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 = 171

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 = 171
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[0.0696007491301596809590448544105, 0.0696007491301596809590448544115]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 = 0.069600748

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 = 171 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": [
    [
      "0.962297463",
      "0.070364972"
    ],
    [
      "0.824022004",
      "0.086394773"
    ],
    [
      "1.227011411",
      "0.09772034"
    ],
    [
      "0.68968574",
      "0.122875251"
    ],
    [
      "1.090451756",
      "0.124711403"
    ],
    [
      "1.460824078",
      "0.191740181"
    ],
    [
      "0.533567473",
      "0.194963688"
    ],
    [
      "1.321845775",
      "0.199619803"
    ],
    [
      "0.898997279",
      "0.203679747"
    ],
    [
      "1.179983901",
      "0.232955702"
    ],
    [
      "1.030196722",
      "0.250196016"
    ],
    [
      "0.657520513",
      "0.258309572"
    ],
    [
      "1.576255192",
      "0.269539378"
    ],
    [
      "0.410813982",
      "0.280898777"
    ],
    [
      "0.792532036",
      "0.295708638"
    ],
    [
      "1.434060124",
      "0.328344529"
    ],
    [
      "0.540684849",
      "0.333983111"
    ],
    [
      "1.295081821",
      "0.336224151"
    ],
    [
      "0.924312731",
      "0.340559932"
    ],
    [
      "1.67878707",
      "0.363689636"
    ],
    [
      "0.303695243",
      "0.382908069"
    ],
    [
      "1.059240887",
      "0.38633379"
    ],
    [
      "0.691016536",
      "0.393420909"
    ],
    [
      "1.547279768",
      "0.409328288"
    ],
    [
      "0.436106184",
      "0.425854544"
    ],
    [
      "1.191626387",
      "0.429358625"
    ],
    [
      "0.823402036",
      "0.436445744"
    ],
    [
      "0.574180871",
      "0.469094449"
    ],
    [
      "1.766124579",
      "0.472083439"
    ],
    [
      "1.324612241",
      "0.472257277"
    ],
    [
      "0.955787537",
      "0.479470579"
    ],
    [
      "0.219421102",
      "0.493700336"
    ],
    [
      "1.456997741",
      "0.515282112"
    ],
    [
      "1.088173037",
      "0.522495414"
    ],
    [
      "0.719948686",
      "0.529582533"
    ],
    [
      "0.351832044",
      "0.53664681"
    ],
    [
      "1.59694997",
      "0.539863855"
    ],
    [
      "1.220558537",
      "0.565520249"
    ],
    [
      "0.852334186",
      "0.572607368"
    ],
    [
      "0.48529044",
      "0.576218435"
    ],
    [
      "1.724063911",
      "0.604778418"
    ],
    [
      "1.352944038",
      "0.608545084"
    ],
    [
      "0.984719687",
      "0.615632203"
    ],
    [
      "0.616495336",
      "0.622719322"
    ],
    [
      "0.248245544",
      "0.629884801"
    ],
    [
      "1.858422348",
      "0.641177149"
    ],
    [
      "1.485329538",
      "0.651569919"
    ],
    [
      "1.117105187",
      "0.658657038"
    ],
    [
      "0.748880836",
      "0.665744157"
    ],
    [
      "0.380656485",
      "0.672831276"
    ],
    [
      "0.122734319",
      "0.690084382"
    ],
    [
      "1.617715038",
      "0.694594754"
    ],
    [
      "1.249490687",
      "0.701681873"
    ],
    [
      "0.881266336",
      "0.708768992"
    ],
    [
      "0.513041986",
      "0.715856111"
    ],
    [
      "1.752996061",
      "0.740940042"
    ],
    [
      "1.381876188",
      "0.744706708"
    ],
    [
      "1.013651837",
      "0.751793827"
    ],
    [
      "0.645427486",
      "0.758880946"
    ],
    [
      "0.277177694",
      "0.766046425"
    ],
    [
      "1.902349416",
      "0.77326602"
    ],
    [
      "1.514261688",
      "0.787731543"
    ],
    [
      "1.146037337",
      "0.794818662"
    ],
    [
      "0.777812986",
      "0.801905781"
    ],
    [
      "0.409588635",
      "0.8089929"
    ],
    [
      "0.151666469",
      "0.826246006"
    ],
    [
      "1.646647188",
      "0.830756378"
    ],
    [
      "1.278422837",
      "0.837843497"
    ],
    [
      "0.910198486",
      "0.844930616"
    ],
    [
      "0.541974136",
      "0.852017735"
    ],
    [
      "1.803058188",
      "0.870827842"
    ],
    [
      "1.410808338",
      "0.880868332"
    ],
    [
      "1.042583987",
      "0.887955451"
    ],
    [
      "0.674359636",
      "0.89504257"
    ],
    [
      "0.306109844",
      "0.902208049"
    ],
    [
      "1.543193838",
      "0.923893167"
    ],
    [
      "1.174969487",
      "0.930980286"
    ],
    [
      "1.927928295",
      "0.932813212"
    ],
    [
      "0.806745136",
      "0.938067405"
    ],
    [
      "0.071532961",
      "0.940069018"
    ],
    [
      "0.438520785",
      "0.945154523"
    ],
    [
      "1.675579338",
      "0.966918002"
    ],
    [
      "1.307354987",
