P83 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入正方形 · n = 33

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 = 33

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	[4.05164, 4.05165)	33. Found by Jake Loyd	Erich Friedman ↗
本站纪录	4.051645095	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。

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
4.051645123→4.051645095
2026-10-31

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。

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.352367660",
      "y": "0.928751129"
    },
    {
      "turn": "0.267949192",
      "x": "0.599180986",
      "y": "0.928751129"
    },
    {
      "turn": "0.267949192",
      "x": "0.845994315",
      "y": "0.928751129"
    },
    {
      "turn": "0.564516019",
      "x": "0.155336674",
      "y": "0.877995633"
    },
    {
      "turn": "-0.267949193",
      "x": "0.475774323",
      "y": "0.857502259"
    },
    {
      "turn": "-0.267949192",
      "x": "0.722587651",
      "y": "0.857502259"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.759531060"
    },
    {
      "turn": "-0.572971036",
      "x": "0.927940501",
      "y": "0.751700176"
    },
    {
      "turn": "-0.039758310",
      "x": "0.294567982",
      "y": "0.739548160"
    },
    {
      "turn": "0.267949192",
      "x": "0.570603140",
      "y": "0.715004518"
    },
    {
      "turn": "0.577350269",
      "x": "0.142497741",
      "y": "0.636124397"
    },
    {
      "turn": "0.003290665",
      "x": "0.855868611",
      "y": "0.628757938"
    },
    {
      "turn": "0.525528726",
      "x": "0.373770597",
      "y": "0.619428003"
    },
    {
      "turn": "-0.566971486",
      "x": "0.713083458",
      "y": "0.574735085"
    },
    {
      "turn": "-0.165461141",
      "x": "0.520674786",
      "y": "0.521373305"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.512717733"
    },
    {
      "turn": "-0.032234690",
      "x": "0.292795263",
      "y": "0.499931544"
    },
    {
      "turn": "-0.577350269",
      "x": "0.928751129",
      "y": "0.505358428"
    },
    {
      "turn": "-0.577350269",
      "x": "0.142497741",
      "y": "0.389311069"
    },
    {
      "turn": "0.000000000",
      "x": "0.857502259",
      "y": "0.381951764"
    },
    {
      "turn": "0.267949192",
      "x": "0.662846724",
      "y": "0.381208794"
    },
    {
      "turn": "0.267949192",
      "x": "0.430446624",
      "y": "0.356244353"
    },
    {
      "turn": "-0.267949192",
      "x": "0.307039960",
      "y": "0.284995482"
    },
    {
      "turn": "-0.267949192",
      "x": "0.553853287",
      "y": "0.284995482"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.265904405"
    },
    {
      "turn": "0.577350269",
      "x": "0.928751129",
      "y": "0.258545101"
    },
    {
      "turn": "0.267949192",
      "x": "0.246813328",
      "y": "0.142497741"
    },
    {
      "turn": "0.267949193",
      "x": "0.493626655",
      "y": "0.142497741"
    },
    {
      "turn": "0.267949192",
      "x": "0.740439983",
      "y": "0.142497741"
    },
    {
      "turn": "-0.267949192",
      "x": "0.123406664",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.370219992",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.617033319",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.863846647",
      "y": "0.071248871"
    }
  ],
  "radius": "0.142497740"
}

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": 33,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.352367660",
      "y": "0.928751129"
    },
    {
      "turn": "0.267949192",
      "x": "0.599180986",
      "y": "0.928751129"
    },
    {
      "turn": "0.267949192",
      "x": "0.845994315",
      "y": "0.928751129"
    },
    {
      "turn": "0.564516019",
      "x": "0.155336674",
      "y": "0.877995633"
    },
    {
      "turn": "-0.267949193",
      "x": "0.475774323",
      "y": "0.857502259"
    },
    {
      "turn": "-0.267949192",
      "x": "0.722587651",
      "y": "0.857502259"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.759531060"
    },
    {
      "turn": "-0.572971036",
      "x": "0.927940501",
      "y": "0.751700176"
    },
    {
      "turn": "-0.039758310",
      "x": "0.294567982",
      "y": "0.739548160"
    },
    {
      "turn": "0.267949192",
      "x": "0.570603140",
      "y": "0.715004518"
    },
    {
      "turn": "0.577350269",
      "x": "0.142497741",
      "y": "0.636124397"
    },
    {
      "turn": "0.003290665",
      "x": "0.855868611",
      "y": "0.628757938"
    },
    {
      "turn": "0.525528726",
      "x": "0.373770597",
      "y": "0.619428003"
    },
    {
      "turn": "-0.566971486",
      "x": "0.713083458",
      "y": "0.574735085"
    },
    {
      "turn": "-0.165461141",
      "x": "0.520674786",
      "y": "0.521373305"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.512717733"
    },
    {
      "turn": "-0.032234690",
      "x": "0.292795263",
      "y": "0.499931544"
    },
    {
      "turn": "-0.577350269",
      "x": "0.928751129",
      "y": "0.505358428"
    },
    {
      "turn": "-0.577350269",
      "x": "0.142497741",
      "y": "0.389311069"
    },
    {
      "turn": "0.000000000",
      "x": "0.857502259",
      "y": "0.381951764"
    },
    {
      "turn": "0.267949192",
      "x": "0.662846724",
      "y": "0.381208794"
    },
    {
      "turn": "0.267949192",
      "x": "0.430446624",
      "y": "0.356244353"
    },
    {
      "turn": "-0.267949192",
      "x": "0.307039960",
      "y": "0.284995482"
    },
    {
      "turn": "-0.267949192",
      "x": "0.553853287",
      "y": "0.284995482"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.265904405"
    },
    {
      "turn": "0.577350269",
      "x": "0.928751129",
      "y": "0.258545101"
    },
    {
      "turn": "0.267949192",
      "x": "0.246813328",
      "y": "0.142497741"
    },
    {
      "turn": "0.267949193",
      "x": "0.493626655",
      "y": "0.142497741"
    },
    {
      "turn": "0.267949192",
      "x": "0.740439983",
      "y": "0.142497741"
    },
    {
      "turn": "-0.267949192",
      "x": "0.123406664",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.370219992",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.617033319",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.863846647",
      "y": "0.071248871"
    }
  ],
  "radius": "0.142497740"
}

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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证书归一化半径居然是中心到顶点的距离NUE_13#312 条回复31 天前