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 = 208

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 = 208
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[0.0632440500977361252199398760355, 0.0632440500977361252199398760365]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 = 0.063244049

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 = 208 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": [
    [
      "1.211433579",
      "0.087417099"
    ],
    [
      "0.896700873",
      "0.41805164"
    ],
    [
      "1.537282213",
      "0.887523781"
    ],
    [
      "1.096579334",
      "1.043067677"
    ],
    [
      "0.937190491",
      "1.729049908"
    ],
    [
      "0.648350997",
      "0.131751943"
    ],
    [
      "1.087304598",
      "1.932678279"
    ],
    [
      "1.362195555",
      "1.565393746"
    ],
    [
      "0.218990222",
      "1.364735758"
    ],
    [
      "1.197582409",
      "0.263396139"
    ],
    [
      "1.33341337",
      "0.799245166"
    ],
    [
      "1.147363019",
      "0.382842529"
    ],
    [
      "1.283479564",
      "1.466383483"
    ],
    [
      "1.66339346",
      "0.877767096"
    ],
    [
      "0.392406902",
      "1.031071704"
    ],
    [
      "1.757998922",
      "1.325973939"
    ],
    [
      "1.635966822",
      "0.312207945"
    ],
    [
      "0.706493431",
      "0.54413637"
    ],
    [
      "1.177766322",
      "0.599818602"
    ],
    [
      "1.144021807",
      "0.925813888"
    ],
    [
      "1.411618631",
      "0.901942648"
    ],
    [
      "0.98837476",
      "0.726387323"
    ],
    [
      "1.012254412",
      "1.830856819"
    ],
    [
      "0.36586216",
      "0.491002217"
    ],
    [
      "1.079517911",
      "1.377288539"
    ],
    [
      "1.531594661",
      "1.771309812"
    ],
    [
      "0.189155206",
      "0.947698952"
    ],
    [
      "0.69038137",
      "0.878736899"
    ],
    [
      "0.773803647",
      "0.090967862"
    ],
    [
      "0.985080653",
      "0.181117541"
    ],
    [
      "0.111779155",
      "1.297616263"
    ],
    [
      "0.54805395",
      "1.230498268"
    ],
    [
      "0.287840457",
      "0.391443927"
    ],
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      "0.828985916",
      "1.412369554"
    ],
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      "0.595496424",
      "1.113244478"
    ],
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      "1.330922038",
      "1.349129693"
    ],
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      "0.111331682",
      "0.84798567"
    ],
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      "0.453169003",
      "1.465005847"
    ],
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      "1.032374304",
      "0.063803646"
    ],
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      "0.585492824",
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      "0.534734323",
      "0.679310335"
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      "0.385617275",
      "1.707139008"
    ],
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      "1.935640803",
      "1.04569462"
    ],
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      "0.923870863",
      "1.177861975"
    ],
    [
      "0.940932286",
      "0.843641113"
    ],
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      "1.066198283",
      "0.826100606"
    ],
    [
      "0.290369885",
      "0.592491901"
    ],
    [
      "0.564117102",
      "0.226112372"
    ],
    [
      "1.682610286",
      "0.537018711"
    ],
    [
      "1.403628514",
      "0.350246601"
    ],
    [
      "1.538400135",
      "1.234002955"
    ],
    [
      "0.818874171",
      "0.318325963"
    ],
    [
      "1.29941829",
      "1.808202638"
    ],
    [
      "0.530992527",
      "1.564719129"
    ],
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      "1.692819813",
      "1.434375615"
    ],
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      "1.350474793",
      "0.465024305"
    ],
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      "0.625877474",
      "1.33021155"
    ],
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      "0.280448714",
      "1.599796948"
    ],
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      "1.332449007",
      "0.124220708"
    ],
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      "0.734100969",
      "1.646877134"
    ],
    [
      "1.378364511",
      "1.231875903"
    ],
    [
      "1.883008416",
      "1.306689246"
    ],
    [
      "1.441355541",
      "1.68257897"
    ],
    [
      "0.439760941",
      "0.249237783"
    ],
    [
      "1.110324292",
      "0.163418097"
    ],
    [
      "0.537311004",
      "0.34972739"
    ],
    [
      "1.130323848",
      "0.717072392"
    ],
    [
      "1.920955297",
      "1.171327321"
    ],
    [
      "0.5076693",
      "1.740343568"
    ],
    [
      "0.267140905",
      "1.048612211"
    ],
    [
      "0.612557847",
      "0.779023617"
    ],
    [
      "1.32580394",
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    ],
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      "0.753942318",
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    ],
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      "0.811924493",
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      "1.052500325",
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      "0.97465788",
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      "1.808721533",
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      "0.83065988",
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    ],
    [
      "0.470230426",
      "1.130784986"
    ],
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      "0.858661413",
      "0.185290688"
    ],
    [
      "0.703719919",
      "1.429910062"
    ],
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      "1.833567832",
      "1.427406574"
    ],
    [
      "0.660737783",
      "0.322067497"
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      "0.785266317",
      "0.644229319"
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      "1.205217247",
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    ],
    [
      "0.768204894",
      "0.978450181"
    ],
    [
      "1.069520574",
      "0.283144017"
    ],
    [
      "0.686658496",
      "1.764130923"
    ],
    [
      "1.732326545",
      "0.415881588"
    ],
    [
      "0.798604866",
