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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 = 93

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 = 93
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[0.0931675346224822907935698380695, 0.0931675346224822907935698380705]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 = 0.093167534

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 = 93 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": [
    [
      "1.062180122",
      "0.095301844"
    ],
    [
      "0.875954645",
      "0.101691662"
    ],
    [
      "1.245780252",
      "0.127109865"
    ],
    [
      "0.633595157",
      "0.170486521"
    ],
    [
      "1.419003138",
      "0.195772737"
    ],
    [
      "0.79226871",
      "0.268177164"
    ],
    [
      "0.462748064",
      "0.27008847"
    ],
    [
      "1.126433633",
      "0.270208232"
    ],
    [
      "1.574535024",
      "0.298391401"
    ],
    [
      "1.299656519",
      "0.338871104"
    ],
    [
      "0.960997917",
      "0.356078903"
    ],
    [
      "0.628329355",
      "0.35674717"
    ],
    [
      "1.705809101",
      "0.43063313"
    ],
    [
      "0.285006109",
      "0.442210743"
    ],
    [
      "1.462740437",
      "0.450163703"
    ],
    [
      "0.793842203",
      "0.454505589"
    ],
    [
      "0.470808794",
      "0.456286262"
    ],
    [
      "1.118363412",
      "0.456368457"
    ],
    [
      "1.291586297",
      "0.525031329"
    ],
    [
      "0.955576973",
      "0.547040027"
    ],
    [
      "0.632837645",
      "0.54830479"
    ],
    [
      "1.807282762",
      "0.58691446"
    ],
    [
      "1.62011734",
      "0.59609515"
    ],
    [
      "0.186092715",
      "0.600124955"
    ],
    [
      "1.438839437",
      "0.639212198"
    ],
    [
      "0.794572415",
      "0.640839228"
    ],
    [
      "0.461770619",
      "0.642402004"
    ],
    [
      "1.119093623",
      "0.642702096"
    ],
    [
      "0.956307185",
      "0.733373665"
    ],
    [
      "0.631934359",
      "0.73463767"
    ],
    [
      "1.280010988",
      "0.736650804"
    ],
    [
      "0.305197423",
      "0.743424734"
    ],
    [
      "1.721591",
      "0.75237648"
    ],
    [
      "0.12154382",
      "0.774922547"
    ],
    [
      "1.540076036",
      "0.795779636"
    ],
    [
      "0.793669129",
      "0.827172107"
    ],
    [
      "0.470972272",
      "0.828509735"
    ],
    [
      "1.118190338",
      "0.829034975"
    ],
    [
      "1.890823582",
      "0.830357832"
    ],
    [
      "1.3801232",
      "0.893807741"
    ],
    [
      "0.955403899",
      "0.919706545"
    ],
    [
      "0.632707042",
      "0.921044173"
    ],
    [
      "0.277738386",
      "0.927725465"
    ],
