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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 = 66

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 = 66
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[0.1099350572982705667637977465085, 0.1099350572982705667637977465095]见来源署名packomania.com ↗
本站纪录0.109935057游侠纪录详情 ↓

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：本站纪录

2y0
0x2
已验证构造r = 0.109935057

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 = 66 的当前纪录构型，0.109935057
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
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. 游侠AI · OpenAI: GPT-5.6 Sol
    0.109935056→0.109935057
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"。

当前第一名的答案

{
  "centers": [
    [
      "1.110385142",
      "0.116806520"
    ],
    [
      "0.722909802",
      "0.154165132"
    ],
    [
      "1.323519266",
      "0.170813117"
    ],
    [
      "0.926462608",
      "0.237286264"
    ],
    [
      "0.520389674",
      "0.255735190"
    ],
    [
      "1.516911480",
      "0.275418656"
    ],
    [
      "1.122762395",
      "0.336327979"
    ],
    [
      "0.706188713",
      "0.373398506"
    ],
    [
      "0.345436538",
      "0.396872919"
    ],
    [
      "1.333980748",
      "0.397399041"
    ],
    [
      "1.678760534",
      "0.424239860"
    ],
    [
      "0.912590455",
      "0.456718328"
    ],
    [
      "0.533940871",
      "0.510049699"
    ],
    [
      "1.495829802",
      "0.546220245"
    ],
    [
      "1.108890242",
      "0.555760043"
    ],
    [
      "0.738408197",
      "0.590895108"
    ],
    [
      "1.799190018",
      "0.608195307"
    ],
    [
      "0.341674738",
      "0.616710850"
    ],
    [
      "1.304932012",
      "0.655311495"
    ],
    [
      "0.934707983",
      "0.689936822"
    ],
    [
      "0.152350702",
      "0.728511013"
    ],
    [
      "0.536100635",
      "0.729909206"
    ],
    [
      "1.616259286",
      "0.730175692"
    ],
    [
      "1.130749753",
      "0.789488274"
    ],
    [
      "0.740567961",
      "0.810754615"
    ],
    [
      "1.870851039",
      "0.816059603"
    ],
    [
      "0.343834501",
      "0.836570357"
    ],
    [
      "1.425361496",
      "0.839266942"
    ],
    [
      "0.936867747",
      "0.909796329"
    ],
    [
      "1.687920307",
      "0.938039988"
    ],
    [
      "0.111661886",
      "0.944583426"
    ],
    [
      "0.532338834",
      "0.949747137"
    ],
    [
      "1.251179237",
      "0.973443722"
    ],
    [
      "0.752945212",
      "1.030276074"
    ],
    [
      "1.889370672",
      "1.035148376"
    ],
    [
      "0.315915920",
      "1.054660752"
    ],
    [
      "1.454470866",
      "1.057201592"
    ],
    [
      "1.064165735",
      "1.089067323"
    ],
    [
      "1.651459565",
      "1.154865913"
    ],
    [
      "0.125181559",
      "1.164037489"
    ],
    [
      "0.578900053",
      "1.164630641"
    ],
    [
      "1.265043229",
      "1.192995955"
    ],
    [
      "0.869149271",
      "1.216929453"
    ],
    [
      "1.853618806",
      "1.252092312"
    ],
    [
      "0.382819082",
      "1.264104859"
    ],
    [
      "1.468385432",
      "1.276630970"
    ],
    [
      "1.068865670",
      "1.308887199"
    ],
    [
      "0.695104112",
      "1.351284020"
    ],
    [
      "0.192084721",
      "1.373481596"
    ],
    [
      "1.670575348",
      "1.373903476"
    ],
    [
      "1.262557357",
      "1.412937177"
    ],
    [
      "0.894820511",
      "1.443241766"
    ],
    [
      "0.499023141",
      "1.450758238"
    ],
    [
      "1.465934405",
      "1.496487423"
    ],
    [
      "1.088512197",
      "1.547291744"
    ],
    [
      "0.308288780",
      "1.560134975"
    ],
    [
      "1.663171609",
      "1.593648901"
    ],
    [
      "0.657437053",
      "1.603230857"
    ],
    [
      "1.291889245",
      "1.630841990"
    ],
    [
      "0.869407425",
      "1.661638292"
    ],
    [
      "0.466702692",
      "1.712607594"
    ],
    [
      "1.496974899",
      "1.738331662"
    ],
    [
      "1.063099112",
      "1.765688270"
    ],
    [
      "0.661386409",
      "1.823094065"
    ],
    [
      "1.266476160",
      "1.849238516"
    ],
    [
      "0.876143345",
      "1.881405202"
    ]
  ],
  "radius": "0.109935057"
}
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": 66
}

