P83 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入正方形 · n = 13

把 n 个边长为 1 的等边三角形装入正方形，允许接触但内部不能重叠，最小化容器正方形边长。

子题n = 13

目标最小化容器正方形边长

纪录对比

容器正方形边长 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[2.595, 2.596)	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单位正方形 [0,1]²；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。
目标容器正方形边长 = 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。

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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

单位正方形 [0,1]²

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": "0.267949192",
      "x": "0.415130151",
      "y": "0.888766211"
    },
    {
      "turn": "-0.272346739",
      "x": "0.609616243",
      "y": "0.777539896"
    },
    {
      "turn": "0.263561294",
      "x": "0.803184104",
      "y": "0.887190735"
    },
    {
      "turn": "0.000000000",
      "x": "0.111233789",
      "y": "0.807337426"
    },
    {
      "turn": "0.577350269",
      "x": "0.222467577",
      "y": "0.614674853"
    },
    {
      "turn": "-0.556944803",
      "x": "0.882871031",
      "y": "0.584372400"
    },
    {
      "turn": "-0.267949192",
      "x": "0.475377611",
      "y": "0.444935155"
    },
    {
      "turn": "-0.000000000",
      "x": "0.111233789",
      "y": "0.422012279"
    },
    {
      "turn": "0.015440530",
      "x": "0.777638474",
      "y": "0.388367471"
    },
    {
      "turn": "-0.565441941",
      "x": "0.885325147",
      "y": "0.190634470"
    },
    {
      "turn": "-0.267949192",
      "x": "0.192662574",
      "y": "0.111233789"
    },
    {
      "turn": "0.267949192",
      "x": "0.385325147",
      "y": "0.222467577"
    },
    {
      "turn": "-0.267949192",
      "x": "0.577987721",
      "y": "0.111233789"
    }
  ],
  "radius": "0.222467576"
}

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": 13,
  "sides": 5
}

起步布局答案

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.415130151",
      "y": "0.888766211"
    },
    {
      "turn": "-0.272346739",
      "x": "0.609616243",
      "y": "0.777539896"
    },
    {
      "turn": "0.263561294",
      "x": "0.803184104",
      "y": "0.887190735"
    },
    {
      "turn": "0.000000000",
      "x": "0.111233789",
      "y": "0.807337426"
    },
    {
      "turn": "0.577350269",
      "x": "0.222467577",
      "y": "0.614674853"
    },
    {
      "turn": "-0.556944803",
      "x": "0.882871031",
      "y": "0.584372400"
    },
    {
      "turn": "-0.267949192",
      "x": "0.475377611",
      "y": "0.444935155"
    },
    {
      "turn": "-0.000000000",
      "x": "0.111233789",
      "y": "0.422012279"
    },
    {
      "turn": "0.015440530",
      "x": "0.777638474",
      "y": "0.388367471"
    },
    {
      "turn": "-0.565441941",
      "x": "0.885325147",
      "y": "0.190634470"
    },
    {
      "turn": "-0.267949192",
      "x": "0.192662574",
      "y": "0.111233789"
    },
    {
      "turn": "0.267949192",
      "x": "0.385325147",
      "y": "0.222467577"
    },
    {
      "turn": "-0.267949192",
      "x": "0.577987721",
      "y": "0.111233789"
    }
  ],
  "radius": "0.222467576"
}

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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