P83 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入正方形 · n = 36

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 = 36

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	[4.20791, 4.20792)	36. Found by Jake Loyd	Erich Friedman ↗
本站纪录	4.207918412	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。

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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
4.207918443→4.207918412
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": [
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      "x": "0.751272528",
      "y": "0.931397164"
    },
    {
      "turn": "0.267949192",
      "x": "0.513625333",
      "y": "0.931397164"
    },
    {
      "turn": "0.267949192",
      "x": "0.275978138",
      "y": "0.931397164"
    },
    {
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      "x": "0.068602836",
      "y": "0.881176402"
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      "x": "0.931397164",
      "y": "0.868529318"
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      "x": "0.632448931",
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    {
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      "x": "0.394801736",
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    },
    {
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      "y": "0.762352805"
    },
    {
      "turn": "0.268114139",
      "x": "0.748395975",
      "y": "0.725587768"
    },
    {
      "turn": "0.268114139",
      "x": "0.510727147",
      "y": "0.725552088"
    },
    {
      "turn": "0.000000000",
      "x": "0.274411344",
      "y": "0.715880349"
    },
    {
      "turn": "-0.267784259",
      "x": "0.881155293",
      "y": "0.681139460"
    },
    {
      "turn": "-0.267784259",
      "x": "0.629571853",
      "y": "0.656985827"
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      "x": "0.345933845",
      "y": "0.598742421"
    },
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      "x": "0.881197523",
      "y": "0.543933794"
    },
    {
      "turn": "-0.577350269",
      "x": "0.137205672",
      "y": "0.524705610"
    },
    {
      "turn": "0.268114139",
      "x": "0.629614083",
      "y": "0.519780161"
    },
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      "y": "0.475294390"
    },
    {
      "turn": "0.000000000",
      "x": "0.068602836",
      "y": "0.405882013"
    },
    {
      "turn": "-0.577350269",
      "x": "0.931397164",
      "y": "0.356470792"
    },
    {
      "turn": "0.268114139",
      "x": "0.440092265",
      "y": "0.343083182"
    },
    {
      "turn": "0.338628999",
      "x": "0.706111404",
      "y": "0.323343479"
    },
    {
      "turn": "0.548891388",
      "x": "0.137077578",
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      "x": "0.321289788",
      "y": "0.274443778"
    },
    {
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      "x": "0.558894764",
      "y": "0.274443763"
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    {
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      "x": "0.862794328",
      "y": "0.237647195"
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    {
      "turn": "0.000000000",
      "x": "0.068602836",
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    },
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      "x": "0.590826538",
      "y": "0.137205672"
    },
    {
      "turn": "0.267923607",
      "x": "0.353172792",
      "y": "0.137205672"
    },
    {
      "turn": "-0.577350269",
      "x": "0.931397164",
      "y": "0.118823597"
    },
    {
      "turn": "-0.276511176",
      "x": "0.736245333",
      "y": "0.074342246"
    },
    {
      "turn": "-0.267974778",
      "x": "0.471999665",
      "y": "0.068608509"
    },
    {
      "turn": "-0.267974778",
      "x": "0.234345919",
      "y": "0.068608509"
    }
  ],
  "radius": "0.137205671"
}

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
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.751272528",
      "y": "0.931397164"
    },
    {
      "turn": "0.267949192",
      "x": "0.513625333",
      "y": "0.931397164"
    },
    {
      "turn": "0.267949192",
      "x": "0.275978138",
      "y": "0.931397164"
    },
    {
      "turn": "0.000000000",
      "x": "0.068602836",
      "y": "0.881176402"
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      "turn": "0.577350269",
      "x": "0.931397164",
      "y": "0.868529318"
    },
    {
      "turn": "-0.267949192",
      "x": "0.632448931",
      "y": "0.862794328"
    },
    {
      "turn": "-0.267949192",
      "x": "0.394801736",
      "y": "0.862794328"
    },
    {
      "turn": "0.577350269",
      "x": "0.137205672",
      "y": "0.762352805"
    },
    {
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      "x": "0.748395975",
      "y": "0.725587768"
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      "x": "0.510727147",
      "y": "0.725552088"
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      "x": "0.274411344",
      "y": "0.715880349"
    },
    {
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      "y": "0.681139460"
    },
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      "x": "0.881197523",
      "y": "0.543933794"
    },
    {
      "turn": "-0.577350269",
      "x": "0.137205672",
      "y": "0.524705610"
    },
    {
      "turn": "0.268114139",
      "x": "0.629614083",
      "y": "0.519780161"
    },
    {
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      "x": "0.493961397",
      "y": "0.480305441"
    },
    {
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      "x": "0.277374298",
      "y": "0.476472507"
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    {
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      "x": "0.762395046",
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    },
    {
      "turn": "0.267923607",
      "x": "0.353172792",
      "y": "0.137205672"
    },
    {
      "turn": "-0.577350269",
      "x": "0.931397164",
      "y": "0.118823597"
    },
    {
      "turn": "-0.276511176",
      "x": "0.736245333",
      "y": "0.074342246"
    },
    {
      "turn": "-0.267974778",
      "x": "0.471999665",
      "y": "0.068608509"
    },
    {
      "turn": "-0.267974778",
      "x": "0.234345919",
      "y": "0.068608509"
    }
  ],
  "radius": "0.137205671"
}

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