P85 · 装箱与覆盖 · 经典问题 · 难

正方形装入等边三角形 · n = 40

把 n 个边长为 1 的正方形装入等边三角形，允许接触但内部不能重叠，最小化容器三角形边长。

子题n = 40

目标最小化容器三角形边长

已知最好（未证明）10.46410来源精度区间：10.4641–10.46411（区间内视为追平）本站展示证书 10.464101789Erich Friedman 编录；Found by Bhavithran Ananthan in July 2026. 原式：s = 7 + 2√3 = 10.46410+。

严格定义

容器编辑器及证书统一缩放到顶点为 (0,0)、(1,0)、(1/2,√3/2) 的等边三角形；仅是坐标约定，分数按文献的单位图形计算。边界接触允许。
提交提交 {radius, placements:[{x,y,turn},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。
约束恰好 n 个同样大的图形，完整位于容器内，任意两图形内部不重叠。多边形可独立旋转，不限直角。
精度输入小数精确表示有理数；正多边形的顶点用代数数定义，不截断成小数。可行性验算没有浮点容差。有限小数证书不等于连续最优性证明。
目标容器三角形边长 = 1/(2 sin(π/4) r)，越小越好。页面、排行榜和外部纪录使用相同单位，自动计算，无需手工换算。

放大来摆，然后提交 ↗

已验证构造已验证构造

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外部来源与证书

外部目标按原始印刷精度换算，本站坐标证书另存。复现损失不降低文献目标；源表的构造不自动视为最优性证明。

查看来源

当前第一名

10.464101789

容器三角形边长

已追平已知最好

答案来源Bhavithran Ananthan

解题方式公开参考构造

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

ANSWER FORMAT

答案怎么写

顶点为 (0,0)、(1,0)、(1/2,√3/2) 的等边三角形

提交 {radius, placements:[{x,y,turn},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.139014541",
      "x": "0.490110712",
      "y": "0.749757764"
    },
    {
      "turn": "-0.123429023",
      "x": "0.561519369",
      "y": "0.660620124"
    },
    {
      "turn": "0.131652498",
      "x": "0.432368715",
      "y": "0.653319760"
    },
    {
      "turn": "-0.131652498",
      "x": "0.611127628",
      "y": "0.577981885"
    },
    {
      "turn": "0.131652498",
      "x": "0.381454130",
      "y": "0.565133113"
    },
    {
      "turn": "0.131652498",
      "x": "0.489902389",
      "y": "0.561841364"
    },
    {
      "turn": "-0.134829526",
      "x": "0.652094646",
      "y": "0.490844535"
    },
    {
      "turn": "0.123879053",
      "x": "0.334120273",
      "y": "0.481238862"
    },
    {
      "turn": "0.131652498",
      "x": "0.443875654",
      "y": "0.478066161"
    },
    {
      "turn": "0.152448479",
      "x": "0.527449966",
      "y": "0.427875972"
    },
    {
      "turn": "-0.133299825",
      "x": "0.706836569",
      "y": "0.411900215"
    },
    {
      "turn": "0.127536167",
      "x": "0.285711072",
      "y": "0.398444519"
    },
    {
      "turn": "0.131652498",
      "x": "0.396333211",
      "y": "0.395166053"
    },
    {
      "turn": "0.197249610",
      "x": "0.604267467",
      "y": "0.360036818"
    },
    {
      "turn": "0.156096206",
      "x": "0.479998137",
      "y": "0.344345038"
    },
    {
      "turn": "-0.130619735",
      "x": "0.748321517",
      "y": "0.325239844"
    },
    {
      "turn": "0.132014560",
      "x": "0.238075541",
      "y": "0.315114613"
    },
    {
      "turn": "0.131652498",
      "x": "0.347534848",
      "y": "0.312991050"
    },
    {
      "turn": "0.245026390",
      "x": "0.552631750",
      "y": "0.260343476"
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    {
      "turn": "0.282386149",
      "x": "0.434417420",
      "y": "0.249024471"
    },
    {
      "turn": "-0.134136147",
      "x": "0.801121920",
      "y": "0.244986703"
    },
    {
      "turn": "0.298251692",
      "x": "0.651528258",
      "y": "0.244202218"
    },
    {
      "turn": "0.131652498",
      "x": "0.189548887",
      "y": "0.232743482"
    },
    {
      "turn": "0.131652498",
      "x": "0.312202526",
      "y": "0.223041426"
    },
    {
      "turn": "-0.105171451",
      "x": "0.737367815",
      "y": "0.162395751"
    },
    {
      "turn": "-0.127593111",
      "x": "0.850573838",
      "y": "0.162488850"
    },
    {
      "turn": "0.371675391",
      "x": "0.611963484",
      "y": "0.147812075"
    },
    {
      "turn": "0.131652498",
      "x": "0.141293211",
      "y": "0.149162199"
    },
    {
      "turn": "0.363879433",
      "x": "0.515428020",
      "y": "0.149004586"
    },
    {
      "turn": "0.131652498",
      "x": "0.250990335",
      "y": "0.148033550"
    },
    {
      "turn": "0.347033730",
      "x": "0.416851128",
      "y": "0.149481748"
    },
    {
      "turn": "0.131652498",
      "x": "0.093323166",
      "y": "0.066075643"
    },
    {
      "turn": "0.131652498",
      "x": "0.203207925",
      "y": "0.065271987"
    },
    {
      "turn": "0.414213562",
      "x": "0.419219108",
      "y": "0.047870714"
    },
    {
      "turn": "0.414213562",
      "x": "0.705913572",
      "y": "0.048517341"
    },
    {
      "turn": "0.414213562",
      "x": "0.801478393",
      "y": "0.048996581"
    },
    {
      "turn": "0.414213562",
      "x": "0.897043214",
      "y": "0.047782411"
    },
    {
      "turn": "0.414213562",
      "x": "0.323654287",
      "y": "0.047782411"
    },
    {
      "turn": "0.414213562",
      "x": "0.514783929",
      "y": "0.047784526"
    },
    {
      "turn": "0.414213562",
      "x": "0.610348751",
      "y": "0.047784542"
    }
  ],
  "radius": "0.067574532"
}

