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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 = 63

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 = 63
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[0.1124561929178359286010067339845, 0.1124561929178359286010067339855]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 = 0.112456192

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 = 63 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": [
    [
      "1.022389862",
      "0.11273865"
    ],
    [
      "0.798642269",
      "0.135599009"
    ],
    [
      "1.244699654",
      "0.146855177"
    ],
    [
      "0.58782518",
      "0.21396824"
    ],
    [
      "1.451295673",
      "0.235757744"
    ],
    [
      "1.046172453",
      "0.336390097"
    ],
    [
      "0.403476547",
      "0.34281374"
    ],
    [
      "0.82242486",
      "0.359250456"
    ],
    [
      "1.268482246",
      "0.370506624"
    ],
    [
      "1.628911028",
      "0.373737333"
    ],
    [
      "0.611607772",
      "0.437619687"
    ],
    [
      "1.447490576",
      "0.506674164"
    ],
    [
      "0.257434609",
      "0.513861492"
    ],
    [
      "0.954096308",
      "0.541591375"
    ],
    [
      "1.76613987",
      "0.551933366"
    ],
    [
      "1.1790332",
      "0.580164482"
    ],
    [
      "0.462576695",
      "0.606069441"
    ],
    [
      "0.74327922",
      "0.619960607"
    ],
    [
      "1.590074384",
      "0.691888194"
    ],
    [
      "0.159077677",
      "0.716127394"
    ],
    [
      "1.366488533",
      "0.71649373"
    ],
    [
      "1.019438193",
      "0.756802945"
    ],
    [
      "1.854169833",
      "0.758902704"
    ],
    [
      "0.594248143",
      "0.78841036"
    ],
    [
      "0.36448159",
      "0.80888035"
    ],
    [
      "0.812157591",
      "0.844097466"
    ],
    [
      "1.209307361",
      "0.877365295"
    ],
    [
      "1.678109989",
      "0.898861672"
    ],
    [
      "1.454545849",
      "0.923451432"
    ],
    [
      "0.114721898",
      "0.936622624"
    ],
    [
      "1.887347933",
      "0.981354485"
    ],
    [
      "0.994964671",
      "0.982427833"
    ],
    [
      "0.543133472",
      "1.007437463"
    ],
    [
      "0.319351795",
      "1.029961719"
    ],
    [
      "0.76104292",
      "1.063124569"
    ],
    [
      "1.297364676",
      "1.084322997"
    ],
    [
      "1.515274124",
      "1.140010102"
    ],
    [
      "1.739908041",
      "1.151198707"
    ],
    [
      "0.127215648",
      "1.161187731"
    ],
    [
      "1.101228525",
      "1.19439657"
    ],
    [
      "0.603861748",
      "1.223996134"
    ],
    [
      "0.380080071",
      "1.24652039"
    ],
    [
      "0.883721048",
      "1.251633609"
    ],
    [
      "1.358092952",
      "1.300881667"
    ],
    [
      "1.583741175",
      "1.357721214"
    ],
    [
      "1.808895688",
      "1.365269456"
    ],
    [
      "0.195756621",
      "1.375401915"
    ],
    [
      "1.161956801",
      "1.41095524"
    ],
    [
      "0.726539876",
      "1.412505174"
    ],
    [
      "0.502758198",
      "1.43502943"
    ],
    [
      "0.944449324",
      "1.46819228"
    ],
    [
      "1.415999107",
      "1.518211966"
    ],
    [
      "1.691106551",
      "1.556871389"
    ],
    [
      "0.315943351",
      "1.565509072"
    ],
    [
      "1.219862956",
      "1.628285539"
    ],
    [
      "0.784446031",
      "1.629835472"
    ],
    [
      "0.520389634",
      "1.659249665"
    ],
    [
      "1.002355479",
      "1.685522578"
    ],
    [
      "1.528936918",
      "1.712712948"
    ],
    [
      "1.332800767",
      "1.822786521"
    ],
    [
      "0.672510143",
      "1.824914785"
    ],
    [
      "1.11529329",
      "1.88002356"
    ],
    [
      "0.890382072",
      "1.8807485"
    ]
  ],
  "radius": "0.112456192"
}
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": 63
}

