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多米诺骨牌装入正方形 · n = 20

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 = 20
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[6.89791, 6.89792)Joshua Wolk2026-09Erich Friedman ↗
本站纪录6.897915799zzzcy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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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 = 20 的当前纪录构型，6.897915799
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
    6.914214826→6.897915799
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": "0.146229388",
      "y": "0.927514337"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.565086020",
      "y": "0.927514337"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.782543010",
      "y": "0.855028673"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.927514337",
      "y": "0.826918455"
    },
    {
      "quarter": 0,
      "turn": "-0.113690227",
      "x": "0.447883934",
      "y": "0.774247785"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.144971327",
      "y": "0.782543010"
    },
    {
      "quarter": 0,
      "turn": "-0.113690227",
      "x": "0.420671593",
      "y": "0.631748319"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.631748319",
      "y": "0.579328407"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.072485663",
      "y": "0.565086020"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.773594364",
      "y": "0.549279519"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.226102746",
      "y": "0.449405616"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.368251681",
      "y": "0.420671593"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.927514337",
      "y": "0.434913980"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.489995189",
      "y": "0.303356172"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.633692653",
      "y": "0.281344423"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.072485663",
      "y": "0.146580671"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.217456990",
      "y": "0.144971327"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.855028673",
      "y": "0.217456990"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.434913980",
      "y": "0.072485663"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.853725182",
      "y": "0.072485663"
    }
  ],
  "radius": "0.144971326"
}
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": 20,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.146229388",
      "y": "0.927514337"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.565086020",
      "y": "0.927514337"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.782543010",
      "y": "0.855028673"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.927514337",
      "y": "0.826918455"
    },
    {
      "quarter": 0,
      "turn": "-0.113690227",
      "x": "0.447883934",
      "y": "0.774247785"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.144971327",
      "y": "0.782543010"
    },
    {
      "quarter": 0,
      "turn": "-0.113690227",
      "x": "0.420671593",
      "y": "0.631748319"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.631748319",
      "y": "0.579328407"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.072485663",
      "y": "0.565086020"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.773594364",
      "y": "0.549279519"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.226102746",
      "y": "0.449405616"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.368251681",
      "y": "0.420671593"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.927514337",
      "y": "0.434913980"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.489995189",
      "y": "0.303356172"
    },
    {
      "quarter": 1,
      "turn": "-0.113690227",
      "x": "0.633692653",
      "y": "0.281344423"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.072485663",
      "y": "0.146580671"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.217456990",
      "y": "0.144971327"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.855028673",
      "y": "0.217456990"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.434913980",
      "y": "0.072485663"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.853725182",
      "y": "0.072485663"
    }
  ],
  "radius": "0.144971326"
}

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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