      "0.974005121"
    ],
    [
      "0.939130636",
      "0.98109224"
    ],
    [
      "0.570906286",
      "0.988179358"
    ],
    [
      "0.200379926",
      "0.99275219"
    ],
    [
      "1.807964839",
      "1.009942837"
    ],
    [
      "1.439740488",
      "1.017029956"
    ],
    [
      "1.071516137",
      "1.024117075"
    ],
    [
      "0.703291786",
      "1.031204194"
    ],
    [
      "0.331658411",
      "1.039044915"
    ],
    [
      "1.572125988",
      "1.060054791"
    ],
    [
      "1.203901637",
      "1.06714191"
    ],
    [
      "0.835677286",
      "1.074229029"
    ],
    [
      "0.072983199",
      "1.079262961"
    ],
    [
      "0.466860541",
      "1.081440676"
    ],
    [
      "1.926729788",
      "1.08255099"
    ],
    [
      "1.704511488",
      "1.103079626"
    ],
    [
      "1.336287137",
      "1.110166745"
    ],
    [
      "0.968062786",
      "1.117253864"
    ],
    [
      "0.599246041",
      "1.124465511"
    ],
    [
      "0.201830164",
      "1.131946134"
    ],
    [
      "1.468672638",
      "1.15319158"
    ],
    [
      "1.100448287",
      "1.160278699"
    ],
    [
      "0.731631542",
      "1.167490346"
    ],
    [
      "0.363407191",
      "1.174577465"
    ],
    [
      "1.601058138",
      "1.196216415"
    ],
    [
      "1.232833787",
      "1.203303534"
    ],
    [
      "0.864017042",
      "1.210515181"
    ],
    [
      "0.495792691",
      "1.2176023"
    ],
    [
      "1.904041313",
      "1.219891043"
    ],
    [
      "0.096840478",
      "1.223411994"
    ],
    [
      "1.765100329",
      "1.228403404"
    ],
    [
      "1.365219287",
      "1.246328369"
    ],
    [
      "0.996402542",
      "1.253540016"
    ],
    [
      "0.628178192",
      "1.260627135"
    ],
    [
      "0.254214973",
      "1.260914692"
    ],
    [
      "1.497604788",
      "1.289353204"
    ],
    [
      "1.128788043",
      "1.296564851"
    ],
    [
      "0.760563692",
      "1.30365197"
    ],
    [
      "0.392339341",
      "1.310739089"
    ],
    [
      "1.661646979",
      "1.321540193"
    ],
    [
      "1.261173543",
      "1.339589686"
    ],
    [
      "0.892949192",
      "1.346676805"
    ],
    [
      "1.861116245",
      "1.352308926"
    ],
    [
      "0.524724841",
      "1.353763924"
    ],
    [
      "0.146527876",
      "1.370388522"
    ],
    [
      "1.393559043",
      "1.382614521"
    ],
    [
      "1.025334692",
      "1.38970164"
    ],
    [
      "0.657110342",
      "1.396788759"
    ],
    [
      "0.283147123",
      "1.397076316"
    ],
    [
      "1.525944544",
      "1.425639356"
    ],
    [
      "1.157720193",
      "1.432726475"
    ],
    [
      "0.789495842",
      "1.439813594"
    ],
    [
      "0.419759343",
      "1.447252546"
    ],
    [
      "1.660648421",
      "1.46073811"
    ],
    [
      "1.290105693",
      "1.47575131"
    ],
    [
      "1.798915444",
      "1.476840518"
    ],
    [
      "0.921881342",
      "1.482838429"
    ],
    [
      "0.552537508",
      "1.490158618"
    ],
    [
      "1.422491193",
      "1.518776145"
    ],
    [
      "1.054266842",
      "1.525863264"
    ],
    [
      "0.233268496",
      "1.527034692"
    ],
    [
      "0.686042492",
      "1.532950383"
    ],
    [
      "1.186652343",
      "1.568888099"
    ],
    [
      "1.576451283",
      "1.571588906"
    ],
    [
      "0.818427992",
      "1.575975218"
    ],
    [
      "0.449849082",
      "1.584138106"
    ],
    [
      "1.71429499",
      "1.595712073"
    ],
    [
      "1.319037843",
      "1.611912934"
    ],
    [
      "0.950921357",
      "1.619119726"
    ],
    [
      "0.582581919",
      "1.626079148"
    ],
    [
      "0.320481227",
      "1.635528916"
    ],
    [
      "1.462827944",
      "1.652005285"
    ],
    [
      "1.083306857",
      "1.662144561"
    ],
    [
      "1.603470092",
      "1.708143075"
    ],
    [
      "0.847360352",
      "1.712136797"
    ],
    [
      "1.212200306",
      "1.720303815"
    ],
    [
      "0.697578774",
      "1.725743366"
    ],
    [
      "0.422904706",
      "1.729797087"
    ],
    [
      "1.359374593",
      "1.745142074"
    ],
    [
      "0.558062736",
      "1.763104207"
    ],
    [
      "1.49106406",
      "1.7902524"
    ],
    [
      "0.957149781",
      "1.797713304"
    ],
    [
      "1.096318444",
      "1.800736609"
    ],
    [
      "0.791298669",
      "1.83955008"
    ],
    [
      "0.654218264",
      "1.863757927"
    ],
    [
      "1.312688466",
      "1.87628117"
    ],
    [
      "1.178451488",
      "1.91312531"
    ],
    [
      "0.901088097",
      "1.925126587"
    ],
    [
      "1.040219946",
      "1.929529516"
    ]
  ],
  "radius": "0.069600748"
}
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": 171
}