当前第一名的答案

{
  "centers": [
    [
      "1.110385142",
      "0.116806520"
    ],
    [
      "0.722909802",
      "0.154165132"
    ],
    [
      "1.323519266",
      "0.170813117"
    ],
    [
      "0.926462608",
      "0.237286264"
    ],
    [
      "0.520389674",
      "0.255735190"
    ],
    [
      "1.516911480",
      "0.275418656"
    ],
    [
      "1.122762395",
      "0.336327979"
    ],
    [
      "0.706188713",
      "0.373398506"
    ],
    [
      "0.345436538",
      "0.396872919"
    ],
    [
      "1.333980748",
      "0.397399041"
    ],
    [
      "1.678760534",
      "0.424239860"
    ],
    [
      "0.912590455",
      "0.456718328"
    ],
    [
      "0.533940871",
      "0.510049699"
    ],
    [
      "1.495829802",
      "0.546220245"
    ],
    [
      "1.108890242",
      "0.555760043"
    ],
    [
      "0.738408197",
      "0.590895108"
    ],
    [
      "1.799190018",
      "0.608195307"
    ],
    [
      "0.341674738",
      "0.616710850"
    ],
    [
      "1.304932012",
      "0.655311495"
    ],
    [
      "0.934707983",
      "0.689936822"
    ],
    [
      "0.152350702",
      "0.728511013"
    ],
    [
      "0.536100635",
      "0.729909206"
    ],
    [
      "1.616259286",
      "0.730175692"
    ],
    [
      "1.130749753",
      "0.789488274"
    ],
    [
      "0.740567961",
      "0.810754615"
    ],
    [
      "1.870851039",
      "0.816059603"
    ],
    [
      "0.343834501",
      "0.836570357"
    ],
    [
      "1.425361496",
      "0.839266942"
    ],
    [
      "0.936867747",
      "0.909796329"
    ],
    [
      "1.687920307",
      "0.938039988"
    ],
    [
      "0.111661886",
      "0.944583426"
    ],
    [
      "0.532338834",
      "0.949747137"
    ],
    [
      "1.251179237",
      "0.973443722"
    ],
    [
      "0.752945212",
      "1.030276074"
    ],
    [
      "1.889370672",
      "1.035148376"
    ],
    [
      "0.315915920",
      "1.054660752"
    ],
    [
      "1.454470866",
      "1.057201592"
    ],
    [
      "1.064165735",
      "1.089067323"
    ],
    [
      "1.651459565",
      "1.154865913"
    ],
    [
      "0.125181559",
      "1.164037489"
    ],
    [
      "0.578900053",
      "1.164630641"
    ],
    [
      "1.265043229",
      "1.192995955"
    ],
    [
      "0.869149271",
      "1.216929453"
    ],
    [
      "1.853618806",
      "1.252092312"
    ],
    [
      "0.382819082",
      "1.264104859"
    ],
    [
      "1.468385432",
      "1.276630970"
    ],
    [
      "1.068865670",
      "1.308887199"
    ],
    [
      "0.695104112",
      "1.351284020"
    ],
    [
      "0.192084721",
      "1.373481596"
    ],
    [
      "1.670575348",
      "1.373903476"
    ],
    [
      "1.262557357",
      "1.412937177"
    ],
    [
      "0.894820511",
      "1.443241766"
    ],
    [
      "0.499023141",
      "1.450758238"
    ],
    [
      "1.465934405",
      "1.496487423"
    ],
    [
      "1.088512197",
      "1.547291744"
    ],
    [
      "0.308288780",
      "1.560134975"
    ],
    [
      "1.663171609",
      "1.593648901"
    ],
    [
      "0.657437053",
      "1.603230857"
    ],
    [
      "1.291889245",
      "1.630841990"
    ],
    [
      "0.869407425",
      "1.661638292"
    ],
    [
      "0.466702692",
      "1.712607594"
    ],
    [
      "1.496974899",
      "1.738331662"
    ],
    [
      "1.063099112",
      "1.765688270"
    ],
    [
      "0.661386409",
      "1.823094065"
    ],
    [
      "1.266476160",
      "1.849238516"
    ],
    [
      "0.876143345",
      "1.881405202"
    ]
  ],
  "radius": "0.109935057"
}

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

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。

还没有帖子。第一个聊聊这道题的思路？