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 = 194

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 = 194
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[0.0656106473024164284022592611995, 0.0656106473024164284022592612005]见来源署名packomania.com ↗
本站纪录0.065610647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 = 0.065610647

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 = 194 的当前纪录构型,0.065610647
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
    0.065610646→0.065610647
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": [
    [
      "1.465127713",
      "1.308955966"
    ],
    [
      "1.724489134",
      "1.348419153"
    ],
    [
      "1.043616391",
      "0.558791706"
    ],
    [
      "1.075545403",
      "1.252415039"
    ],
    [
      "1.573499656",
      "0.262313341"
    ],
    [
      "1.894950897",
      "1.268600734"
    ],
    [
      "0.586943995",
      "0.161866240"
    ],
    [
      "0.377683813",
      "1.035947931"
    ],
    [
      "1.238050093",
      "1.046336563"
    ],
    [
      "1.254729506",
      "0.474599443"
    ],
    [
      "0.900432492",
      "0.191785528"
    ],
    [
      "0.913315601",
      "1.458709686"
    ],
    [
      "0.199214618",
      "0.895214741"
    ],
    [
      "1.077590222",
      "0.310054751"
    ],
    [
      "0.279555599",
      "0.791453128"
    ],
    [
      "0.588796929",
      "0.951755670"
    ],
    [
      "0.475917592",
      "0.359196497"
    ],
    [
      "0.361719309",
      "0.689136265"
    ],
    [
      "0.556102322",
      "1.177027134"
    ],
    [
      "0.865197884",
      "0.417712502"
    ],
    [
      "0.215454011",
      "1.242242580"
    ],
    [
      "1.091784796",
      "1.599442876"
    ],
    [
      "0.313537854",
      "0.564511846"
    ],
    [
      "0.605476344",
      "0.380018550"
    ],
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      "1.031552039",
      "0.186619043"
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      "1.709574008",
      "0.994247131"
    ],
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      "1.302897911",
      "1.515250614"
    ],
    [
      "1.092002349",
      "0.070151071"
    ],
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      "1.798212011",
      "1.178767463"
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      "1.776266435",
      "0.479909734"
    ],
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      "1.706635395",
      "1.611350865"
    ],
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      "0.987224305",
      "1.934302009"
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      "1.054794907"
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      "1.579023978",
      "0.980991276"
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      "1.008642611"
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      "0.523733290",
      "1.402168758"
    ],
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      "1.346344935",
      "0.132170151"
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      "0.183456739",
      "0.545752971"
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      "1.432758681",
      "1.534097589"
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      "0.945684632",
      "1.233568062"
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      "0.843164525"
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      "1.660965584",
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      "0.370485552"
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      "0.393872520",
      "1.383321781"
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      "1.026936978",
      "1.130528825"
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      "0.829697326",
      "0.081261484"
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      "0.654084768",
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      "1.277069883",
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      "0.994567946",
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      "0.685963092",
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      "0.848467781",
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      "1.271262015"
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      "0.088929855",
      "1.207447954"
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      "1.654927671"
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      "1.065183538"
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      "0.620784283",
      "1.647093297"
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    [
      "1.286658518",
      "1.168222777"
    ]
  ],
  "radius": "0.065610647"
}
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": 194
}

当前第一名的答案

{
  "centers": [
    [
      "1.465127713",
      "1.308955966"
    ],
    [
      "1.724489134",
      "1.348419153"
    ],
    [
      "1.043616391",
      "0.558791706"
    ],
    [
      "1.075545403",
      "1.252415039"
    ],
    [
      "1.573499656",
      "0.262313341"
    ],
    [
      "1.894950897",
      "1.268600734"
    ],
    [
      "0.586943995",
      "0.161866240"
    ],
    [
      "0.377683813",
      "1.035947931"
    ],
    [
      "1.238050093",
      "1.046336563"
    ],
    [
      "1.254729506",
      "0.474599443"
    ],
    [
      "0.900432492",
      "0.191785528"
    ],
    [
      "0.913315601",
      "1.458709686"
    ],
    [
      "0.199214618",
      "0.895214741"
    ],
    [
      "1.077590222",
      "0.310054751"
    ],
    [
      "0.279555599",
      "0.791453128"
    ],
    [
      "0.588796929",
      "0.951755670"
    ],
    [
      "0.475917592",
      "0.359196497"
    ],
    [
      "0.361719309",
      "0.689136265"
    ],
    [
      "0.556102322",
      "1.177027134"
    ],
    [
      "0.865197884",
      "0.417712502"
    ],
    [
      "0.215454011",
      "1.242242580"
    ],
    [
      "1.091784796",
      "1.599442876"
    ],
    [
      "0.313537854",
      "0.564511846"
    ],
    [
      "0.605476344",
      "0.380018550"
    ],
    [
      "1.031552039",
      "0.186619043"
    ],
    [
      "1.709574008",
      "0.994247131"
    ],
    [
      "1.302897911",
      "1.515250614"
    ],
    [
      "1.092002349",
      "0.070151071"
    ],
    [
      "1.798212011",
      "1.178767463"
    ],
    [
      "1.776266435",
      "0.479909734"
    ],
    [
      "1.706635395",
      "1.611350865"
    ],
    [
      "0.987224305",
      "1.934302009"
    ],
    [
      "1.456132122",
      "0.320998007"
    ],
    [
      "0.653594061",
      "1.421015734"
    ],
    [
      "0.572832424",
      "0.604944004"
    ],
    [
      "1.416519288",
      "1.187069752"
    ],
    [
      "1.695752862",
      "0.376292202"
    ],
    [
      "0.720151842",
      "1.891494397"
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  "radius": "0.065610647"
}

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

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