极值竞技场
中/EN
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 = 78

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 = 78
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[0.1014434397193697258715657384075, 0.1014434397193697258715657384085]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 = 0.101443439

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 = 78 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": [
    [
      "1.028559715",
      "0.101897425"
    ],
    [
      "1.229319633",
      "0.131198298"
    ],
    [
      "0.679449819",
      "0.160564789"
    ],
    [
      "1.418388371",
      "0.204792441"
    ],
    [
      "0.870438429",
      "0.229022344"
    ],
    [
      "0.499294967",
      "0.253877784"
    ],
    [
      "1.059592481",
      "0.302396935"
    ],
    [
      "1.586126824",
      "0.318927876"
    ],
    [
      "1.26031201",
      "0.331973217"
    ],
    [
      "0.676077591",
      "0.363423641"
    ],
    [
      "0.344667071",
      "0.3852296"
    ],
    [
      "0.891017558",
      "0.43094961"
    ],
    [
      "1.428025112",
      "0.446145898"
    ],
    [
      "0.518025059",
      "0.49063403"
    ],
    [
      "1.080213856",
      "0.504233118"
    ],
    [
      "1.617218807",
      "0.519418211"
    ],
    [
      "0.223449376",
      "0.547923657"
    ],
    [
      "0.73296763",
      "0.558171194"
    ],
    [
      "1.270654923",
      "0.574199438"
    ],
    [
      "1.803970546",
      "0.59870802"
    ],
    [
      "0.922161324",
      "0.631443507"
    ],
    [
      "1.459848617",
      "0.647471751"
    ],
    [
      "0.574915098",
      "0.685381583"
    ],
    [
      "0.37202942",
      "0.68607957"
    ],
    [
      "1.111355019",
      "0.70471582"
    ],
    [
      "0.141821804",
      "0.733665481"
    ],
    [
      "1.668540411",
      "0.74977676"
    ],
    [
      "0.764108793",
      "0.758653896"
    ],
    [
      "1.301796085",
      "0.77468214"
    ],
    [
      "1.873505762",
      "0.789306917"
    ],
    [
      "0.953302487",
      "0.831926209"
    ],
    [
      "1.49098978",
      "0.847954453"
    ],
    [
      "0.493287527",
      "0.871123407"
    ],
    [
      "0.290401848",
      "0.871821395"
    ],
    [
      "1.142496182",
      "0.905198522"
    ],
    [
      "0.682481221",
      "0.94439572"
    ],
    [
      "0.102876913",
      "0.949264823"
    ],
    [
      "1.699681573",
      "0.950259462"
    ],
    [
      "1.332937248",
      "0.975164842"
    ],
    [
      "1.898507886",
      "0.990647374"
    ],
    [
      "0.885930561",
      "1.023300474"
    ],
    [
      "1.522130943",
      "1.048437155"
    ],
    [
      "0.504773744",
      "1.073684886"
    ],
    [
      "0.301888065",
      "1.074382874"
    ],
    [
      "1.075124255",
      "1.096572787"
    ],
    [
      "0.722354871",
      "1.143325803"
    ],
    [
      "0.11436313",
      "1.151826302"
    ],
    [
      "1.267384581",
      "1.167169878"
    ],
    [
      "1.675100257",
      "1.181716842"
    ],
    [
      "1.877702257",
      "1.192464647"
    ],
    [
      "0.916901406",
      "1.225429861"
    ],
    [
      "1.456578275",
      "1.24044219"
    ],
    [
      "0.561411522",
      "1.268505952"
    ],
    [
      "0.358525843",
      "1.26920394"
    ],
    [
      "1.110883014",
      "1.29628357"
    ],
    [
      "0.749577703",
      "1.344378043"
    ],
    [
      "0.171000908",
      "1.346647368"
    ],
    [
      "1.609547589",
      "1.373721877"
    ],
    [
      "1.313065858",
      "1.384026186"
    ],
    [
      "1.812149589",
      "1.384469682"
    ],
    [
      "0.95414844",
      "1.425114307"
    ],
    [
      "0.457427675",
      "1.446352221"
    ],
    [
      "1.144086418",
      "1.496435062"
    ],
    [
      "1.469457804",
      "1.52047997"
    ],
    [
      "0.26990274",
      "1.523795649"
    ],
    [
      "0.789270442",
      "1.543344303"
    ],
    [
      "1.705191891",
      "1.556873674"
    ],
    [
      "0.593551343",
      "1.596796339"
    ],
    [
      "1.297881714",
      "1.628760769"
    ],
    [
      "0.406026408",
      "1.674239768"
    ],
    [
      "1.106386643",
      "1.695788549"
    ],
    [
      "1.562282081",
      "1.700887119"
    ],
    [
      "0.904044532",
      "1.710646407"
    ],
    [
      "0.590176204",
      "1.799655143"
    ],
    [
      "1.390705992",
      "1.809167918"
    ],
    [
      "0.780024432",
      "1.871214463"
    ],
    [
      "1.19921092",
      "1.876195698"
    ],
    [
      "0.981231848",
      "1.898355032"
    ]
  ],
  "radius": "0.101443439"
}
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": 78
}