起步布局答案

{
  "centers": [
    [
      "0.962297463",
      "0.070364972"
    ],
    [
      "0.824022004",
      "0.086394773"
    ],
    [
      "1.227011411",
      "0.09772034"
    ],
    [
      "0.68968574",
      "0.122875251"
    ],
    [
      "1.090451756",
      "0.124711403"
    ],
    [
      "1.460824078",
      "0.191740181"
    ],
    [
      "0.533567473",
      "0.194963688"
    ],
    [
      "1.321845775",
      "0.199619803"
    ],
    [
      "0.898997279",
      "0.203679747"
    ],
    [
      "1.179983901",
      "0.232955702"
    ],
    [
      "1.030196722",
      "0.250196016"
    ],
    [
      "0.657520513",
      "0.258309572"
    ],
    [
      "1.576255192",
      "0.269539378"
    ],
    [
      "0.410813982",
      "0.280898777"
    ],
    [
      "0.792532036",
      "0.295708638"
    ],
    [
      "1.434060124",
      "0.328344529"
    ],
    [
      "0.540684849",
      "0.333983111"
    ],
    [
      "1.295081821",
      "0.336224151"
    ],
    [
      "0.924312731",
      "0.340559932"
    ],
    [
      "1.67878707",
      "0.363689636"
    ],
    [
      "0.303695243",
      "0.382908069"
    ],
    [
      "1.059240887",
      "0.38633379"
    ],
    [
      "0.691016536",
      "0.393420909"
    ],
    [
      "1.547279768",
      "0.409328288"
    ],
    [
      "0.436106184",
      "0.425854544"
    ],
    [
      "1.191626387",
      "0.429358625"
    ],
    [
      "0.823402036",
      "0.436445744"
    ],
    [
      "0.574180871",
      "0.469094449"
    ],
    [
      "1.766124579",
      "0.472083439"
    ],
    [
      "1.324612241",
      "0.472257277"
    ],
    [
      "0.955787537",
      "0.479470579"
    ],
    [
      "0.219421102",
      "0.493700336"
    ],
    [
      "1.456997741",
      "0.515282112"
    ],
    [
      "1.088173037",
      "0.522495414"
    ],
    [
      "0.719948686",
      "0.529582533"
    ],
    [
      "0.351832044",
      "0.53664681"
    ],
    [
      "1.59694997",
      "0.539863855"
    ],
    [
      "1.220558537",
      "0.565520249"
    ],
    [
      "0.852334186",
      "0.572607368"
    ],
    [
      "0.48529044",
      "0.576218435"
    ],
    [
      "1.724063911",
      "0.604778418"
    ],
    [
      "1.352944038",
      "0.608545084"
    ],
    [
      "0.984719687",
      "0.615632203"
    ],
    [
      "0.616495336",
      "0.622719322"
    ],
    [
      "0.248245544",
      "0.629884801"
    ],
    [
      "1.858422348",
      "0.641177149"
    ],
    [
      "1.485329538",
      "0.651569919"
    ],
    [
      "1.117105187",
      "0.658657038"
    ],
    [
      "0.748880836",
      "0.665744157"
    ],
    [
      "0.380656485",
      "0.672831276"
    ],
    [
      "0.122734319",
      "0.690084382"
    ],
    [
      "1.617715038",
      "0.694594754"
    ],
    [
      "1.249490687",
      "0.701681873"
    ],
    [
      "0.881266336",
      "0.708768992"
    ],
    [
      "0.513041986",
      "0.715856111"
    ],
    [
      "1.752996061",
      "0.740940042"
    ],
    [
      "1.381876188",
      "0.744706708"
    ],
    [
      "1.013651837",
      "0.751793827"
    ],
    [
      "0.645427486",
      "0.758880946"
    ],
    [
      "0.277177694",
      "0.766046425"
    ],
    [
      "1.902349416",
      "0.77326602"
    ],
    [
      "1.514261688",
      "0.787731543"
    ],
    [
      "1.146037337",
      "0.794818662"
    ],
    [
      "0.777812986",
      "0.801905781"
    ],
    [
      "0.409588635",
      "0.8089929"
    ],
    [
      "0.151666469",
      "0.826246006"
    ],
    [
      "1.646647188",
      "0.830756378"
    ],
    [
      "1.278422837",
      "0.837843497"
    ],
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  "radius": "0.069600748"
}

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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