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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 = 64

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 = 64
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[0.1115825958257262504168364364895, 0.1115825958257262504168364364905]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 = 0.111582595

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 = 64 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": [
    [
      "1.091870212",
      "0.116345459"
    ],
    [
      "0.868760684",
      "0.121329569"
    ],
    [
      "1.30918287",
      "0.167118594"
    ],
    [
      "0.653932155",
      "0.181756435"
    ],
    [
      "1.506986564",
      "0.270445267"
    ],
    [
      "0.460939977",
      "0.293813215"
    ],
    [
      "1.149043041",
      "0.332062798"
    ],
    [
      "0.903316649",
      "0.348698758"
    ],
    [
      "0.679936944",
      "0.403401317"
    ],
    [
      "1.672800184",
      "0.419805725"
    ],
    [
      "1.350018489",
      "0.429076053"
    ],
    [
      "0.301961659",
      "0.450429297"
    ],
    [
      "1.045276382",
      "0.529636033"
    ],
    [
      "0.826165839",
      "0.571983087"
    ],
    [
      "1.515832109",
      "0.578436511"
    ],
    [
      "1.796161144",
      "0.605775551"
    ],
    [
      "0.607055297",
      "0.61433014"
    ],
    [
      "1.24625183",
      "0.626649289"
    ],
    [
      "0.387889145",
      "0.656388443"
    ],
    [
      "0.166608478",
      "0.69798671"
    ],
    [
      "0.972394735",
      "0.740564856"
    ],
    [
      "1.63919307",
      "0.764406337"
    ],
    [
      "1.412434334",
      "0.778072348"
    ],
    [
      "0.753284192",
      "0.782911909"
    ],
    [
      "1.869285559",
      "0.816620335"
    ],
    [
      "0.53417365",
      "0.825258963"
    ],
    [
      "1.173370183",
      "0.837578111"
    ],
    [
      "0.315007498",
      "0.867317265"
    ],
    [
      "0.899513087",
      "0.951493679"
    ],
    [
      "0.112440683",
      "0.960962278"
    ],
    [
      "1.539059452",
      "0.963845454"
    ],
    [
      "0.680107785",
      "0.994095958"
    ],
    [
      "1.319893376",
      "1.005904154"
    ],
    [
      "0.460941634",
      "1.03615426"
    ],
    [
      "1.887559388",
      "1.039036094"
    ],
    [
      "1.100488536",
      "1.048506934"
    ],
    [
      "1.684993306",
      "1.132682691"
    ],
    [
      "0.826630698",
      "1.162422245"
    ],
    [
      "1.465827231",
      "1.174741392"
    ],
    [
      "0.130714105",
      "1.183378071"
    ],
    [
      "1.246716838",
      "1.217089218"
    ],
    [
      "0.587565666",
      "1.221927652"
    ],
    [
      "0.360806754",
      "1.235592744"
    ],
    [
      "1.027606147",
      "1.2594355"
    ],
    [
      "1.833391522",
      "1.30201329"
    ],
    [
      "1.612112403",
      "1.343611771"
    ],
    [
      "0.753748308",
      "1.373350811"
    ],
    [
      "1.392946328",
      "1.385670472"
    ],
    [
      "0.203838132",
      "1.394222989"
    ],
    [
      "0.484167374",
      "1.421562796"
    ],
    [
      "1.173834448",
      "1.428017784"
    ],
    [
      "0.954723757",
      "1.470364067"
    ],
    [
      "1.698037333",
      "1.549571984"
    ],
    [
      "0.64998072",
      "1.570923558"
    ],
    [
      "0.327198752",
      "1.580193041"
    ],
    [
      "1.320063939",
      "1.596599038"
    ],
    [
      "1.096684007",
      "1.651301048"
    ],
    [
      "0.850956168",
      "1.667936814"
    ],
    [
      "1.539058727",
      "1.706187774"
    ],
    [
      "0.493012098",
      "1.729553803"
    ],
    [
      "1.346066343",
      "1.818244199"
    ],
    [
      "0.690815602",
      "1.832880839"
    ],
    [
      "1.131237704",
      "1.878670672"
    ],
    [
      "0.908128167",
      "1.883654373"
    ]
  ],
  "radius": "0.111582595"
}
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": 64
}

