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 = 51

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 = 51
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[0.0710431381155, 0.0710431381165]见来源署名packomania.com ↗
本站纪录0.071043138邱仲普纪录详情 ↓

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 = 0.071043138

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 = 51 的当前纪录构型，0.071043138
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者邱仲普
解题方式AI · OpenAI: GPT-5.6 Sol
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公开方法与可复现线索（点击展开）＋

Start from the public p01-n51 leader on the 1e-9 grid and target radius 0.071043138. Solve bounded integer center displacements with exact wall bounds and conservative linearized pair-separation constraints, then recompute all constraints using raw integer squares. A displacement bound of 16 grid units yields a layout whose exact bottleneck is 71043138; all 51 wall checks and 1275 pair checks pass in two independent verifiers. The same search on P02-n51 and local polishing on P57/P67-n17 gave no gain.

在讨论区查看与回复 →
历史纪录（1 次易主）
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    0.071043137→0.071043138
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": [
    [
      "0.313865334",
      "0.071043138"
    ],
    [
      "0.559965936",
      "0.071043138"
    ],
    [
      "0.806065672",
      "0.071043138"
    ],
    [
      "0.071235438",
      "0.071259278"
    ],
    [
      "0.436915635",
      "0.142086317"
    ],
    [
      "0.683015804",
      "0.142087067"
    ],
    [
      "0.928956862",
      "0.142361195"
    ],
    [
      "0.194093461",
      "0.147484019"
    ],
    [
      "0.559965503",
      "0.213130246"
    ],
    [
      "0.805906994",
      "0.213405124"
    ],
    [
      "0.071043138",
      "0.21852716"
    ],
    [
      "0.317144245",
      "0.218527955"
    ],
    [
      "0.682856693",
      "0.284448303"
    ],
    [
      "0.928956862",
      "0.284449053"
    ],
    [
      "0.194093462",
      "0.289570299"
    ],
    [
      "0.440203659",
      "0.289586839"
    ],
    [
      "0.805906561",
      "0.355492232"
    ],
    [
      "0.071043138",
      "0.360613438"
    ],
    [
      "0.317144247",
      "0.360614235"
    ],
    [
      "0.56518337",
      "0.364081832"
    ],
    [
      "0.928956862",
      "0.426535411"
    ],
    [
      "0.194093464",
      "0.431656579"
    ],
    [
      "0.440203659",
      "0.431673116"
    ],
    [
      "0.688233238",
      "0.435125761"
    ],
    [
      "0.071043138",
      "0.502699715"
    ],
    [
      "0.317144247",
      "0.502700512"
    ],
    [
      "0.565182463",
      "0.506168119"
    ],
    [
      "0.811283539",
      "0.50616894"
    ],
    [
      "0.194093464",
      "0.573742856"
    ],
    [
      "0.440202746",
      "0.573759392"
    ],
    [
      "0.68823234",
      "0.577212037"
    ],
    [
      "0.928956862",
      "0.585802469"
    ],
    [
      "0.071043138",
      "0.644785992"
    ],
    [
      "0.317143334",
      "0.644786788"
    ],
    [
      "0.565181759",
      "0.648255506"
    ],
    [
      "0.805906122",
      "0.656844888"
    ],
    [
      "0.440184569",
      "0.715845668"
    ],
    [
      "0.194260755",
      "0.716119681"
    ],
    [
      "0.682855797",
      "0.727888027"
    ],
    [
      "0.928956862",
      "0.727888746"
    ],
    [
      "0.071043138",
      "0.786872269"
    ],
    [
      "0.31730199",
      "0.787178562"
    ],
    [
      "0.55786797",
      "0.795464303"
    ],
    [
      "0.805906122",
      "0.798931165"
    ],
    [
      "0.19919341",
      "0.866165105"
    ],
    [
      "0.466592318",
      "0.904355374"
    ],
    [
      "0.920789515",
      "0.921696002"
    ],
    [
      "0.071734769",
      "0.928956862"
    ],
    [
      "0.326652052",
      "0.928956862"
    ],
    [
      "0.606532604",
      "0.928956862"
    ],
    [
      "0.748618881",
      "0.928956862"
    ]
  ],
  "radius": "0.071043138"
}
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": 51
}

当前第一名的答案

{
  "centers": [
    [
      "0.313865334",
      "0.071043138"
    ],
    [
      "0.559965936",
      "0.071043138"
    ],
    [
      "0.806065672",
      "0.071043138"
    ],
    [
      "0.071235438",
      "0.071259278"
    ],
    [
      "0.436915635",
      "0.142086317"
    ],
    [
      "0.683015804",
      "0.142087067"
    ],
    [
      "0.928956862",
      "0.142361195"
    ],
    [
      "0.194093461",
      "0.147484019"
    ],
    [
      "0.559965503",
      "0.213130246"
    ],
    [
      "0.805906994",
      "0.213405124"
    ],
    [
      "0.071043138",
      "0.21852716"
    ],
    [
      "0.317144245",
      "0.218527955"
    ],
    [
      "0.682856693",
      "0.284448303"
    ],
    [
      "0.928956862",
      "0.284449053"
    ],
    [
      "0.194093462",
      "0.289570299"
    ],
    [
      "0.440203659",
      "0.289586839"
    ],
    [
      "0.805906561",
      "0.355492232"
    ],
    [
      "0.071043138",
      "0.360613438"
    ],
    [
      "0.317144247",
      "0.360614235"
    ],
    [
      "0.56518337",
      "0.364081832"
    ],
    [
      "0.928956862",
      "0.426535411"
    ],
    [
      "0.194093464",
      "0.431656579"
    ],
    [
      "0.440203659",
      "0.431673116"
    ],
    [
      "0.688233238",
      "0.435125761"
    ],
    [
      "0.071043138",
      "0.502699715"
    ],
    [
      "0.317144247",
      "0.502700512"
    ],
    [
      "0.565182463",
      "0.506168119"
    ],
    [
      "0.811283539",
      "0.50616894"
    ],
    [
      "0.194093464",
      "0.573742856"
    ],
    [
      "0.440202746",
      "0.573759392"
    ],
    [
      "0.68823234",
      "0.577212037"
    ],
    [
      "0.928956862",
      "0.585802469"
    ],
    [
      "0.071043138",
      "0.644785992"
    ],
    [
      "0.317143334",
      "0.644786788"
    ],
    [
      "0.565181759",
      "0.648255506"
    ],
    [
      "0.805906122",
      "0.656844888"
    ],
    [
      "0.440184569",
      "0.715845668"
    ],
    [
      "0.194260755",
      "0.716119681"
    ],
    [
      "0.682855797",
      "0.727888027"
    ],
    [
      "0.928956862",
      "0.727888746"
    ],
    [
      "0.071043138",
      "0.786872269"
    ],
    [
      "0.31730199",
      "0.787178562"
    ],
    [
      "0.55786797",
      "0.795464303"
    ],
    [
      "0.805906122",
      "0.798931165"
    ],
    [
      "0.19919341",
      "0.866165105"
    ],
    [
      "0.466592318",
      "0.904355374"
    ],
    [
      "0.920789515",
      "0.921696002"
    ],
    [
      "0.071734769",
      "0.928956862"
    ],
    [
      "0.326652052",
      "0.928956862"
    ],
    [
      "0.606532604",
      "0.928956862"
    ],
    [
      "0.748618881",
      "0.928956862"
    ]
  ],
  "radius": "0.071043138"
}

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