      "1.195402482"
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      "0.213000992",
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      "0.297521955",
      "1.265579283"
    ],
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      "0.751162392",
      "1.312656272"
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    [
      "1.563543677",
      "1.64892313"
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    [
      "0.737470551",
      "0.221512428"
    ],
    [
      "0.565115373",
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    ],
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      "1.534906765",
      "1.430619024"
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    [
      "0.236597679",
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    [
      "1.126960384",
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    [
      "0.581227434",
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      "1.174402858",
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      "0.859366967",
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      "1.584724687",
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    ],
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      "0.893489813",
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    ],
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      "1.768388723",
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      "0.66000032",
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    [
      "1.607288869",
      "0.435402159"
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    ],
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      "1.436719792",
      "1.119653353"
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      "1.1103323",
      "1.597692346"
    ],
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      "1.447403056",
      "0.176991971"
    ],
    [
      "0.944143346",
      "0.30079785"
    ],
    [
      "0.642938897",
      "0.995990688"
    ],
    [
      "0.608834972",
      "1.664417641"
    ],
    [
      "1.42974767",
      "0.565333452"
    ],
    [
      "0.971313337",
      "1.060608185"
    ],
    [
      "1.255589846",
      "0.699531884"
    ],
    [
      "1.062890859",
      "1.714946554"
    ],
    [
      "1.459061105",
      "0.784688858"
    ],
    [
      "0.578435001",
      "1.44746534"
    ],
    [
      "1.457080502",
      "1.330886167"
    ],
    [
      "1.380855843",
      "0.681991376"
    ],
    [
      "1.221845331",
      "1.02552717"
    ],
    [
      "0.230826814",
      "0.704088775"
    ],
    [
      "0.412954842",
      "0.372852801"
    ],
    [
      "0.500611477",
      "1.347752058"
    ],
    [
      "1.506503578",
      "0.667435068"
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    [
      "1.897188855",
      "0.925192806"
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    [
      "0.5176729",
      "1.013531196"
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    [
      "1.208147372",
      "0.816785674"
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    [
      "1.487064478",
      "1.003616066"
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    [
      "1.613175725",
      "0.993859381"
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    [
      "1.87986534",
      "0.678521084"
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    [
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      "0.444423088"
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    [
      "0.20163635",
      "1.490027747"
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    [
      "0.375345479",
      "1.365292565"
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    [
      "1.409638028",
      "1.448139957"
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    [
      "0.495160117",
      "0.46898569"
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    [
      "0.673319947",
      "1.21295776"
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      "0.067976792"
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    [
      "1.688494621",
      "1.095477802"
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      "1.494542329"
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      "0.492933611"
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      "1.333724561"
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    [
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      "0.449824144"
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      "0.846047339",
      "1.078148693"
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    [
      "1.001694387",
      "1.277575257"
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    [
      "1.799447017",
      "1.206469567"
    ],
    [
      "0.90680944",
      "1.512082836"
    ],
    [
      "0.327903006",
      "1.482546355"
    ],
    [
      "1.285970896",
      "0.916498956"
    ],
    [
      "0.831759429",
      "0.526595863"
    ],
    [
      "0.656277445",
      "1.547163852"
    ],
    [
      "0.40572653",
      "1.582259637"
    ],
    [
      "1.613016126",
      "1.532511291"
    ],
    [
      "0.87642839",
      "1.295115764"
    ],
    [
      "1.157774774",
      "1.480438556"
    ],
    [
      "0.105560816",
      "0.721629282"
    ],
    [
      "1.223830989",
      "1.909621569"
    ],
    [
      "1.300102194",
      "1.128677187"
    ],
    [
      "1.422837869",
      "1.835894638"
    ],
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      "0.063889208",
      "0.965239459"
    ],
    [
      "1.689199961",
      "1.634440796"
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      "1.758379726",
      "0.643300563"
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    [
      "1.236037091",
      "1.583637272"
    ],
    [
      "1.741216984",
      "0.977480378"
    ],
    [
      "0.079781623",
      "1.175242257"
    ],
    [
      "0.705393882",
      "1.889223788"
    ],
    [
      "0.314583378",
      "0.931358422"
    ],
    [
      "0.781543443",
      "1.529623344"
    ],
    [
      "0.910532315",
      "0.626688812"
    ],
    [
      "0.954251914",
      "1.394829047"
    ],
    [
      "1.673809115",
      "1.221110503"
    ],
    [
      "0.141712732",
      "1.064952741"
    ],
    [
      "0.984632964",
      "1.611796118"
    ]
  ],
  "radius": "0.063244049"
}
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": 208
}

