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多米诺骨牌装入正方形 · n = 25

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 = 25
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[7.64575, 7.64576)Joshua Wolk2026-09Erich Friedman ↗
本站纪录7.64575136zzzcy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 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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当前展示：本站纪录

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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
多米诺骨牌装入正方形 n = 25 的当前纪录构型，7.64575136
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zzzcy
解题方式AI · OpenAI: GPT-6 Astra
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）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    7.707106998→7.64575136
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。

当前第一名的答案

{
  "placements": [
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.130856258",
      "y": "0.934604203"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.393012253",
      "y": "0.934604203"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.673021015",
      "y": "0.869208406"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.869208406",
      "y": "0.934604203"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.065395797",
      "y": "0.738164347"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.196187391",
      "y": "0.738416812"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.392374781",
      "y": "0.803812609"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.803812609",
      "y": "0.738416812"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.934604203",
      "y": "0.738416812"
    },
    {
      "quarter": 0,
      "turn": "0.215250437",
      "x": "0.501025832",
      "y": "0.656672066"
    },
    {
      "quarter": 0,
      "turn": "0.215250437",
      "x": "0.614442645",
      "y": "0.564369965"
    },
    {
      "quarter": 0,
      "turn": "0.215250437",
      "x": "0.267374781",
      "y": "0.538489492"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.934604203",
      "y": "0.476833625"
    },
    {
      "quarter": 0,
      "turn": "0.215250437",
      "x": "0.380791594",
      "y": "0.446187391"
    },
    {
      "quarter": 3,
      "turn": "0.215250437",
      "x": "0.767310739",
      "y": "0.412826341"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.065395797",
      "y": "0.392374781"
    },
    {
      "quarter": 3,
      "turn": "0.215250437",
      "x": "0.646114711",
      "y": "0.363416812"
    },
    {
      "quarter": 3,
      "turn": "0.215250437",
      "x": "0.537152702",
      "y": "0.286905867"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.196187391",
      "y": "0.261583188"
    },
    {
      "quarter": 3,
      "turn": "0.215250437",
      "x": "0.412463660",
      "y": "0.245234238"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.065395797",
      "y": "0.130791594"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.803812609",
      "y": "0.130791594"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.934604203",
      "y": "0.131897680"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.261583188",
      "y": "0.065395797"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.607625219",
      "y": "0.065395797"
    }
  ],
  "radius": "0.130791593"
}
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": 25,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.130856258",
      "y": "0.934604203"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.393012253",
      "y": "0.934604203"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.673021015",
      "y": "0.869208406"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.869208406",
      "y": "0.934604203"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.065395797",
      "y": "0.738164347"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.196187391",
      "y": "0.738416812"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.392374781",
      "y": "0.803812609"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.803812609",
      "y": "0.738416812"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.934604203",
      "y": "0.738416812"
    },
    {
      "quarter": 0,
      "turn": "0.215250437",
      "x": "0.501025832",
      "y": "0.656672066"
    },
    {
      "quarter": 0,
      "turn": "0.215250437",
      "x": "0.614442645",
      "y": "0.564369965"
    },
    {
      "quarter": 0,
      "turn": "0.215250437",
      "x": "0.267374781",
      "y": "0.538489492"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.934604203",
      "y": "0.476833625"
    },
    {
      "quarter": 0,
      "turn": "0.215250437",
      "x": "0.380791594",
      "y": "0.446187391"
    },
    {
      "quarter": 3,
      "turn": "0.215250437",
      "x": "0.767310739",
      "y": "0.412826341"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.065395797",
      "y": "0.392374781"
    },
    {
      "quarter": 3,
      "turn": "0.215250437",
      "x": "0.646114711",
      "y": "0.363416812"
    },
    {
      "quarter": 3,
      "turn": "0.215250437",
      "x": "0.537152702",
      "y": "0.286905867"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.196187391",
      "y": "0.261583188"
    },
    {
      "quarter": 3,
      "turn": "0.215250437",
      "x": "0.412463660",
      "y": "0.245234238"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.065395797",
      "y": "0.130791594"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.803812609",
      "y": "0.130791594"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.934604203",
      "y": "0.131897680"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.261583188",
      "y": "0.065395797"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.607625219",
      "y": "0.065395797"
    }
  ],
  "radius": "0.130791593"
}

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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本题n=2，4，8，12，18，24，32，40时的证明NUE_13#313 条回复外部记录更新（这次不想抄数字了）NUE_13#310 条回复外部记录更新（9.24更新）NUE_13#310 条回复n=1的证明LittleChasa#1352 条回复