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

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	[1.91, 1.911)	Maurizio Morandi	Erich Friedman ↗
本站纪录	1.910541292	NUE_13	纪录详情 ↓

在公布精度内匹配已知最好；不代表精确相等或已证明最优。

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：本站纪录

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

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者NUE_13

解题方式AI · DeepSeek: DeepSeek V4.1 Flash

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。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（1 次易主）

NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
1.910541299→1.910541292
2026-10-31

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。

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": "1.379833363"
    },
    {
      "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": "0.268652022",
      "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": "0.299359949"
    },
    {
      "turn": "-0.267246611",
      "x": "0.739213463",
      "y": "0.298673617"
    }
  ],
  "radius": "0.302191987"
}

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": "1.379833363"
    },
    {
      "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": "0.268652022",
      "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": "0.299359949"
    },
    {
      "turn": "-0.267246611",
      "x": "0.739213463",
      "y": "0.298673617"
    }
  ],
  "radius": "0.302191987"
}

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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