起步布局答案

{
  "centers": [
    [
      "1.211433579",
      "0.087417099"
    ],
    [
      "0.896700873",
      "0.41805164"
    ],
    [
      "1.537282213",
      "0.887523781"
    ],
    [
      "1.096579334",
      "1.043067677"
    ],
    [
      "0.937190491",
      "1.729049908"
    ],
    [
      "0.648350997",
      "0.131751943"
    ],
    [
      "1.087304598",
      "1.932678279"
    ],
    [
      "1.362195555",
      "1.565393746"
    ],
    [
      "0.218990222",
      "1.364735758"
    ],
    [
      "1.197582409",
      "0.263396139"
    ],
    [
      "1.33341337",
      "0.799245166"
    ],
    [
      "1.147363019",
      "0.382842529"
    ],
    [
      "1.283479564",
      "1.466383483"
    ],
    [
      "1.66339346",
      "0.877767096"
    ],
    [
      "0.392406902",
      "1.031071704"
    ],
    [
      "1.757998922",
      "1.325973939"
    ],
    [
      "1.635966822",
      "0.312207945"
    ],
    [
      "0.706493431",
      "0.54413637"
    ],
    [
      "1.177766322",
      "0.599818602"
    ],
    [
      "1.144021807",
      "0.925813888"
    ],
    [
      "1.411618631",
      "0.901942648"
    ],
    [
      "0.98837476",
      "0.726387323"
    ],
    [
      "1.012254412",
      "1.830856819"
    ],
    [
      "0.36586216",
      "0.491002217"
    ],
    [
      "1.079517911",
      "1.377288539"
    ],
    [
      "1.531594661",
      "1.771309812"
    ],
    [
      "0.189155206",
      "0.947698952"
    ],
    [
      "0.69038137",
      "0.878736899"
    ],
    [
      "0.773803647",
      "0.090967862"
    ],
    [
      "0.985080653",
      "0.181117541"
    ],
    [
      "0.111779155",
      "1.297616263"
    ],
    [
      "0.54805395",
      "1.230498268"
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  "radius": "0.063244049"
}

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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