P84 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入圆盘 · n = 36

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 = 36

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	[2.43969, 2.4397)	Thomas Schadt	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": "0.119843188",
      "x": "1.260454368",
      "y": "1.815577958"
    },
    {
      "turn": "0.119843188",
      "x": "0.839511951",
      "y": "1.774913997"
    },
    {
      "turn": "-0.427900037",
      "x": "0.609564381",
      "y": "1.718995575"
    },
    {
      "turn": "-0.427900037",
      "x": "1.056082732",
      "y": "1.654486257"
    },
    {
      "turn": "-0.427900037",
      "x": "1.423854926",
      "y": "1.644396732"
    },
    {
      "turn": "0.119843188",
      "x": "0.535804239",
      "y": "1.494009591"
    },
    {
      "turn": "-0.237398095",
      "x": "1.719247112",
      "y": "1.472288459"
    },
    {
      "turn": "-0.427900037",
      "x": "0.305856670",
      "y": "1.438091169"
    },
    {
      "turn": "0.119843188",
      "x": "0.989535721",
      "y": "1.427386609"
    },
    {
      "turn": "0.119843188",
      "x": "1.357307915",
      "y": "1.417297084"
    },
    {
      "turn": "-0.427900037",
      "x": "0.759588151",
      "y": "1.371468187"
    },
    {
      "turn": "-0.427900037",
      "x": "1.152936279",
      "y": "1.256205383"
    },
    {
      "turn": "0.119843188",
      "x": "0.239309659",
      "y": "1.210991521"
    },
    {
      "turn": "0.134412340",
      "x": "1.742157271",
      "y": "1.178986598"
    },
    {
      "turn": "0.119843188",
      "x": "0.676404601",
      "y": "1.137487116"
    },
    {
      "turn": "-0.411039993",
      "x": "1.513907437",
      "y": "1.116498456"
    },
    {
      "turn": "-0.427900037",
      "x": "0.446457031",
      "y": "1.081568694"
    },
    {
      "turn": "0.119843188",
      "x": "1.066674459",
      "y": "1.034882766"
    },
    {
      "turn": "-0.427900037",
      "x": "0.839805160",
      "y": "0.966305890"
    },
    {
      "turn": "-0.411039993",
      "x": "1.794262577",
      "y": "0.895157930"
    },
    {
      "turn": "0.134412340",
      "x": "1.453898838",
      "y": "0.887584230"
    },
    {
      "turn": "0.119843188",
      "x": "0.163990476",
      "y": "0.863162424"
    },
    {
      "turn": "-0.411039993",
      "x": "1.219935209",
      "y": "0.845966813"
    },
    {
      "turn": "0.119843188",
      "x": "0.601085418",
      "y": "0.789658018"
    },
    {
      "turn": "-0.427900037",
      "x": "0.371137849",
      "y": "0.733739596"
    },
    {
      "turn": "0.134412340",
      "x": "1.734253978",
      "y": "0.666243705"
    },
    {
      "turn": "0.119843188",
      "x": "0.998366812",
      "y": "0.658220851"
    },
    {
      "turn": "-0.427900037",
      "x": "0.764485977",
      "y": "0.618476792"
    },
    {
      "turn": "-0.411039993",
      "x": "1.506004144",
      "y": "0.603755563"
    },
    {
      "turn": "-0.427900037",
      "x": "1.167774162",
      "y": "0.492773391"
    },
    {
      "turn": "0.119843188",
      "x": "0.404964190",
      "y": "0.477227545"
    },
    {
      "turn": "0.134412340",
      "x": "1.538214718",
      "y": "0.350666591"
    },
    {
      "turn": "0.119843188",
      "x": "0.802245584",
      "y": "0.345790377"
    },
    {
      "turn": "-0.427900037",
      "x": "0.568364748",
      "y": "0.306046319"
    },
    {
      "turn": "0.119843188",
      "x": "1.201600503",
      "y": "0.236261339"
    },
    {
      "turn": "-0.427900037",
      "x": "0.971652934",
      "y": "0.180342917"
    }
  ],
  "radius": "0.236649010"
}

