P84 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入圆盘 · n = 21

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 = 21

目标最小化容器圆半径

已知最好（未证明）1.910来源精度区间：1.91–1.911（区间内视为追平）本站展示证书 1.910541299Erich Friedman 编录；Found by Maurizio Morandi in May 2008. 原式：r = 1.910+。

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

1.910541299

容器圆半径

已追平已知最好

答案来源Maurizio Morandi

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": "0.222940118",
      "x": "1.067078279",
      "y": "1.810764193"
    },
    {
      "turn": "-0.313994495",
      "x": "0.793503235",
      "y": "1.682402623"
    },
    {
      "turn": "0.268652021",
      "x": "1.442551256",
      "y": "1.558702609"
    },
    {
      "turn": "-0.267246610",
      "x": "1.704455096",
      "y": "1.407949911"
    },
    {
      "turn": "0.268652021",
      "x": "0.525980629",
      "y": "1.379833362"
    },
    {
      "turn": "0.268652021",
      "x": "1.035701657",
      "y": "1.356735259"
    },
    {
      "turn": "-0.267246610",
      "x": "0.264473042",
      "y": "1.228394333"
    },
    {
      "turn": "-0.267246610",
      "x": "0.774194071",
      "y": "1.205296229"
    },
    {
      "turn": "-0.267246610",
      "x": "1.297605497",
      "y": "1.205982561"
    },
    {
      "turn": "0.268652021",
      "x": "1.704851350",
      "y": "1.105758183"
    },
    {
      "turn": "0.268652021",
      "x": "0.264869296",
      "y": "0.926202605"
    },
    {
      "turn": "0.268652021",
      "x": "1.298001751",
      "y": "0.903790833"
    },
    {
      "turn": "0.268652021",
      "x": "0.774590325",
      "y": "0.903104501"
    },
    {
      "turn": "-0.267246610",
      "x": "1.036494165",
      "y": "0.752351803"
    },
    {
      "turn": "-0.267246610",
      "x": "1.559905591",
      "y": "0.753038135"
    },
    {
      "turn": "-0.267246610",
      "x": "0.513082738",
      "y": "0.751665471"
    },
    {
      "turn": "0.268652021",
      "x": "1.524132475",
      "y": "0.450798979"
    },
    {
      "turn": "0.268652021",
      "x": "1.000721049",
      "y": "0.450112647"
    },
    {
      "turn": "0.268652021",
      "x": "0.477309623",
      "y": "0.449426315"
    },
    {
      "turn": "-0.267246610",
      "x": "1.262624889",
      "y": "0.299359950"
    },
    {
      "turn": "-0.267246610",
      "x": "0.739213463",
      "y": "0.298673617"
    }
  ],
  "radius": "0.302191986"
}

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": 21,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.222940118",
      "x": "1.067078279",
      "y": "1.810764193"
    },
    {
      "turn": "-0.313994495",
      "x": "0.793503235",
      "y": "1.682402623"
    },
    {
      "turn": "0.268652021",
      "x": "1.442551256",
      "y": "1.558702609"
    },
    {
      "turn": "-0.267246610",
      "x": "1.704455096",
      "y": "1.407949911"
    },
    {
      "turn": "0.268652021",
      "x": "0.525980629",
      "y": "1.379833362"
    },
    {
      "turn": "0.268652021",
      "x": "1.035701657",
      "y": "1.356735259"
    },
    {
      "turn": "-0.267246610",
      "x": "0.264473042",
      "y": "1.228394333"
    },
    {
      "turn": "-0.267246610",
      "x": "0.774194071",
      "y": "1.205296229"
    },
    {
      "turn": "-0.267246610",
      "x": "1.297605497",
      "y": "1.205982561"
    },
    {
      "turn": "0.268652021",
      "x": "1.704851350",
      "y": "1.105758183"
    },
    {
      "turn": "0.268652021",
      "x": "0.264869296",
      "y": "0.926202605"
    },
    {
      "turn": "0.268652021",
      "x": "1.298001751",
      "y": "0.903790833"
    },
    {
      "turn": "0.268652021",
      "x": "0.774590325",
      "y": "0.903104501"
    },
    {
      "turn": "-0.267246610",
      "x": "1.036494165",
      "y": "0.752351803"
    },
    {
      "turn": "-0.267246610",
      "x": "1.559905591",
      "y": "0.753038135"
    },
    {
      "turn": "-0.267246610",
      "x": "0.513082738",
      "y": "0.751665471"
    },
    {
      "turn": "0.268652021",
      "x": "1.524132475",
      "y": "0.450798979"
    },
    {
      "turn": "0.268652021",
      "x": "1.000721049",
      "y": "0.450112647"
    },
    {
      "turn": "0.268652021",
      "x": "0.477309623",
      "y": "0.449426315"
    },
    {
      "turn": "-0.267246610",
      "x": "1.262624889",
      "y": "0.299359950"
    },
    {
      "turn": "-0.267246610",
      "x": "0.739213463",
      "y": "0.298673617"
    }
  ],
  "radius": "0.302191986"
}

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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