P84 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入圆盘 · n = 27

把 n 个边长为 1 的等边三角形装入圆盘，允许接触但内部不能重叠，最小化容器圆半径。

子题n = 27

目标最小化容器圆半径

纪录对比

容器圆半径 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[2.138, 2.139)	David W. Cantrell	Erich Friedman ↗
本站纪录	暂无选手纪录	—	—

严格定义

容器编辑器与证书将容器统一归一化为圆心 (1,1)、半径 1 的闭圆盘；公开得分按文献的单位边长图形规范折算。允许接触边界。
提交提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。
约束恰好 n 个全等图形完全位于容器内，任意两图形内部互不重叠。多边形允许任意角度独立旋转。
精度输入的十进制小数按精确有理数解析，正多边形顶点由精确代数数定义，可行性检验不设浮点容差。
目标容器圆半径 = 1/(2 sin(π/3) r)，越小越好。页面、排行榜与文献纪录统一使用该单位并自动换算。

放大编辑并提交 ↗

当前展示：起步布局

已验证构造已验证构造

帮助理解

文献来源与坐标证书

文献目标值按原始发表精度换算，九位小数坐标证书单独存储并与文献目标对照展示。

查看来源

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 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

ANSWER FORMAT

答案怎么写

圆心 (1,1)、半径 1 的闭圆盘

提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。

起步布局答案

{
  "placements": [
    {
      "turn": "0.399178434",
      "x": "0.804231102",
      "y": "1.814081697"
    },
    {
      "turn": "0.144800318",
      "x": "1.195768898",
      "y": "1.814081697"
    },
    {
      "turn": "0.366295857",
      "x": "0.491652134",
      "y": "1.555777406"
    },
    {
      "turn": "0.174211903",
      "x": "1.508347866",
      "y": "1.555777406"
    },
    {
      "turn": "-0.174211903",
      "x": "0.745730445",
      "y": "1.464479605"
    },
    {
      "turn": "-0.366295857",
      "x": "1.254269555",
      "y": "1.464479605"
    },
    {
      "turn": "-0.238231847",
      "x": "0.252645603",
      "y": "1.377512751"
    },
    {
      "turn": "-0.298114758",
      "x": "1.747354397",
      "y": "1.377512751"
    },
    {
      "turn": "0.267949192",
      "x": "1.000000000",
      "y": "1.196006996"
    },
    {
      "turn": "0.267949192",
      "x": "0.480994160",
      "y": "1.107012935"
    },
    {
      "turn": "0.267949192",
      "x": "1.519005840",
      "y": "1.107012935"
    },
    {
      "turn": "-0.267949192",
      "x": "0.766187452",
      "y": "1.061015258"
    },
    {
      "turn": "-0.267949192",
      "x": "1.233812548",
      "y": "1.061015258"
    },
    {
      "turn": "-0.267949192",
      "x": "0.247181612",
      "y": "0.972021198"
    },
    {
      "turn": "-0.267949192",
      "x": "1.752818388",
      "y": "0.972021198"
    },
    {
      "turn": "0.267949192",
      "x": "0.766187452",
      "y": "0.791031784"
    },
    {
      "turn": "0.267949192",
      "x": "1.233812548",
      "y": "0.791031784"
    },
    {
      "turn": "0.267949192",
      "x": "0.247181612",
      "y": "0.702037723"
    },
    {
      "turn": "0.267949192",
      "x": "1.752818388",
      "y": "0.702037723"
    },
    {
      "turn": "-0.267949192",
      "x": "1.000000000",
      "y": "0.656040046"
    },
    {
      "turn": "-0.267949192",
      "x": "0.532374905",
      "y": "0.656040046"
    },
    {
      "turn": "-0.267949192",
      "x": "1.467625095",
      "y": "0.656040046"
    },
    {
      "turn": "0.267949192",
      "x": "1.000000000",
      "y": "0.386056572"
    },
    {
      "turn": "0.267949192",
      "x": "0.532374905",
      "y": "0.386056572"
    },
    {
      "turn": "0.267949192",
      "x": "1.467625095",
      "y": "0.386056572"
    },
    {
      "turn": "-0.267949192",
      "x": "0.766187452",
      "y": "0.251064834"
    },
    {
      "turn": "-0.267949192",
      "x": "1.233812548",
      "y": "0.251064834"
    }
  ],
  "radius": "0.269983473"
}

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": 27,
  "sides": 5
}

起步布局答案

{
  "placements": [
    {
      "turn": "0.399178434",
      "x": "0.804231102",
      "y": "1.814081697"
    },
    {
      "turn": "0.144800318",
      "x": "1.195768898",
      "y": "1.814081697"
    },
    {
      "turn": "0.366295857",
      "x": "0.491652134",
      "y": "1.555777406"
    },
    {
      "turn": "0.174211903",
      "x": "1.508347866",
      "y": "1.555777406"
    },
    {
      "turn": "-0.174211903",
      "x": "0.745730445",
      "y": "1.464479605"
    },
    {
      "turn": "-0.366295857",
      "x": "1.254269555",
      "y": "1.464479605"
    },
    {
      "turn": "-0.238231847",
      "x": "0.252645603",
      "y": "1.377512751"
    },
    {
      "turn": "-0.298114758",
      "x": "1.747354397",
      "y": "1.377512751"
    },
    {
      "turn": "0.267949192",
      "x": "1.000000000",
      "y": "1.196006996"
    },
    {
      "turn": "0.267949192",
      "x": "0.480994160",
      "y": "1.107012935"
    },
    {
      "turn": "0.267949192",
      "x": "1.519005840",
      "y": "1.107012935"
    },
    {
      "turn": "-0.267949192",
      "x": "0.766187452",
      "y": "1.061015258"
    },
    {
      "turn": "-0.267949192",
      "x": "1.233812548",
      "y": "1.061015258"
    },
    {
      "turn": "-0.267949192",
      "x": "0.247181612",
      "y": "0.972021198"
    },
    {
      "turn": "-0.267949192",
      "x": "1.752818388",
      "y": "0.972021198"
    },
    {
      "turn": "0.267949192",
      "x": "0.766187452",
      "y": "0.791031784"
    },
    {
      "turn": "0.267949192",
      "x": "1.233812548",
      "y": "0.791031784"
    },
    {
      "turn": "0.267949192",
      "x": "0.247181612",
      "y": "0.702037723"
    },
    {
      "turn": "0.267949192",
      "x": "1.752818388",
      "y": "0.702037723"
    },
    {
      "turn": "-0.267949192",
      "x": "1.000000000",
      "y": "0.656040046"
    },
    {
      "turn": "-0.267949192",
      "x": "0.532374905",
      "y": "0.656040046"
    },
    {
      "turn": "-0.267949192",
      "x": "1.467625095",
      "y": "0.656040046"
    },
    {
      "turn": "0.267949192",
      "x": "1.000000000",
      "y": "0.386056572"
    },
    {
      "turn": "0.267949192",
      "x": "0.532374905",
      "y": "0.386056572"
    },
    {
      "turn": "0.267949192",
      "x": "1.467625095",
      "y": "0.386056572"
    },
    {
      "turn": "-0.267949192",
      "x": "0.766187452",
      "y": "0.251064834"
    },
    {
      "turn": "-0.267949192",
      "x": "1.233812548",
      "y": "0.251064834"
    }
  ],
  "radius": "0.269983473"
}

提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。 · 验证器 v1.0.0

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