起步布局答案

{
  "centers": [
    [
      "1.091870212",
      "0.116345459"
    ],
    [
      "0.868760684",
      "0.121329569"
    ],
    [
      "1.30918287",
      "0.167118594"
    ],
    [
      "0.653932155",
      "0.181756435"
    ],
    [
      "1.506986564",
      "0.270445267"
    ],
    [
      "0.460939977",
      "0.293813215"
    ],
    [
      "1.149043041",
      "0.332062798"
    ],
    [
      "0.903316649",
      "0.348698758"
    ],
    [
      "0.679936944",
      "0.403401317"
    ],
    [
      "1.672800184",
      "0.419805725"
    ],
    [
      "1.350018489",
      "0.429076053"
    ],
    [
      "0.301961659",
      "0.450429297"
    ],
    [
      "1.045276382",
      "0.529636033"
    ],
    [
      "0.826165839",
      "0.571983087"
    ],
    [
      "1.515832109",
      "0.578436511"
    ],
    [
      "1.796161144",
      "0.605775551"
    ],
    [
      "0.607055297",
      "0.61433014"
    ],
    [
      "1.24625183",
      "0.626649289"
    ],
    [
      "0.387889145",
      "0.656388443"
    ],
    [
      "0.166608478",
      "0.69798671"
    ],
    [
      "0.972394735",
      "0.740564856"
    ],
    [
      "1.63919307",
      "0.764406337"
    ],
    [
      "1.412434334",
      "0.778072348"
    ],
    [
      "0.753284192",
      "0.782911909"
    ],
    [
      "1.869285559",
      "0.816620335"
    ],
    [
      "0.53417365",
      "0.825258963"
    ],
    [
      "1.173370183",
      "0.837578111"
    ],
    [
      "0.315007498",
      "0.867317265"
    ],
    [
      "0.899513087",
      "0.951493679"
    ],
    [
      "0.112440683",
      "0.960962278"
    ],
    [
      "1.539059452",
      "0.963845454"
    ],
    [
      "0.680107785",
      "0.994095958"
    ],
    [
      "1.319893376",
      "1.005904154"
    ],
    [
      "0.460941634",
      "1.03615426"
    ],
    [
      "1.887559388",
      "1.039036094"
    ],
    [
      "1.100488536",
      "1.048506934"
    ],
    [
      "1.684993306",
      "1.132682691"
    ],
    [
      "0.826630698",
      "1.162422245"
    ],
    [
      "1.465827231",
      "1.174741392"
    ],
    [
      "0.130714105",
      "1.183378071"
    ],
    [
      "1.246716838",
      "1.217089218"
    ],
    [
      "0.587565666",
      "1.221927652"
    ],
    [
      "0.360806754",
      "1.235592744"
    ],
    [
      "1.027606147",
      "1.2594355"
    ],
    [
      "1.833391522",
      "1.30201329"
    ],
    [
      "1.612112403",
      "1.343611771"
    ],
    [
      "0.753748308",
      "1.373350811"
    ],
    [
      "1.392946328",
      "1.385670472"
    ],
    [
      "0.203838132",
      "1.394222989"
    ],
    [
      "0.484167374",
      "1.421562796"
    ],
    [
      "1.173834448",
      "1.428017784"
    ],
    [
      "0.954723757",
      "1.470364067"
    ],
    [
      "1.698037333",
      "1.549571984"
    ],
    [
      "0.64998072",
      "1.570923558"
    ],
    [
      "0.327198752",
      "1.580193041"
    ],
    [
      "1.320063939",
      "1.596599038"
    ],
    [
      "1.096684007",
      "1.651301048"
    ],
    [
      "0.850956168",
      "1.667936814"
    ],
    [
      "1.539058727",
      "1.706187774"
    ],
    [
      "0.493012098",
      "1.729553803"
    ],
    [
      "1.346066343",
      "1.818244199"
    ],
    [
      "0.690815602",
      "1.832880839"
    ],
    [
      "1.131237704",
      "1.878670672"
    ],
    [
      "0.908128167",
      "1.883654373"
    ]
  ],
  "radius": "0.111582595"
}

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。

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