起步布局答案

{
  "centers": [
    [
      "1.028559715",
      "0.101897425"
    ],
    [
      "1.229319633",
      "0.131198298"
    ],
    [
      "0.679449819",
      "0.160564789"
    ],
    [
      "1.418388371",
      "0.204792441"
    ],
    [
      "0.870438429",
      "0.229022344"
    ],
    [
      "0.499294967",
      "0.253877784"
    ],
    [
      "1.059592481",
      "0.302396935"
    ],
    [
      "1.586126824",
      "0.318927876"
    ],
    [
      "1.26031201",
      "0.331973217"
    ],
    [
      "0.676077591",
      "0.363423641"
    ],
    [
      "0.344667071",
      "0.3852296"
    ],
    [
      "0.891017558",
      "0.43094961"
    ],
    [
      "1.428025112",
      "0.446145898"
    ],
    [
      "0.518025059",
      "0.49063403"
    ],
    [
      "1.080213856",
      "0.504233118"
    ],
    [
      "1.617218807",
      "0.519418211"
    ],
    [
      "0.223449376",
      "0.547923657"
    ],
    [
      "0.73296763",
      "0.558171194"
    ],
    [
      "1.270654923",
      "0.574199438"
    ],
    [
      "1.803970546",
      "0.59870802"
    ],
    [
      "0.922161324",
      "0.631443507"
    ],
    [
      "1.459848617",
      "0.647471751"
    ],
    [
      "0.574915098",
      "0.685381583"
    ],
    [
      "0.37202942",
      "0.68607957"
    ],
    [
      "1.111355019",
      "0.70471582"
    ],
    [
      "0.141821804",
      "0.733665481"
    ],
    [
      "1.668540411",
      "0.74977676"
    ],
    [
      "0.764108793",
      "0.758653896"
    ],
    [
      "1.301796085",
      "0.77468214"
    ],
    [
      "1.873505762",
      "0.789306917"
    ],
    [
      "0.953302487",
      "0.831926209"
    ],
    [
      "1.49098978",
      "0.847954453"
    ],
    [
      "0.493287527",
      "0.871123407"
    ],
    [
      "0.290401848",
      "0.871821395"
    ],
    [
      "1.142496182",
      "0.905198522"
    ],
    [
      "0.682481221",
      "0.94439572"
    ],
    [
      "0.102876913",
      "0.949264823"
    ],
    [
      "1.699681573",
      "0.950259462"
    ],
    [
      "1.332937248",
      "0.975164842"
    ],
    [
      "1.898507886",
      "0.990647374"
    ],
    [
      "0.885930561",
      "1.023300474"
    ],
    [
      "1.522130943",
      "1.048437155"
    ],
    [
      "0.504773744",
      "1.073684886"
    ],
    [
      "0.301888065",
      "1.074382874"
    ],
    [
      "1.075124255",
      "1.096572787"
    ],
    [
      "0.722354871",
      "1.143325803"
    ],
    [
      "0.11436313",
      "1.151826302"
    ],
    [
      "1.267384581",
      "1.167169878"
    ],
    [
      "1.675100257",
      "1.181716842"
    ],
    [
      "1.877702257",
      "1.192464647"
    ],
    [
      "0.916901406",
      "1.225429861"
    ],
    [
      "1.456578275",
      "1.24044219"
    ],
    [
      "0.561411522",
      "1.268505952"
    ],
    [
      "0.358525843",
      "1.26920394"
    ],
    [
      "1.110883014",
      "1.29628357"
    ],
    [
      "0.749577703",
      "1.344378043"
    ],
    [
      "0.171000908",
      "1.346647368"
    ],
    [
      "1.609547589",
      "1.373721877"
    ],
    [
      "1.313065858",
      "1.384026186"
    ],
    [
      "1.812149589",
      "1.384469682"
    ],
    [
      "0.95414844",
      "1.425114307"
    ],
    [
      "0.457427675",
      "1.446352221"
    ],
    [
      "1.144086418",
      "1.496435062"
    ],
    [
      "1.469457804",
      "1.52047997"
    ],
    [
      "0.26990274",
      "1.523795649"
    ],
    [
      "0.789270442",
      "1.543344303"
    ],
    [
      "1.705191891",
      "1.556873674"
    ],
    [
      "0.593551343",
      "1.596796339"
    ],
    [
      "1.297881714",
      "1.628760769"
    ],
    [
      "0.406026408",
      "1.674239768"
    ],
    [
      "1.106386643",
      "1.695788549"
    ],
    [
      "1.562282081",
      "1.700887119"
    ],
    [
      "0.904044532",
      "1.710646407"
    ],
    [
      "0.590176204",
      "1.799655143"
    ],
    [
      "1.390705992",
      "1.809167918"
    ],
    [
      "0.780024432",
      "1.871214463"
    ],
    [
      "1.19921092",
      "1.876195698"
    ],
    [
      "0.981231848",
      "1.898355032"
    ]
  ],
  "radius": "0.101443439"
}

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

讨论区

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？