.415787078"
    ],
    [
      "1.205853268",
      "1.896126386"
    ],
    [
      "0.461400425",
      "0.394778772"
    ],
    [
      "0.288085708",
      "1.309871673"
    ],
    [
      "0.890874811",
      "0.381381133"
    ],
    [
      "0.782967019",
      "1.745494297"
    ],
    [
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      "0.522872778"
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      "0.080591779"
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      "1.85691189"
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      "1.309528883"
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      "1.251066876"
    ],
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      "0.561967663"
    ],
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      "1.115605588"
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      "0.158444679"
    ],
    [
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      "1.592873893"
    ],
    [
      "0.984951091",
      "1.535514119"
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      "0.754942582"
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      "1.135039913"
    ],
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      "0.259071755"
    ],
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      "1.843454662"
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      "0.976260551"
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      "0.553563685"
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      "0.40318275"
    ],
    [
      "0.772758817",
      "1.030933861"
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      "0.814268453"
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      "0.260033847",
      "0.773440881"
    ],
    [
      "1.579456354",
      "0.286103368"
    ],
    [
      "0.392477716",
      "1.690170081"
    ],
    [
      "1.734262492",
      "1.256368577"
    ],
    [
      "0.98143158",
      "1.919278674"
    ],
    [
      "0.386277678",
      "0.873467258"
    ],
    [
      "0.936796123",
      "0.956820675"
    ],
    [
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      "1.523109266"
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    [
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    ],
    [
      "1.215335126",
      "0.999227519"
    ],
    [
      "1.1115865",
      "1.635044286"
    ],
    [
      "0.96381708",
      "0.798035761"
    ],
    [
      "1.673616805",
      "0.832054129"
    ],
    [
      "1.064312878",
      "1.05521914"
    ],
    [
      "0.236530379",
      "0.932784434"
    ],
    [
      "1.161293211",
      "1.316797347"
    ],
    [
      "1.463337706",
      "1.204814104"
    ],
    [
      "0.560145014",
      "0.654190761"
    ],
    [
      "0.912030899",
      "0.221708966"
    ],
    [
      "1.892744903",
      "1.218223787"
    ],
    [
      "1.393378331",
      "0.784450985"
    ]
  ],
  "radius": "0.080533813"
}

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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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?