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P83 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入正方形 · n = 32

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 = 32
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[3.9788, 3.97881)32. Found by Jake LoydErich 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：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
等边三角形装入正方形 n = 32 的当前构型
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

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。

起步布局答案

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.273403183",
      "y": "0.927446691"
    },
    {
      "turn": "0.267949192",
      "x": "0.725944708",
      "y": "0.927446691"
    },
    {
      "turn": "0.005249269",
      "x": "0.073868583",
      "y": "0.875102591"
    },
    {
      "turn": "0.568642410",
      "x": "0.925805363",
      "y": "0.875296006"
    },
    {
      "turn": "0.267949192",
      "x": "0.499673946",
      "y": "0.884039297"
    },
    {
      "turn": "-0.267949192",
      "x": "0.374007929",
      "y": "0.811485989"
    },
    {
      "turn": "-0.267949192",
      "x": "0.625339962",
      "y": "0.811485989"
    },
    {
      "turn": "-0.570372392",
      "x": "0.147737167",
      "y": "0.750205183"
    },
    {
      "turn": "-0.006555613",
      "x": "0.851610725",
      "y": "0.750592012"
    },
    {
      "turn": "0.001903605",
      "x": "0.294145047",
      "y": "0.629126025"
    },
    {
      "turn": "0.000000000",
      "x": "0.072553309",
      "y": "0.624539166"
    },
    {
      "turn": "0.267949192",
      "x": "0.501838684",
      "y": "0.666379372"
    },
    {
      "turn": "-0.568866686",
      "x": "0.704887299",
      "y": "0.623505006"
    },
    {
      "turn": "0.568642410",
      "x": "0.925805363",
      "y": "0.625830845"
    },
    {
      "turn": "-0.574814916",
      "x": "0.367176265",
      "y": "0.503737144"
    },
    {
      "turn": "0.573887845",
      "x": "0.148399784",
      "y": "0.500020836"
    },
    {
      "turn": "-0.002600717",
      "x": "0.851600216",
      "y": "0.499979164"
    },
    {
      "turn": "0.001903605",
      "x": "0.632823735",
      "y": "0.496262856"
    },
    {
      "turn": "0.006386147",
      "x": "0.295112701",
      "y": "0.376494994"
    },
    {
      "turn": "-0.006555613",
      "x": "0.074194637",
      "y": "0.374169155"
    },
    {
      "turn": "-0.574814916",
      "x": "0.705854953",
      "y": "0.370873975"
    },
    {
      "turn": "-0.577350269",
      "x": "0.927446691",
      "y": "0.375460834"
    },
    {
      "turn": "-0.267949192",
      "x": "0.498161316",
      "y": "0.333620628"
    },
    {
      "turn": "0.568642410",
      "x": "0.148389275",
      "y": "0.249407988"
    },
    {
      "turn": "0.005249269",
      "x": "0.852262833",
      "y": "0.249794817"
    },
    {
      "turn": "0.267949192",
      "x": "0.374660038",
      "y": "0.188514011"
    },
    {
      "turn": "0.267949192",
      "x": "0.625992071",
      "y": "0.188514011"
    },
    {
      "turn": "-0.267949192",
      "x": "0.500326054",
      "y": "0.115960703"
    },
    {
      "turn": "-0.006555613",
      "x": "0.074194637",
      "y": "0.124703994"
    },
    {
      "turn": "-0.570372392",
      "x": "0.926131417",
      "y": "0.124897409"
    },
    {
      "turn": "-0.267949192",
      "x": "0.274055292",
      "y": "0.072553309"
    },
    {
      "turn": "-0.267949192",
      "x": "0.726596817",
      "y": "0.072553309"
    }
  ],
  "radius": "0.145106616"
}
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": 32,
  "sides": 5
}

起步布局答案

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.273403183",
      "y": "0.927446691"
    },
    {
      "turn": "0.267949192",
      "x": "0.725944708",
      "y": "0.927446691"
    },
    {
      "turn": "0.005249269",
      "x": "0.073868583",
      "y": "0.875102591"
    },
    {
      "turn": "0.568642410",
      "x": "0.925805363",
      "y": "0.875296006"
    },
    {
      "turn": "0.267949192",
      "x": "0.499673946",
      "y": "0.884039297"
    },
    {
      "turn": "-0.267949192",
      "x": "0.374007929",
      "y": "0.811485989"
    },
    {
      "turn": "-0.267949192",
      "x": "0.625339962",
      "y": "0.811485989"
    },
    {
      "turn": "-0.570372392",
      "x": "0.147737167",
      "y": "0.750205183"
    },
    {
      "turn": "-0.006555613",
      "x": "0.851610725",
      "y": "0.750592012"
    },
    {
      "turn": "0.001903605",
      "x": "0.294145047",
      "y": "0.629126025"
    },
    {
      "turn": "0.000000000",
      "x": "0.072553309",
      "y": "0.624539166"
    },
    {
      "turn": "0.267949192",
      "x": "0.501838684",
      "y": "0.666379372"
    },
    {
      "turn": "-0.568866686",
      "x": "0.704887299",
      "y": "0.623505006"
    },
    {
      "turn": "0.568642410",
      "x": "0.925805363",
      "y": "0.625830845"
    },
    {
      "turn": "-0.574814916",
      "x": "0.367176265",
      "y": "0.503737144"
    },
    {
      "turn": "0.573887845",
      "x": "0.148399784",
      "y": "0.500020836"
    },
    {
      "turn": "-0.002600717",
      "x": "0.851600216",
      "y": "0.499979164"
    },
    {
      "turn": "0.001903605",
      "x": "0.632823735",
      "y": "0.496262856"
    },
    {
      "turn": "0.006386147",
      "x": "0.295112701",
      "y": "0.376494994"
    },
    {
      "turn": "-0.006555613",
      "x": "0.074194637",
      "y": "0.374169155"
    },
    {
      "turn": "-0.574814916",
      "x": "0.705854953",
      "y": "0.370873975"
    },
    {
      "turn": "-0.577350269",
      "x": "0.927446691",
      "y": "0.375460834"
    },
    {
      "turn": "-0.267949192",
      "x": "0.498161316",
      "y": "0.333620628"
    },
    {
      "turn": "0.568642410",
      "x": "0.148389275",
      "y": "0.249407988"
    },
    {
      "turn": "0.005249269",
      "x": "0.852262833",
      "y": "0.249794817"
    },
    {
      "turn": "0.267949192",
      "x": "0.374660038",
      "y": "0.188514011"
    },
    {
      "turn": "0.267949192",
      "x": "0.625992071",
      "y": "0.188514011"
    },
    {
      "turn": "-0.267949192",
      "x": "0.500326054",
      "y": "0.115960703"
    },
    {
      "turn": "-0.006555613",
      "x": "0.074194637",
      "y": "0.124703994"
    },
    {
      "turn": "-0.570372392",
      "x": "0.926131417",
      "y": "0.124897409"
    },
    {
      "turn": "-0.267949192",
      "x": "0.274055292",
      "y": "0.072553309"
    },
    {
      "turn": "-0.267949192",
      "x": "0.726596817",
      "y": "0.072553309"
    }
  ],
  "radius": "0.145106616"
}

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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