    [
      "1.723298435",
      "0.938703727"
    ],
    [
      "0.094084784",
      "0.959223278"
    ],
    [
      "1.542747433",
      "0.984769915"
    ],
    [
      "1.214865143",
      "0.988329483"
    ],
    [
      "0.443513236",
      "1.012810466"
    ],
    [
      "0.794441812",
      "1.01357861"
    ],
    [
      "1.90669102",
      "1.016016073"
    ],
    [
      "1.377024942",
      "1.080117051"
    ],
    [
      "1.008247353",
      "1.09839154"
    ],
    [
      "0.628324257",
      "1.107327691"
    ],
    [
      "1.753797495",
      "1.122525829"
    ],
    [
      "0.29069062",
      "1.131126817"
    ],
    [
      "0.104874972",
      "1.145245669"
    ],
    [
      "1.539649174",
      "1.171079225"
    ],
    [
      "1.17971814",
      "1.171319773"
    ],
    [
      "0.847285266",
      "1.192263605"
    ],
    [
      "0.465454345",
      "1.197849234"
    ],
    [
      "1.341877939",
      "1.263107341"
    ],
    [
      "1.865085918",
      "1.271976975"
    ],
    [
      "0.685482735",
      "1.284679506"
    ],
    [
      "1.009836887",
      "1.284719829"
    ],
    [
      "1.680013218",
      "1.293630003"
    ],
    [
      "0.153458805",
      "1.32513555"
    ],
    [
      "0.339522521",
      "1.33518799"
    ],
    [
      "1.504037738",
      "1.354894909"
    ],
    [
      "1.181307674",
      "1.357648062"
    ],
    [
      "0.522907191",
      "1.376803171"
    ],
    [
      "0.8488748",
      "1.378591894"
    ],
    [
      "1.791156412",
      "1.443189183"
    ],
    [
      "1.343467473",
      "1.44943563"
    ],
    [
      "0.687072269",
      "1.471007796"
    ],
    [
      "1.011426421",
      "1.471048119"
    ],
    [
      "0.237784994",
      "1.491297675"
    ],
    [
      "1.505627272",
      "1.541223198"
    ],
    [
      "1.182897208",
      "1.543976352"
    ],
    [
      "0.850464334",
      "1.564920184"
    ],
    [
      "0.526502004",
      "1.565548517"
    ],
    [
      "1.683870257",
      "1.595538909"
    ],
    [
      "0.354293149",
      "1.636716407"
    ],
    [
      "1.344762232",
      "1.637167225"
    ],
    [
      "0.688661802",
      "1.657336085"
    ],
    [
      "1.120574163",
      "1.719579871"
    ],
    [
      "1.547709979",
      "1.722744007"
    ],
    [
      "0.934600438",
      "1.731178639"
    ],
    [
      "0.498064106",
      "1.75525193"
    ],
    [
      "1.388424489",
      "1.819433668"
    ],
    [
      "0.663027617",
      "1.84189948"
    ],
    [
      "1.212739081",
      "1.881525498"
    ],
    [
      "0.842218655",
      "1.893000654"
    ],
    [
      "1.02807148",
      "1.906397877"
    ]
  ],
  "radius": "0.093167534"
}
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": 93
}

