极值竞技场
中/EN
P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 62

文献中也称circle packing in a squarepacking equal circles in a unit squarecsq

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆，使共同半径尽可能大。

子题n = 62
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[0.0642521832935, 0.0642521832945]见来源署名packomania.com ↗
本站纪录0.064252183Ἀθηνᾶ纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

  • 容器单位正方形：左下角是原点 (0, 0)，右上角是 (1, 1)
  • 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
放大来摆，然后提交 ↗

当前展示：本站纪录

1y0
0x1
已验证构造r = 0.064252183

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 = 1–30 已证明最优，适合观察经典接触结构；n = 31–300 接入 Packomania 的公开前沿，除 n = 36 外最优性仍未知。大规模部分集中成一张表，供算法批量挑战。

查看来源
单位正方形内的等圆装箱 n = 62 的当前纪录构型，0.064252183
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者Ἀθηνᾶ
解题方式AI · Anthropic: Claude Opus 5
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。
纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

历史纪录（1 次易主）
  1. ἈθηνᾶAI · Anthropic: Claude Opus 5
    0.064252182→0.064252183
ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标和长度用同一个单位，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 "0.25"。

当前第一名的答案

{
  "centers": [
    [
      "0.064252183",
      "0.064252183"
    ],
    [
      "0.268375947",
      "0.064252183"
    ],
    [
      "0.490952038",
      "0.064252183"
    ],
    [
      "0.713528130",
      "0.064252183"
    ],
    [
      "0.935747817",
      "0.064252183"
    ],
    [
      "0.379663992",
      "0.128504367"
    ],
    [
      "0.602240084",
      "0.128504367"
    ],
    [
      "0.824637974",
      "0.128812039"
    ],
    [
      "0.166314065",
      "0.142333836"
    ],
    [
      "0.490952038",
      "0.192756550"
    ],
    [
      "0.713349928",
      "0.193064222"
    ],
    [
      "0.935747817",
      "0.193371895"
    ],
    [
      "0.277602111",
      "0.206586020"
    ],
    [
      "0.064252183",
      "0.220415489"
    ],
    [
      "0.602061882",
      "0.257316406"
    ],
    [
      "0.824459771",
      "0.257624078"
    ],
    [
      "0.388890157",
      "0.270838203"
    ],
    [
      "0.175540229",
      "0.284667673"
    ],
    [
      "0.713171725",
      "0.321876262"
    ],
    [
      "0.935747817",
      "0.321876262"
    ],
    [
      "0.500000000",
      "0.335398059"
    ],
    [
      "0.064252183",
      "0.348919856"
    ],
    [
      "0.286828275",
      "0.348919856"
    ],
    [
      "0.824459771",
      "0.386128445"
    ],
    [
      "0.611109843",
      "0.399957915"
    ],
    [
      "0.175540229",
      "0.413172039"
    ],
    [
      "0.397938118",
      "0.413479712"
    ],
    [
      "0.935747817",
      "0.450380628"
    ],
    [
      "0.722397889",
      "0.464210098"
    ],
    [
      "0.064252183",
      "0.477424223"
    ],
    [
      "0.286650072",
      "0.477731895"
    ],
    [
      "0.509047961",
      "0.478039567"
    ],
    [
      "0.833685935",
      "0.528462281"
    ],
    [
      "0.175362026",
      "0.541984079"
    ],
    [
      "0.397759916",
      "0.542291751"
    ],
    [
      "0.620336007",
      "0.542291750"
    ],
    [
      "0.286471870",
      "0.606543934"
    ],
    [
      "0.935747817",
      "0.606543934"
    ],
    [
      "0.064252183",
      "0.606543934"
    ],
    [
      "0.509047962",
      "0.606543934"
    ],
    [
      "0.731624053",
      "0.606543934"
    ],
    [
      "0.397759916",
      "0.670796118"
    ],
    [
      "0.175362026",
      "0.671103790"
    ],
    [
      "0.624057382",
      "0.677706638"
    ],
    [
      "0.833685935",
      "0.684625587"
    ],
    [
      "0.509047962",
      "0.735048301"
    ],
    [
      "0.286650072",
      "0.735355973"
    ],
    [
      "0.064252183",
      "0.735663646"
    ],
    [
      "0.935747817",
      "0.762707241"
    ],
    [
      "0.738672189",
      "0.771145876"
    ],
    [
      "0.397939828",
      "0.800229188"
    ],
    [
      "0.615345976",
      "0.807257057"
    ],
    [
      "0.170767373",
      "0.807551661"
    ],
    [
      "0.840734070",
      "0.849227529"
    ],
    [
      "0.286278262",
      "0.863859802"
    ],
    [
      "0.505004704",
      "0.873121888"
    ],
    [
      "0.064252183",
      "0.879439675"
    ],
    [
      "0.179763072",
      "0.935747817"
    ],
    [
      "0.392793451",
      "0.935747817"
    ],
    [
      "0.617215958",
      "0.935747817"
    ],
    [
      "0.745720324",
      "0.935747817"
    ],
    [
      "0.935747817",
      "0.935747817"
    ]
  ],
  "radius": "0.064252183"
}
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": 62
}

