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 = 163

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 = 163
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[0.0710974416398281166158454337795, 0.0710974416398281166158454337805]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 = 0.071097441

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 = 163 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": [
    [
      "0.839369354",
      "0.822753482"
    ],
    [
      "0.110842204",
      "1.268809182"
    ],
    [
      "1.563684758",
      "0.261677945"
    ],
    [
      "0.083882132",
      "0.846416098"
    ],
    [
      "1.258324197",
      "1.614804931"
    ],
    [
      "1.012910846",
      "0.487699398"
    ],
    [
      "1.037336058",
      "0.1162157"
    ],
    [
      "1.279317525",
      "1.092737634"
    ],
    [
      "0.444250245",
      "1.333754332"
    ],
    [
      "1.020649251",
      "1.798703748"
    ],
    [
      "0.373174183",
      "1.21059761"
    ],
    [
      "0.266667398",
      "1.304808489"
    ],
    [
      "1.20289436",
      "1.327152818"
    ],
    [
      "1.734715454",
      "0.431622339"
    ],
    [
      "1.046194422",
      "0.62748392"
    ],
    [
      "0.47371581",
      "1.472862798"
    ],
    [
      "1.12708146",
      "1.703761576"
    ],
    [
      "1.355212482",
      "0.858434177"
    ],
    [
      "1.710270061",
      "1.374671774"
    ],
    [
      "0.962201481",
      "1.378064281"
    ],
    [
      "1.443587239",
      "1.276241355"
    ],
    [
      "0.312913524",
      "0.374882302"
    ],
    [
      "1.789261962",
      "1.140213758"
    ],
    [
      "0.361514974",
      "0.528712368"
    ],
    [
      "0.87847326",
      "1.801021621"
    ],
    [
      "0.460187163",
      "0.833126918"
    ],
    [
      "0.428264042",
      "0.694561774"
    ],
    [
      "1.492641581",
      "0.904695681"
    ],
    [
      "0.766013644",
      "1.054398018"
    ],
    [
      "0.797236214",
      "1.195185591"
    ],
    [
      "1.469902715",
      "1.424603144"
    ],
    [
      "0.968384689",
      "0.352655678"
    ],
    [
      "0.63682245",
      "1.854963175"
    ],
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      "0.071174414",
      "0.988042009"
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    ],
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    ],
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      "1.384902585",
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      "0.225552479",
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      "0.798097346",
      "0.174722052"
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      "0.596181901",
      "0.874657821"
    ],
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      "0.978716231",
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      "1.598226642",
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      "0.849797651",
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      "1.575487776",
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      "0.778237216",
      "1.902042618"
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      "1.537257381",
      "0.529612215"
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      "1.187578798"
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      "1.427965211"
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      "0.902821274",
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      "0.826926317",
      "1.334246307"
    ],
    [
      "0.831876004",
      "0.312846586"
    ],
    [
      "0.398251584",
      "1.707643135"
    ],
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      "1.176715889",
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    ],
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      "0.991363376",
      "1.517236723"
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      "0.661581461",
      "0.134937658"
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      "0.740914122",
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      "1.568536538",
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      "0.49747747",
      "0.570348706"
    ],
    [
      "1.733338125",
      "0.853773046"
    ],
    [
      "1.812858261",
      "0.550420852"
    ],
    [
      "1.252413969",
      "0.431652815"
    ],
    [
      "1.401280709",
      "0.488022198"
    ],
    [
      "1.258934462",
      "1.756998505"
    ],
    [
      "1.656922854",
      "1.08819698"
    ],
    [
      "0.991973642",
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      "0.56425878",
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      "1.927213572"
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    [
      "1.36427012",
      "1.519963768"
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      "1.07164136"
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      "1.226341732",
      "1.900904758"
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      "0.853750293",
      "1.481261736"
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    [
      "0.292301546",
      "0.652925436"
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      "0.356147789",
      "0.930055724"
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      "0.579454493",
      "1.377790637"
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      "1.143760824"
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    [
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    ]
  ],
  "radius": "0.071097441"
}
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": 163
}

起步布局答案

{
  "centers": [
    [
      "0.839369354",
      "0.822753482"
    ],
    [
      "0.110842204",
      "1.268809182"
    ],
    [
      "1.563684758",
      "0.261677945"
    ],
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      "0.846416098"
    ],
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      "1.258324197",
      "1.614804931"
    ],
    [
      "1.012910846",
      "0.487699398"
    ],
    [
      "1.037336058",
      "0.1162157"
    ],
    [
      "1.279317525",
      "1.092737634"
    ],
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      "0.444250245",
      "1.333754332"
    ],
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      "1.020649251",
      "1.798703748"
    ],
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      "0.373174183",
      "1.21059761"
    ],
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      "1.327152818"
    ],
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      "0.431622339"
    ],
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      "0.62748392"
    ],
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      "1.472862798"
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      "1.12708146",
      "1.703761576"
    ],
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      "0.858434177"
    ],
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    ],
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  "radius": "0.071097441"
}

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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