P85 · 装箱与覆盖 · 经典问题 · 难

正方形装入等边三角形 · n = 20

将 n 个边长为 1 的正方形装入等边三角形容器，允许边界接触但内部不得重叠，最小化容器等边三角形的边长。

子题n = 20

目标最小化容器三角形边长

纪录对比

容器三角形边长 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[7.61695, 7.61696)	Haowei Lin	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顶点为 (0,0)、(1,0)、(1/2,√3/2) 的等边三角形；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。
目标容器三角形边长 = 1/(2 sin(π/4) 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

顶点为 (0,0)、(1,0)、(1/2,√3/2) 的等边三角形

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": "0.131652498",
      "x": "0.486123621",
      "y": "0.710704738"
    },
    {
      "turn": "0.131652498",
      "x": "0.420470681",
      "y": "0.596990509"
    },
    {
      "turn": "0.131652498",
      "x": "0.561926044",
      "y": "0.579426240"
    },
    {
      "turn": "0.131652498",
      "x": "0.354825969",
      "y": "0.483290533"
    },
    {
      "turn": "0.131652498",
      "x": "0.496283007",
      "y": "0.465729166"
    },
    {
      "turn": "-0.122541828",
      "x": "0.662727702",
      "y": "0.441408945"
    },
    {
      "turn": "-0.068598303",
      "x": "0.571706786",
      "y": "0.332644473"
    },
    {
      "turn": "-0.131652498",
      "x": "0.734036221",
      "y": "0.329376706"
    },
    {
      "turn": "-0.395459928",
      "x": "0.414609275",
      "y": "0.328467898"
    },
    {
      "turn": "0.131652498",
      "x": "0.258855152",
      "y": "0.317064203"
    },
    {
      "turn": "-0.068598303",
      "x": "0.637864501",
      "y": "0.218055949"
    },
    {
      "turn": "-0.131652498",
      "x": "0.799999854",
      "y": "0.215124342"
    },
    {
      "turn": "0.131652498",
      "x": "0.193212116",
      "y": "0.203367129"
    },
    {
      "turn": "-0.414105290",
      "x": "0.476337674",
      "y": "0.197004097"
    },
    {
      "turn": "0.131652498",
      "x": "0.127569080",
      "y": "0.089670055"
    },
    {
      "turn": "-0.203095735",
      "x": "0.710810947",
      "y": "0.086310777"
    },
    {
      "turn": "-0.304463964",
      "x": "0.859248093",
      "y": "0.077102556"
    },
    {
      "turn": "0.414213562",
      "x": "0.424323257",
      "y": "0.065697223"
    },
    {
      "turn": "0.414213562",
      "x": "0.293037184",
      "y": "0.065643036"
    },
    {
      "turn": "-0.414213562",
      "x": "0.555609329",
      "y": "0.065643036"
    }
  ],
  "radius": "0.092833271"
}

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": 20,
  "sides": 5
}

起步布局答案

{
  "placements": [
    {
      "turn": "0.131652498",
      "x": "0.486123621",
      "y": "0.710704738"
    },
    {
      "turn": "0.131652498",
      "x": "0.420470681",
      "y": "0.596990509"
    },
    {
      "turn": "0.131652498",
      "x": "0.561926044",
      "y": "0.579426240"
    },
    {
      "turn": "0.131652498",
      "x": "0.354825969",
      "y": "0.483290533"
    },
    {
      "turn": "0.131652498",
      "x": "0.496283007",
      "y": "0.465729166"
    },
    {
      "turn": "-0.122541828",
      "x": "0.662727702",
      "y": "0.441408945"
    },
    {
      "turn": "-0.068598303",
      "x": "0.571706786",
      "y": "0.332644473"
    },
    {
      "turn": "-0.131652498",
      "x": "0.734036221",
      "y": "0.329376706"
    },
    {
      "turn": "-0.395459928",
      "x": "0.414609275",
      "y": "0.328467898"
    },
    {
      "turn": "0.131652498",
      "x": "0.258855152",
      "y": "0.317064203"
    },
    {
      "turn": "-0.068598303",
      "x": "0.637864501",
      "y": "0.218055949"
    },
    {
      "turn": "-0.131652498",
      "x": "0.799999854",
      "y": "0.215124342"
    },
    {
      "turn": "0.131652498",
      "x": "0.193212116",
      "y": "0.203367129"
    },
    {
      "turn": "-0.414105290",
      "x": "0.476337674",
      "y": "0.197004097"
    },
    {
      "turn": "0.131652498",
      "x": "0.127569080",
      "y": "0.089670055"
    },
    {
      "turn": "-0.203095735",
      "x": "0.710810947",
      "y": "0.086310777"
    },
    {
      "turn": "-0.304463964",
      "x": "0.859248093",
      "y": "0.077102556"
    },
    {
      "turn": "0.414213562",
      "x": "0.424323257",
      "y": "0.065697223"
    },
    {
      "turn": "0.414213562",
      "x": "0.293037184",
      "y": "0.065643036"
    },
    {
      "turn": "-0.414213562",
      "x": "0.555609329",
      "y": "0.065643036"
    }
  ],
  "radius": "0.092833271"
}

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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