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多米诺骨牌装入正方形 · n = 27

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 = 27
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[7.95595, 7.95596)Yanis Gomes2026-10Erich Friedman ↗
本站纪录7.951240513NUE_13纪录详情 ↓

数值超出外部纪录区间 · 待审核。

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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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 = 27 的当前纪录构型，7.951240513
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（6 次易主）
  1. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    7.951242789→7.951240513
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    7.951243484→7.951242789
  3. DeepSeek HarnessAI · DeepSeek: DeepSeek V4.1 Flash 1017
    7.9540592→7.951243484
  4. DeepSeek HarnessAI · DeepSeek: DeepSeek V4.1 Flash 1017
    7.984804534→7.9540592
  5. 邱仲普AI · OpenAI: GPT-5.6 Sol
    7.984804598→7.984804534
  6. zzzcyAI · OpenAI: GPT-6 Astra
    8→7.984804598
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": "-0.000002477",
      "x": "0.380866305",
      "y": "0.937115366"
    },
    {
      "quarter": 0,
      "turn": "-0.000000154",
      "x": "0.748466859",
      "y": "0.937116434"
    },
    {
      "quarter": 3,
      "turn": "-0.000000329",
      "x": "0.937116647",
      "y": "0.870518462"
    },
    {
      "quarter": 3,
      "turn": "0.000000870",
      "x": "0.188651588",
      "y": "0.874230990"
    },
    {
      "quarter": 3,
      "turn": "-0.000002508",
      "x": "0.062884198",
      "y": "0.874226434"
    },
    {
      "quarter": 0,
      "turn": "0.250720014",
      "x": "0.517859718",
      "y": "0.803592565"
    },
    {
      "quarter": 0,
      "turn": "0.245495657",
      "x": "0.718083518",
      "y": "0.760216443"
    },
    {
      "quarter": 1,
      "turn": "0.000003125",
      "x": "0.314418963",
      "y": "0.748465104"
    },
    {
      "quarter": 0,
      "turn": "0.000010308",
      "x": "0.125768510",
      "y": "0.685574501"
    },
    {
      "quarter": 0,
      "turn": "0.245611087",
      "x": "0.779872017",
      "y": "0.650542467"
    },
    {
      "quarter": 1,
      "turn": "0.250718495",
      "x": "0.577556006",
      "y": "0.621573775"
    },
    {
      "quarter": 0,
      "turn": "0.000034873",
      "x": "0.251537789",
      "y": "0.559806840"
    },
    {
      "quarter": 3,
      "turn": "0.000002072",
      "x": "0.937115216",
      "y": "0.503097655"
    },
    {
      "quarter": 1,
      "turn": "0.251340060",
      "x": "0.499720798",
      "y": "0.500241012"
    },
    {
      "quarter": 3,
      "turn": "0.000029960",
      "x": "0.062891131",
      "y": "0.496881881"
    },
    {
      "quarter": 0,
      "turn": "0.000053443",
      "x": "0.748459112",
      "y": "0.440211901"
    },
    {
      "quarter": 1,
      "turn": "0.251374775",
      "x": "0.422341357",
      "y": "0.378366773"
    },
    {
      "quarter": 0,
      "turn": "0.247553736",
      "x": "0.222429606",
      "y": "0.350977369"
    },
    {
      "quarter": 0,
      "turn": "0.000004540",
      "x": "0.874232232",
      "y": "0.314443974"
    },
    {
      "quarter": 1,
      "turn": "0.000003868",
      "x": "0.685582189",
      "y": "0.251541980"
    },
    {
      "quarter": 0,
      "turn": "0.247473535",
      "x": "0.281853345",
      "y": "0.240085910"
    },
    {
      "quarter": 0,
      "turn": "0.251375347",
      "x": "0.482146710",
      "y": "0.196362522"
    },
    {
      "quarter": 3,
      "turn": "0.000000014",
      "x": "0.811349799",
      "y": "0.125783024"
    },
    {
      "quarter": 3,
      "turn": "0.000000185",