起步布局答案

{
  "centers": [
    [
      "1.022389862",
      "0.11273865"
    ],
    [
      "0.798642269",
      "0.135599009"
    ],
    [
      "1.244699654",
      "0.146855177"
    ],
    [
      "0.58782518",
      "0.21396824"
    ],
    [
      "1.451295673",
      "0.235757744"
    ],
    [
      "1.046172453",
      "0.336390097"
    ],
    [
      "0.403476547",
      "0.34281374"
    ],
    [
      "0.82242486",
      "0.359250456"
    ],
    [
      "1.268482246",
      "0.370506624"
    ],
    [
      "1.628911028",
      "0.373737333"
    ],
    [
      "0.611607772",
      "0.437619687"
    ],
    [
      "1.447490576",
      "0.506674164"
    ],
    [
      "0.257434609",
      "0.513861492"
    ],
    [
      "0.954096308",
      "0.541591375"
    ],
    [
      "1.76613987",
      "0.551933366"
    ],
    [
      "1.1790332",
      "0.580164482"
    ],
    [
      "0.462576695",
      "0.606069441"
    ],
    [
      "0.74327922",
      "0.619960607"
    ],
    [
      "1.590074384",
      "0.691888194"
    ],
    [
      "0.159077677",
      "0.716127394"
    ],
    [
      "1.366488533",
      "0.71649373"
    ],
    [
      "1.019438193",
      "0.756802945"
    ],
    [
      "1.854169833",
      "0.758902704"
    ],
    [
      "0.594248143",
      "0.78841036"
    ],
    [
      "0.36448159",
      "0.80888035"
    ],
    [
      "0.812157591",
      "0.844097466"
    ],
    [
      "1.209307361",
      "0.877365295"
    ],
    [
      "1.678109989",
      "0.898861672"
    ],
    [
      "1.454545849",
      "0.923451432"
    ],
    [
      "0.114721898",
      "0.936622624"
    ],
    [
      "1.887347933",
      "0.981354485"
    ],
    [
      "0.994964671",
      "0.982427833"
    ],
    [
      "0.543133472",
      "1.007437463"
    ],
    [
      "0.319351795",
      "1.029961719"
    ],
    [
      "0.76104292",
      "1.063124569"
    ],
    [
      "1.297364676",
      "1.084322997"
    ],
    [
      "1.515274124",
      "1.140010102"
    ],
    [
      "1.739908041",
      "1.151198707"
    ],
    [
      "0.127215648",
      "1.161187731"
    ],
    [
      "1.101228525",
      "1.19439657"
    ],
    [
      "0.603861748",
      "1.223996134"
    ],
    [
      "0.380080071",
      "1.24652039"
    ],
    [
      "0.883721048",
      "1.251633609"
    ],
    [
      "1.358092952",
      "1.300881667"
    ],
    [
      "1.583741175",
      "1.357721214"
    ],
    [
      "1.808895688",
      "1.365269456"
    ],
    [
      "0.195756621",
      "1.375401915"
    ],
    [
      "1.161956801",
      "1.41095524"
    ],
    [
      "0.726539876",
      "1.412505174"
    ],
    [
      "0.502758198",
      "1.43502943"
    ],
    [
      "0.944449324",
      "1.46819228"
    ],
    [
      "1.415999107",
      "1.518211966"
    ],
    [
      "1.691106551",
      "1.556871389"
    ],
    [
      "0.315943351",
      "1.565509072"
    ],
    [
      "1.219862956",
      "1.628285539"
    ],
    [
      "0.784446031",
      "1.629835472"
    ],
    [
      "0.520389634",
      "1.659249665"
    ],
    [
      "1.002355479",
      "1.685522578"
    ],
    [
      "1.528936918",
      "1.712712948"
    ],
    [
      "1.332800767",
      "1.822786521"
    ],
    [
      "0.672510143",
      "1.824914785"
    ],
    [
      "1.11529329",
      "1.88002356"
    ],
    [
      "0.890382072",
      "1.8807485"
    ]
  ],
  "radius": "0.112456192"
}

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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