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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 = 53

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 = 53
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[0.1222556236876199177417565489175, 0.1222556236876199177417565489185]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 = 0.122255623

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 = 53 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": [
    [
      "0.997479698",
      "0.124972233"
    ],
    [
      "1.240060884",
      "0.155721632"
    ],
    [
      "0.754172259",
      "0.157382701"
    ],
    [
      "1.366858011",
      "0.364786786"
    ],
    [
      "0.877898976",
      "0.368279476"
    ],
    [
      "0.388882409",
      "0.369944066"
    ],
    [
      "0.633393653",
      "0.369981569"
    ],
    [
      "1.122403713",
      "0.370063743"
    ],
    [
      "1.611310836",
      "0.37013156"
    ],
    [
      "0.236012527",
      "0.568663171"
    ],
    [
      "1.742022264",
      "0.576780721"
    ],
    [
      "0.511041663",
      "0.581752627"
    ],
    [
      "1.000064154",
      "0.581768212"
    ],
    [
      "0.755552908",
      "0.58179013"
    ],
    [
      "1.244575383",
      "0.581865143"
    ],
    [
      "1.489086627",
      "0.581902646"
    ],
    [
      "0.388753549",
      "0.793486817"
    ],
    [
      "0.633264793",
      "0.793524336"
    ],
    [
      "0.877776038",
      "0.793561839"
    ],
    [
      "1.122287282",
      "0.793599341"
    ],
    [
      "1.366798527",
      "0.793636844"
    ],
    [
      "1.611309771",
      "0.793674346"
    ],
    [
      "0.144460635",
      "0.80381747"
    ],
    [
      "1.855599505",
      "0.804079922"
    ],
    [
      "0.510976693",
      "1.005258534"
    ],
    [
      "0.755487937",
      "1.005296037"
    ],
    [
      "0.999999182",
      "1.005333539"
    ],
    [
      "1.244510426",
      "1.005371042"
    ],
    [
      "1.489021671",
      "1.005408544"
    ],
    [
      "0.266683778",
      "1.015589171"
    ],
    [
      "1.733311405",
      "1.01581412"
    ],
    [
      "0.633199837",
      "1.217030235"
    ],
    [
      "0.877711081",
      "1.217067737"
    ],
    [
      "1.122222326",
      "1.21710524"
    ],
    [
      "1.36673357",
      "1.217142742"
    ],
    [
      "0.153570868",
      "1.232363753"
    ],
    [
      "1.846357814",
      "1.23262339"
    ],
    [
      "0.394889526",
      "1.271747194"
    ],
    [
      "1.605027086",
      "1.271932802"
    ],
    [
      "0.755422981",
      "1.428801935"
    ],
    [
      "0.999934226",
      "1.428839438"
    ],
    [
      "1.24444547",
      "1.42887694"
    ],
    [
      "1.727729647",
      "1.483430395"
    ],
    [
      "0.51711267",
      "1.483518894"
    ],
    [
      "1.482738985",
      "1.483667"
    ],
    [
      "0.268458273",
      "1.485055"
    ],
    [
      "0.877646125",
      "1.640573636"
    ],
    [
      "1.12215737",
      "1.640611138"
    ],
    [
      "0.509511078",
      "1.72791195"
    ],
    [
      "1.490265608",
      "1.728062376"
    ],
    [
      "0.729338284",
      "1.834971512"
    ],
    [
      "1.270405572",
      "1.835054499"
    ],
    [
      "1.004164481",
      "1.854788907"
    ]
  ],
  "radius": "0.122255623"
}
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": 53
}

起步布局答案

{
  "centers": [
    [
      "0.997479698",
      "0.124972233"
    ],
    [
      "1.240060884",
      "0.155721632"
    ],
    [
      "0.754172259",
      "0.157382701"
    ],
    [
      "1.366858011",
      "0.364786786"
    ],
    [
      "0.877898976",
      "0.368279476"
    ],
    [
      "0.388882409",
      "0.369944066"
    ],
    [
      "0.633393653",
      "0.369981569"
    ],
    [
      "1.122403713",
      "0.370063743"
    ],
    [
      "1.611310836",
      "0.37013156"
    ],
    [
      "0.236012527",
      "0.568663171"
    ],
    [
      "1.742022264",
      "0.576780721"
    ],
    [
      "0.511041663",
      "0.581752627"
    ],
    [
      "1.000064154",
      "0.581768212"
    ],
    [
      "0.755552908",
      "0.58179013"
    ],
    [
      "1.244575383",
      "0.581865143"
    ],
    [
      "1.489086627",
      "0.581902646"
    ],
    [
      "0.388753549",
      "0.793486817"
    ],
    [
      "0.633264793",
      "0.793524336"
    ],
    [
      "0.877776038",
      "0.793561839"
    ],
    [
      "1.122287282",
      "0.793599341"
    ],
    [
      "1.366798527",
      "0.793636844"
    ],
    [
      "1.611309771",
      "0.793674346"
    ],
    [
      "0.144460635",
      "0.80381747"
    ],
    [
      "1.855599505",
      "0.804079922"
    ],
    [
      "0.510976693",
      "1.005258534"
    ],
    [
      "0.755487937",
      "1.005296037"
    ],
    [
      "0.999999182",
      "1.005333539"
    ],
    [
      "1.244510426",
      "1.005371042"
    ],
    [
      "1.489021671",
      "1.005408544"
    ],
    [
      "0.266683778",
      "1.015589171"
    ],
    [
      "1.733311405",
      "1.01581412"
    ],
    [
      "0.633199837",
      "1.217030235"
    ],
    [
      "0.877711081",
      "1.217067737"
    ],
    [
      "1.122222326",
      "1.21710524"
    ],
    [
      "1.36673357",
      "1.217142742"
    ],
    [
      "0.153570868",
      "1.232363753"
    ],
    [
      "1.846357814",
      "1.23262339"
    ],
    [
      "0.394889526",
      "1.271747194"
    ],
    [
      "1.605027086",
      "1.271932802"
    ],
    [
      "0.755422981",
      "1.428801935"
    ],
    [
      "0.999934226",
      "1.428839438"
    ],
    [
      "1.24444547",
      "1.42887694"
    ],
    [
      "1.727729647",
      "1.483430395"
    ],
    [
      "0.51711267",
      "1.483518894"
    ],
    [
      "1.482738985",
      "1.483667"
    ],
    [
      "0.268458273",
      "1.485055"
    ],
    [
      "0.877646125",
      "1.640573636"
    ],
    [
      "1.12215737",
      "1.640611138"
    ],
    [
      "0.509511078",
      "1.72791195"
    ],
    [
      "1.490265608",
      "1.728062376"
    ],
    [
      "0.729338284",
      "1.834971512"
    ],
    [
      "1.270405572",
      "1.835054499"
    ],
    [
      "1.004164481",
      "1.854788907"
    ]
  ],
  "radius": "0.122255623"
}

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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