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

已知最好（未证明）2.138来源精度区间：2.138–2.139（区间内视为追平）本站展示证书 2.138465229Erich Friedman 编录；Found by David W. Cantrell in July 2005. 原式：r = 2.138+。

严格定义

容器编辑器及证书统一缩放到圆心 (1,1)、半径 1 的闭圆盘；仅是坐标约定，分数按文献的单位图形计算。边界接触允许。
提交提交 {radius, placements:[{x,y,turn},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。
约束恰好 n 个同样大的图形，完整位于容器内，任意两图形内部不重叠。多边形可独立旋转，不限直角。
精度输入小数精确表示有理数；正多边形的顶点用代数数定义，不截断成小数。可行性验算没有浮点容差。有限小数证书不等于连续最优性证明。
目标容器圆半径 = 1/(2 sin(π/3) r)，越小越好。页面、排行榜和外部纪录使用相同单位，自动计算，无需手工换算。

放大来摆，然后提交 ↗

已验证构造已验证构造

帮助理解

外部来源与证书

外部目标按原始印刷精度换算，本站坐标证书另存。复现损失不降低文献目标；源表的构造不自动视为最优性证明。

查看来源

当前第一名

2.138465229

容器圆半径

已追平已知最好

答案来源David W. Cantrell

解题方式公开参考构造

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。

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},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。

当前第一名的答案

{
  "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
}

当前第一名的答案

{
  "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},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。 · 验证器 v1.0.0

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