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多米诺骨牌装入正方形 · n = 23

将 n 个 1 × 2 的多米诺骨牌装入正方形容器，允许边界接触但内部不得重叠，最小化容器正方形的边长。

子题n = 23
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
外部已知最好待核实以原始来源为准见来源署名Erich 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 1 × 2 单位骨牌规范折算。允许接触边界。
  • 提交提交 {radius, placements:[{x,y,turn,quarter},…]}，其中 x、y、turn 为最多九位小数的十进制字符串。radius 为单位正方形容器内多米诺骨牌的短边长，对应容器正方形边长 = 1/radius。x、y 为骨牌中心坐标；quarter 为 0–3 的整数，表示先绕中心旋转 quarter 个直角，再旋转由 turn = tan(θ/2) 给出的有理角。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。
  • 约束恰好 n 个全等骨牌完全位于容器内，任意两骨牌内部互不重叠。每个骨牌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，可行性检验不设浮点容差。
  • 目标容器正方形边长 = 1/r（其中 r 为骨牌短边在单位正方形容器内的长度），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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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。

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多米诺骨牌装入正方形 n = 23 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,quarter},…]}，其中 x、y、turn 为最多九位小数的十进制字符串。radius 为单位正方形容器内多米诺骨牌的短边长，对应容器正方形边长 = 1/radius。x、y 为骨牌中心坐标；quarter 为 0–3 的整数，表示先绕中心旋转 quarter 个直角，再旋转由 turn = tan(θ/2) 给出的有理角。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。

起步布局答案

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.142857143",
      "y": "0.071428571"
    },
    {
      "turn": "0.000000000",
      "x": "0.428571429",
      "y": "0.071428571"
    },
    {
      "turn": "0.000000000",
      "x": "0.142857143",
      "y": "0.214285714"
    },
    {
      "turn": "0.000000000",
      "x": "0.428571429",
      "y": "0.214285714"
    },
    {
      "turn": "0.000000000",
      "x": "0.142857143",
      "y": "0.357142857"
    },
    {
      "turn": "0.000000000",
      "x": "0.428571429",
      "y": "0.357142857"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.928571429",
      "y": "0.142857143"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.928571429",
      "y": "0.428571429"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.785714286",
      "y": "0.142857143"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.785714286",
      "y": "0.428571429"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.642857143",
      "y": "0.142857143"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.642857143",
      "y": "0.428571429"
    },
    {
      "turn": "0.000000000",
      "x": "0.857142857",
      "y": "0.928571429"
    },
    {
      "turn": "0.000000000",
      "x": "0.571428571",
      "y": "0.928571429"
    },
    {
      "turn": "0.000000000",
      "x": "0.857142857",
      "y": "0.785714286"
    },
    {
      "turn": "0.000000000",
      "x": "0.571428571",
      "y": "0.785714286"
    },
    {
      "turn": "0.000000000",
      "x": "0.857142857",
      "y": "0.642857143"
    },
    {
      "turn": "0.000000000",
      "x": "0.571428571",
      "y": "0.642857143"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.071428571",
      "y": "0.857142857"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.071428571",
      "y": "0.571428571"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.214285714",
      "y": "0.857142857"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.214285714",
      "y": "0.571428571"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.357142857",
      "y": "0.857142857"
    }
  ],
  "radius": "0.142857142"
}
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": 23,
  "sides": 5
}

起步布局答案

{
  "placements": [
    {
      "turn": "0.000000000",
      "x": "0.142857143",
      "y": "0.071428571"
    },
    {
      "turn": "0.000000000",
      "x": "0.428571429",
      "y": "0.071428571"
    },
    {
      "turn": "0.000000000",
      "x": "0.142857143",
      "y": "0.214285714"
    },
    {
      "turn": "0.000000000",
      "x": "0.428571429",
      "y": "0.214285714"
    },
    {
      "turn": "0.000000000",
      "x": "0.142857143",
      "y": "0.357142857"
    },
    {
      "turn": "0.000000000",
      "x": "0.428571429",
      "y": "0.357142857"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.928571429",
      "y": "0.142857143"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.928571429",
      "y": "0.428571429"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.785714286",
      "y": "0.142857143"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.785714286",
      "y": "0.428571429"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.642857143",
      "y": "0.142857143"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.642857143",
      "y": "0.428571429"
    },
    {
      "turn": "0.000000000",
      "x": "0.857142857",
      "y": "0.928571429"
    },
    {
      "turn": "0.000000000",
      "x": "0.571428571",
      "y": "0.928571429"
    },
    {
      "turn": "0.000000000",
      "x": "0.857142857",
      "y": "0.785714286"
    },
    {
      "turn": "0.000000000",
      "x": "0.571428571",
      "y": "0.785714286"
    },
    {
      "turn": "0.000000000",
      "x": "0.857142857",
      "y": "0.642857143"
    },
    {
      "turn": "0.000000000",
      "x": "0.571428571",
      "y": "0.642857143"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.071428571",
      "y": "0.857142857"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.071428571",
      "y": "0.571428571"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.214285714",
      "y": "0.857142857"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.214285714",
      "y": "0.571428571"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.357142857",
      "y": "0.857142857"
    }
  ],
  "radius": "0.142857142"
}

提交 {radius, placements:[{x,y,turn,quarter},…]}，其中 x、y、turn 为最多九位小数的十进制字符串。radius 为单位正方形容器内多米诺骨牌的短边长，对应容器正方形边长 = 1/radius。x、y 为骨牌中心坐标；quarter 为 0–3 的整数，表示先绕中心旋转 quarter 个直角，再旋转由 turn = tan(θ/2) 给出的有理角。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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