P85 · 装箱与覆盖 · 经典问题 · 难

正方形装入等边三角形 · n = 23

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 = 23

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	[8.301, 8.302)	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。

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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

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": "0.487271459",
      "y": "0.723514723"
    },
    {
      "turn": "0.131652498",
      "x": "0.427039359",
      "y": "0.619189664"
    },
    {
      "turn": "0.131652498",
      "x": "0.556821499",
      "y": "0.603050522"
    },
    {
      "turn": "0.131652498",
      "x": "0.367494001",
      "y": "0.514468114"
    },
    {
      "turn": "0.131652498",
      "x": "0.498355296",
      "y": "0.497705921"
    },
    {
      "turn": "0.414210431",
      "x": "0.627737227",
      "y": "0.422158883"
    },
    {
      "turn": "0.131652498",
      "x": "0.306575157",
      "y": "0.410539547"
    },
    {
      "turn": "0.131652498",
      "x": "0.437138539",
      "y": "0.393949354"
    },
    {
      "turn": "0.131652498",
      "x": "0.246343056",
      "y": "0.306214488"
    },
    {
      "turn": "0.414208565",
      "x": "0.569329776",
      "y": "0.301679017"
    },
    {
      "turn": "0.414209957",
      "x": "0.689794313",
      "y": "0.301678380"
    },
    {
      "turn": "0.131652498",
      "x": "0.376125196",
      "y": "0.290075345"
    },
    {
      "turn": "0.131652498",
      "x": "0.186110955",
      "y": "0.201889429"
    },
    {
      "turn": "0.414213562",
      "x": "0.508825255",
      "y": "0.181198622"
    },
    {
      "turn": "0.414207456",
      "x": "0.629290085",
      "y": "0.181197051"
    },
    {
      "turn": "0.414200653",
      "x": "0.749754986",
      "y": "0.181196175"
    },
    {
      "turn": "0.414213562",
      "x": "0.388361054",
      "y": "0.147564664"
    },
    {
      "turn": "-0.231047658",
      "x": "0.268143194",
      "y": "0.076547495"
    },
    {
      "turn": "0.414224323",
      "x": "0.130700788",
      "y": "0.060871247"
    },
    {
      "turn": "0.414213562",
      "x": "0.508825255",
      "y": "0.060715696"
    },
    {
      "turn": "0.414213562",
      "x": "0.629289457",
      "y": "0.060714821"
    },
    {
      "turn": "0.414213562",
      "x": "0.749753658",
      "y": "0.060714821"
    },
    {
      "turn": "0.414213562",
      "x": "0.870217860",
      "y": "0.060232101"
    }
  ],
  "radius": "0.085181053"
}

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": 23,
  "sides": 5
}

起步布局答案

{
  "placements": [
    {
      "turn": "0.131652498",
      "x": "0.487271459",
      "y": "0.723514723"
    },
    {
      "turn": "0.131652498",
      "x": "0.427039359",
      "y": "0.619189664"
    },
    {
      "turn": "0.131652498",
      "x": "0.556821499",
      "y": "0.603050522"
    },
    {
      "turn": "0.131652498",
      "x": "0.367494001",
      "y": "0.514468114"
    },
    {
      "turn": "0.131652498",
      "x": "0.498355296",
      "y": "0.497705921"
    },
    {
      "turn": "0.414210431",
      "x": "0.627737227",
      "y": "0.422158883"
    },
    {
      "turn": "0.131652498",
      "x": "0.306575157",
      "y": "0.410539547"
    },
    {
      "turn": "0.131652498",
      "x": "0.437138539",
      "y": "0.393949354"
    },
    {
      "turn": "0.131652498",
      "x": "0.246343056",
      "y": "0.306214488"
    },
    {
      "turn": "0.414208565",
      "x": "0.569329776",
      "y": "0.301679017"
    },
    {
      "turn": "0.414209957",
      "x": "0.689794313",
      "y": "0.301678380"
    },
    {
      "turn": "0.131652498",
      "x": "0.376125196",
      "y": "0.290075345"
    },
    {
      "turn": "0.131652498",
      "x": "0.186110955",
      "y": "0.201889429"
    },
    {
      "turn": "0.414213562",
      "x": "0.508825255",
      "y": "0.181198622"
    },
    {
      "turn": "0.414207456",
      "x": "0.629290085",
      "y": "0.181197051"
    },
    {
      "turn": "0.414200653",
      "x": "0.749754986",
      "y": "0.181196175"
    },
    {
      "turn": "0.414213562",
      "x": "0.388361054",
      "y": "0.147564664"
    },
    {
      "turn": "-0.231047658",
      "x": "0.268143194",
      "y": "0.076547495"
    },
    {
      "turn": "0.414224323",
      "x": "0.130700788",
      "y": "0.060871247"
    },
    {
      "turn": "0.414213562",
      "x": "0.508825255",
      "y": "0.060715696"
    },
    {
      "turn": "0.414213562",
      "x": "0.629289457",
      "y": "0.060714821"
    },
    {
      "turn": "0.414213562",
      "x": "0.749753658",
      "y": "0.060714821"
    },
    {
      "turn": "0.414213562",
      "x": "0.870217860",
      "y": "0.060232101"
    }
  ],
  "radius": "0.085181053"
}

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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