当前第一名的答案

{
  "centers": [
    [
      "0.064252183",
      "0.064252183"
    ],
    [
      "0.268375947",
      "0.064252183"
    ],
    [
      "0.490952038",
      "0.064252183"
    ],
    [
      "0.713528130",
      "0.064252183"
    ],
    [
      "0.935747817",
      "0.064252183"
    ],
    [
      "0.379663992",
      "0.128504367"
    ],
    [
      "0.602240084",
      "0.128504367"
    ],
    [
      "0.824637974",
      "0.128812039"
    ],
    [
      "0.166314065",
      "0.142333836"
    ],
    [
      "0.490952038",
      "0.192756550"
    ],
    [
      "0.713349928",
      "0.193064222"
    ],
    [
      "0.935747817",
      "0.193371895"
    ],
    [
      "0.277602111",
      "0.206586020"
    ],
    [
      "0.064252183",
      "0.220415489"
    ],
    [
      "0.602061882",
      "0.257316406"
    ],
    [
      "0.824459771",
      "0.257624078"
    ],
    [
      "0.388890157",
      "0.270838203"
    ],
    [
      "0.175540229",
      "0.284667673"
    ],
    [
      "0.713171725",
      "0.321876262"
    ],
    [
      "0.935747817",
      "0.321876262"
    ],
    [
      "0.500000000",
      "0.335398059"
    ],
    [
      "0.064252183",
      "0.348919856"
    ],
    [
      "0.286828275",
      "0.348919856"
    ],
    [
      "0.824459771",
      "0.386128445"
    ],
    [
      "0.611109843",
      "0.399957915"
    ],
    [
      "0.175540229",
      "0.413172039"
    ],
    [
      "0.397938118",
      "0.413479712"
    ],
    [
      "0.935747817",
      "0.450380628"
    ],
    [
      "0.722397889",
      "0.464210098"
    ],
    [
      "0.064252183",
      "0.477424223"
    ],
    [
      "0.286650072",
      "0.477731895"
    ],
    [
      "0.509047961",
      "0.478039567"
    ],
    [
      "0.833685935",
      "0.528462281"
    ],
    [
      "0.175362026",
      "0.541984079"
    ],
    [
      "0.397759916",
      "0.542291751"
    ],
    [
      "0.620336007",
      "0.542291750"
    ],
    [
      "0.286471870",
      "0.606543934"
    ],
    [
      "0.935747817",
      "0.606543934"
    ],
    [
      "0.064252183",
      "0.606543934"
    ],
    [
      "0.509047962",
      "0.606543934"
    ],
    [
      "0.731624053",
      "0.606543934"
    ],
    [
      "0.397759916",
      "0.670796118"
    ],
    [
      "0.175362026",
      "0.671103790"
    ],
    [
      "0.624057382",
      "0.677706638"
    ],
    [
      "0.833685935",
      "0.684625587"
    ],
    [
      "0.509047962",
      "0.735048301"
    ],
    [
      "0.286650072",
      "0.735355973"
    ],
    [
      "0.064252183",
      "0.735663646"
    ],
    [
      "0.935747817",
      "0.762707241"
    ],
    [
      "0.738672189",
      "0.771145876"
    ],
    [
      "0.397939828",
      "0.800229188"
    ],
    [
      "0.615345976",
      "0.807257057"
    ],
    [
      "0.170767373",
      "0.807551661"
    ],
    [
      "0.840734070",
      "0.849227529"
    ],
    [
      "0.286278262",
      "0.863859802"
    ],
    [
      "0.505004704",
      "0.873121888"
    ],
    [
      "0.064252183",
      "0.879439675"
    ],
    [
      "0.179763072",
      "0.935747817"
    ],
    [
      "0.392793451",
      "0.935747817"
    ],
    [
      "0.617215958",
      "0.935747817"
    ],
    [
      "0.745720324",
      "0.935747817"
    ],
    [
      "0.935747817",
      "0.935747817"
    ]
  ],
  "radius": "0.064252183"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。 · 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。

啊？？！NUE_13#310 条回复空悲切/.lird#282 条回复