起步布局答案

{
  "centers": [
    [
      "1.062180122",
      "0.095301844"
    ],
    [
      "0.875954645",
      "0.101691662"
    ],
    [
      "1.245780252",
      "0.127109865"
    ],
    [
      "0.633595157",
      "0.170486521"
    ],
    [
      "1.419003138",
      "0.195772737"
    ],
    [
      "0.79226871",
      "0.268177164"
    ],
    [
      "0.462748064",
      "0.27008847"
    ],
    [
      "1.126433633",
      "0.270208232"
    ],
    [
      "1.574535024",
      "0.298391401"
    ],
    [
      "1.299656519",
      "0.338871104"
    ],
    [
      "0.960997917",
      "0.356078903"
    ],
    [
      "0.628329355",
      "0.35674717"
    ],
    [
      "1.705809101",
      "0.43063313"
    ],
    [
      "0.285006109",
      "0.442210743"
    ],
    [
      "1.462740437",
      "0.450163703"
    ],
    [
      "0.793842203",
      "0.454505589"
    ],
    [
      "0.470808794",
      "0.456286262"
    ],
    [
      "1.118363412",
      "0.456368457"
    ],
    [
      "1.291586297",
      "0.525031329"
    ],
    [
      "0.955576973",
      "0.547040027"
    ],
    [
      "0.632837645",
      "0.54830479"
    ],
    [
      "1.807282762",
      "0.58691446"
    ],
    [
      "1.62011734",
      "0.59609515"
    ],
    [
      "0.186092715",
      "0.600124955"
    ],
    [
      "1.438839437",
      "0.639212198"
    ],
    [
      "0.794572415",
      "0.640839228"
    ],
    [
      "0.461770619",
      "0.642402004"
    ],
    [
      "1.119093623",
      "0.642702096"
    ],
    [
      "0.956307185",
      "0.733373665"
    ],
    [
      "0.631934359",
      "0.73463767"
    ],
    [
      "1.280010988",
      "0.736650804"
    ],
    [
      "0.305197423",
      "0.743424734"
    ],
    [
      "1.721591",
      "0.75237648"
    ],
    [
      "0.12154382",
      "0.774922547"
    ],
    [
      "1.540076036",
      "0.795779636"
    ],
    [
      "0.793669129",
      "0.827172107"
    ],
    [
      "0.470972272",
      "0.828509735"
    ],
    [
      "1.118190338",
      "0.829034975"
    ],
    [
      "1.890823582",
      "0.830357832"
    ],
    [
      "1.3801232",
      "0.893807741"
    ],
    [
      "0.955403899",
      "0.919706545"
    ],
    [
      "0.632707042",
      "0.921044173"
    ],
    [
      "0.277738386",
      "0.927725465"
    ],
    [
      "1.723298435",
      "0.938703727"
    ],
    [
      "0.094084784",
      "0.959223278"
    ],
    [
      "1.542747433",
      "0.984769915"
    ],
    [
      "1.214865143",
      "0.988329483"
    ],
    [
      "0.443513236",
      "1.012810466"
    ],
    [
      "0.794441812",
      "1.01357861"
    ],
    [
      "1.90669102",
      "1.016016073"
    ],
    [
      "1.377024942",
      "1.080117051"
    ],
    [
      "1.008247353",
      "1.09839154"
    ],
    [
      "0.628324257",
      "1.107327691"
    ],
    [
      "1.753797495",
      "1.122525829"
    ],
    [
      "0.29069062",
      "1.131126817"
    ],
    [
      "0.104874972",
      "1.145245669"
    ],
    [
      "1.539649174",
      "1.171079225"
    ],
    [
      "1.17971814",
      "1.171319773"
    ],
    [
      "0.847285266",
      "1.192263605"
    ],
    [
      "0.465454345",
      "1.197849234"
    ],
    [
      "1.341877939",
      "1.263107341"
    ],
    [
      "1.865085918",
      "1.271976975"
    ],
    [
      "0.685482735",
      "1.284679506"
    ],
    [
      "1.009836887",
      "1.284719829"
    ],
    [
      "1.680013218",
      "1.293630003"
    ],
    [
      "0.153458805",
      "1.32513555"
    ],
    [
      "0.339522521",
      "1.33518799"
    ],
    [
      "1.504037738",
      "1.354894909"
    ],
    [
      "1.181307674",
      "1.357648062"
    ],
    [
      "0.522907191",
      "1.376803171"
    ],
    [
      "0.8488748",
      "1.378591894"
    ],
    [
      "1.791156412",
      "1.443189183"
    ],
    [
      "1.343467473",
      "1.44943563"
    ],
    [
      "0.687072269",
      "1.471007796"
    ],
    [
      "1.011426421",
      "1.471048119"
    ],
    [
      "0.237784994",
      "1.491297675"
    ],
    [
      "1.505627272",
      "1.541223198"
    ],
    [
      "1.182897208",
      "1.543976352"
    ],
    [
      "0.850464334",
      "1.564920184"
    ],
    [
      "0.526502004",
      "1.565548517"
    ],
    [
      "1.683870257",
      "1.595538909"
    ],
    [
      "0.354293149",
      "1.636716407"
    ],
    [
      "1.344762232",
      "1.637167225"
    ],
    [
      "0.688661802",
      "1.657336085"
    ],
    [
      "1.120574163",
      "1.719579871"
    ],
    [
      "1.547709979",
      "1.722744007"
    ],
    [
      "0.934600438",
      "1.731178639"
    ],
    [
      "0.498064106",
      "1.75525193"
    ],
    [
      "1.388424489",
      "1.819433668"
    ],
    [
      "0.663027617",
      "1.84189948"
    ],
    [
      "1.212739081",
      "1.881525498"
    ],
    [
      "0.842218655",
      "1.893000654"
    ],
    [
      "1.02807148",
      "1.906397877"
    ]
  ],
  "radius": "0.093167534"
}

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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