      "x": "0.937116683",
      "y": "0.125767145"
    },
    {
      "quarter": 3,
      "turn": "0.000000063",
      "x": "0.062883342",
      "y": "0.134099953"
    },
    {
      "quarter": 0,
      "turn": "0.000000192",
      "x": "0.251533287",
      "y": "0.062883410"
    },
    {
      "quarter": 0,
      "turn": "0.000007401",
      "x": "0.622057129",
      "y": "0.062890084"
    }
  ],
  "radius": "0.125766539"
}
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": 27,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "quarter": 0,
      "turn": "-0.000002477",
      "x": "0.380866305",
      "y": "0.937115366"
    },
    {
      "quarter": 0,
      "turn": "-0.000000154",
      "x": "0.748466859",
      "y": "0.937116434"
    },
    {
      "quarter": 3,
      "turn": "-0.000000329",
      "x": "0.937116647",
      "y": "0.870518462"
    },
    {
      "quarter": 3,
      "turn": "0.000000870",
      "x": "0.188651588",
      "y": "0.874230990"
    },
    {
      "quarter": 3,
      "turn": "-0.000002508",
      "x": "0.062884198",
      "y": "0.874226434"
    },
    {
      "quarter": 0,
      "turn": "0.250720014",
      "x": "0.517859718",
      "y": "0.803592565"
    },
    {
      "quarter": 0,
      "turn": "0.245495657",
      "x": "0.718083518",
      "y": "0.760216443"
    },
    {
      "quarter": 1,
      "turn": "0.000003125",
      "x": "0.314418963",
      "y": "0.748465104"
    },
    {
      "quarter": 0,
      "turn": "0.000010308",
      "x": "0.125768510",
      "y": "0.685574501"
    },
    {
      "quarter": 0,
      "turn": "0.245611087",
      "x": "0.779872017",
      "y": "0.650542467"
    },
    {
      "quarter": 1,
      "turn": "0.250718495",
      "x": "0.577556006",
      "y": "0.621573775"
    },
    {
      "quarter": 0,
      "turn": "0.000034873",
      "x": "0.251537789",
      "y": "0.559806840"
    },
    {
      "quarter": 3,
      "turn": "0.000002072",
      "x": "0.937115216",
      "y": "0.503097655"
    },
    {
      "quarter": 1,
      "turn": "0.251340060",
      "x": "0.499720798",
      "y": "0.500241012"
    },
    {
      "quarter": 3,
      "turn": "0.000029960",
      "x": "0.062891131",
      "y": "0.496881881"
    },
    {
      "quarter": 0,
      "turn": "0.000053443",
      "x": "0.748459112",
      "y": "0.440211901"
    },
    {
      "quarter": 1,
      "turn": "0.251374775",
      "x": "0.422341357",
      "y": "0.378366773"
    },
    {
      "quarter": 0,
      "turn": "0.247553736",
      "x": "0.222429606",
      "y": "0.350977369"
    },
    {
      "quarter": 0,
      "turn": "0.000004540",
      "x": "0.874232232",
      "y": "0.314443974"
    },
    {
      "quarter": 1,
      "turn": "0.000003868",
      "x": "0.685582189",
      "y": "0.251541980"
    },
    {
      "quarter": 0,
      "turn": "0.247473535",
      "x": "0.281853345",
      "y": "0.240085910"
    },
    {
      "quarter": 0,
      "turn": "0.251375347",
      "x": "0.482146710",
      "y": "0.196362522"
    },
    {
      "quarter": 3,
      "turn": "0.000000014",
      "x": "0.811349799",
      "y": "0.125783024"
    },
    {
      "quarter": 3,
      "turn": "0.000000185",
      "x": "0.937116683",
      "y": "0.125767145"
    },
    {
      "quarter": 3,
      "turn": "0.000000063",
      "x": "0.062883342",
      "y": "0.134099953"
    },
    {
      "quarter": 0,
      "turn": "0.000000192",
      "x": "0.251533287",
      "y": "0.062883410"
    },
    {
      "quarter": 0,
      "turn": "0.000007401",
      "x": "0.622057129",
      "y": "0.062890084"
    }
  ],
  "radius": "0.125766539"
}

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