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": 40,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.139014541",
      "x": "0.490110712",
      "y": "0.749757764"
    },
    {
      "turn": "-0.123429023",
      "x": "0.561519369",
      "y": "0.660620124"
    },
    {
      "turn": "0.131652498",
      "x": "0.432368715",
      "y": "0.653319760"
    },
    {
      "turn": "-0.131652498",
      "x": "0.611127628",
      "y": "0.577981885"
    },
    {
      "turn": "0.131652498",
      "x": "0.381454130",
      "y": "0.565133113"
    },
    {
      "turn": "0.131652498",
      "x": "0.489902389",
      "y": "0.561841364"
    },
    {
      "turn": "-0.134829526",
      "x": "0.652094646",
      "y": "0.490844535"
    },
    {
      "turn": "0.123879053",
      "x": "0.334120273",
      "y": "0.481238862"
    },
    {
      "turn": "0.131652498",
      "x": "0.443875654",
      "y": "0.478066161"
    },
    {
      "turn": "0.152448479",
      "x": "0.527449966",
      "y": "0.427875972"
    },
    {
      "turn": "-0.133299825",
      "x": "0.706836569",
      "y": "0.411900215"
    },
    {
      "turn": "0.127536167",
      "x": "0.285711072",
      "y": "0.398444519"
    },
    {
      "turn": "0.131652498",
      "x": "0.396333211",
      "y": "0.395166053"
    },
    {
      "turn": "0.197249610",
      "x": "0.604267467",
      "y": "0.360036818"
    },
    {
      "turn": "0.156096206",
      "x": "0.479998137",
      "y": "0.344345038"
    },
    {
      "turn": "-0.130619735",
      "x": "0.748321517",
      "y": "0.325239844"
    },
    {
      "turn": "0.132014560",
      "x": "0.238075541",
      "y": "0.315114613"
    },
    {
      "turn": "0.131652498",
      "x": "0.347534848",
      "y": "0.312991050"
    },
    {
      "turn": "0.245026390",
      "x": "0.552631750",
      "y": "0.260343476"
    },
    {
      "turn": "0.282386149",
      "x": "0.434417420",
      "y": "0.249024471"
    },
    {
      "turn": "-0.134136147",
      "x": "0.801121920",
      "y": "0.244986703"
    },
    {
      "turn": "0.298251692",
      "x": "0.651528258",
      "y": "0.244202218"
    },
    {
      "turn": "0.131652498",
      "x": "0.189548887",
      "y": "0.232743482"
    },
    {
      "turn": "0.131652498",
      "x": "0.312202526",
      "y": "0.223041426"
    },
    {
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      "x": "0.737367815",
      "y": "0.162395751"
    },
    {
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      "x": "0.850573838",
      "y": "0.162488850"
    },
    {
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      "x": "0.611963484",
      "y": "0.147812075"
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    {
      "turn": "0.131652498",
      "x": "0.141293211",
      "y": "0.149162199"
    },
    {
      "turn": "0.363879433",
      "x": "0.515428020",
      "y": "0.149004586"
    },
    {
      "turn": "0.131652498",
      "x": "0.250990335",
      "y": "0.148033550"
    },
    {
      "turn": "0.347033730",
      "x": "0.416851128",
      "y": "0.149481748"
    },
    {
      "turn": "0.131652498",
      "x": "0.093323166",
      "y": "0.066075643"
    },
    {
      "turn": "0.131652498",
      "x": "0.203207925",
      "y": "0.065271987"
    },
    {
      "turn": "0.414213562",
      "x": "0.419219108",
      "y": "0.047870714"
    },
    {
      "turn": "0.414213562",
      "x": "0.705913572",
      "y": "0.048517341"
    },
    {
      "turn": "0.414213562",
      "x": "0.801478393",
      "y": "0.048996581"
    },
    {
      "turn": "0.414213562",
      "x": "0.897043214",
      "y": "0.047782411"
    },
    {
      "turn": "0.414213562",
      "x": "0.323654287",
      "y": "0.047782411"
    },
    {
      "turn": "0.414213562",
      "x": "0.514783929",
      "y": "0.047784526"
    },
    {
      "turn": "0.414213562",
      "x": "0.610348751",
      "y": "0.047784542"
    }
  ],
  "radius": "0.067574532"
}

提交 {radius, placements:[{x,y,turn},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。 · 验证器 v1.0.0

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