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": 36,
  "sides": 5
}

起步布局答案

{
  "placements": [
    {
      "turn": "0.119843188",
      "x": "1.260454368",
      "y": "1.815577958"
    },
    {
      "turn": "0.119843188",
      "x": "0.839511951",
      "y": "1.774913997"
    },
    {
      "turn": "-0.427900037",
      "x": "0.609564381",
      "y": "1.718995575"
    },
    {
      "turn": "-0.427900037",
      "x": "1.056082732",
      "y": "1.654486257"
    },
    {
      "turn": "-0.427900037",
      "x": "1.423854926",
      "y": "1.644396732"
    },
    {
      "turn": "0.119843188",
      "x": "0.535804239",
      "y": "1.494009591"
    },
    {
      "turn": "-0.237398095",
      "x": "1.719247112",
      "y": "1.472288459"
    },
    {
      "turn": "-0.427900037",
      "x": "0.305856670",
      "y": "1.438091169"
    },
    {
      "turn": "0.119843188",
      "x": "0.989535721",
      "y": "1.427386609"
    },
    {
      "turn": "0.119843188",
      "x": "1.357307915",
      "y": "1.417297084"
    },
    {
      "turn": "-0.427900037",
      "x": "0.759588151",
      "y": "1.371468187"
    },
    {
      "turn": "-0.427900037",
      "x": "1.152936279",
      "y": "1.256205383"
    },
    {
      "turn": "0.119843188",
      "x": "0.239309659",
      "y": "1.210991521"
    },
    {
      "turn": "0.134412340",
      "x": "1.742157271",
      "y": "1.178986598"
    },
    {
      "turn": "0.119843188",
      "x": "0.676404601",
      "y": "1.137487116"
    },
    {
      "turn": "-0.411039993",
      "x": "1.513907437",
      "y": "1.116498456"
    },
    {
      "turn": "-0.427900037",
      "x": "0.446457031",
      "y": "1.081568694"
    },
    {
      "turn": "0.119843188",
      "x": "1.066674459",
      "y": "1.034882766"
    },
    {
      "turn": "-0.427900037",
      "x": "0.839805160",
      "y": "0.966305890"
    },
    {
      "turn": "-0.411039993",
      "x": "1.794262577",
      "y": "0.895157930"
    },
    {
      "turn": "0.134412340",
      "x": "1.453898838",
      "y": "0.887584230"
    },
    {
      "turn": "0.119843188",
      "x": "0.163990476",
      "y": "0.863162424"
    },
    {
      "turn": "-0.411039993",
      "x": "1.219935209",
      "y": "0.845966813"
    },
    {
      "turn": "0.119843188",
      "x": "0.601085418",
      "y": "0.789658018"
    },
    {
      "turn": "-0.427900037",
      "x": "0.371137849",
      "y": "0.733739596"
    },
    {
      "turn": "0.134412340",
      "x": "1.734253978",
      "y": "0.666243705"
    },
    {
      "turn": "0.119843188",
      "x": "0.998366812",
      "y": "0.658220851"
    },
    {
      "turn": "-0.427900037",
      "x": "0.764485977",
      "y": "0.618476792"
    },
    {
      "turn": "-0.411039993",
      "x": "1.506004144",
      "y": "0.603755563"
    },
    {
      "turn": "-0.427900037",
      "x": "1.167774162",
      "y": "0.492773391"
    },
    {
      "turn": "0.119843188",
      "x": "0.404964190",
      "y": "0.477227545"
    },
    {
      "turn": "0.134412340",
      "x": "1.538214718",
      "y": "0.350666591"
    },
    {
      "turn": "0.119843188",
      "x": "0.802245584",
      "y": "0.345790377"
    },
    {
      "turn": "-0.427900037",
      "x": "0.568364748",
      "y": "0.306046319"
    },
    {
      "turn": "0.119843188",
      "x": "1.201600503",
      "y": "0.236261339"
    },
    {
      "turn": "-0.427900037",
      "x": "0.971652934",
      "y": "0.180342917"
    }
  ],
  "radius": "0.236649010"
}

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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