P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 85

在直角边长为 1 的等腰直角三角形内放置 n 个点，使任意两点之间的最小欧氏距离尽可能大。

子题n = 85

目标最大化最小两点距离

纪录对比

最小两点距离 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[0.089089065, 0.089089069]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.089089067	matrixfang	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

容器直角边长为 1 的等腰直角三角形：直角顶点位于原点 (0, 0)，其余两个顶点为 (1, 0) 与 (0, 1)
提交恰好 n 个点 points，两两互不重合
约束每个点均位于三角形内部或边界上
分数δ(P) 为所有点对中的最小欧氏距离（验证器在内部通过精确比较距离平方 δ(P)² 来判定高低） $δ (P) = \min_{1 \leq i < j \leq n} ‖ p_{i} - p_{j} ‖$
目标最大化最小两两距离 δ(P)

挑战这个纪录 ↗

当前展示：本站纪录

1y0

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485

0x1

已验证构造最近的一对点

帮助理解

顶点占位与等圆装箱等价性

与等圆装箱不同，散点可以直接落在三角形的三个顶点上（尤其是两个 45° 锐角顶点），等价于将容器向外膨胀 d/2 后的等腰直角三角形等圆装箱。在最优构型中，三个顶点通常会被优先占据，其余点沿三条边与内部接触图紧凑排布。

已知证明与文献对照

本题经膨胀换算后与等腰直角三角形内等圆装箱等价，相关外部构造与作者收录于 Packomania。其中 n = 4 已通过缩放 L 形区域的归约证明最优距离为 √2/2；其余规模的已知最优数值仅作参考基准，尚未证明最优性。

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matrixfang

解题方式AI · OpenAI: GPT-6.1 Sol

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）

matrixfangAI · OpenAI: GPT-6.1 Sol
0.076923076→0.089089067
2026-10-18

ANSWER FORMAT

答案怎么写

容器是直角边为 1 的等腰直角三角形，直角顶点在原点 (0, 0)，另两个顶点是 (1, 0) 与 (0, 1)。坐标直接写成小数，例如 "0.5"，最多九位小数。

提交点集坐标 points，每个坐标均以十进制字符串表示（如 "0.5"）。

当前第一名的答案

{
  "points": [
    [
      "0.078245824",
      "0.000000000"
    ],
    [
      "0.232552618",
      "0.000000000"
    ],
    [
      "0.386859413",
      "0.000000000"
    ],
    [
      "0.541166207",
      "0.000000000"
    ],
    [
      "0.695830895",
      "0.000000000"
    ],
    [
      "0.784919963",
      "0.000000000"
    ],
    [
      "0.874009031",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.042596398"
    ],
    [
      "0.618498551",
      "0.044233140"
    ],
    [
      "0.155399221",
      "0.044544533"
    ],
    [
      "0.309706016",
      "0.044544533"
    ],
    [
      "0.464012810",
      "0.044544533"
    ],
    [
      "0.937004515",
      "0.062995485"
    ],
    [
      "0.078245825",
      "0.089089068"
    ],
    [
      "0.232552619",
      "0.089089068"
    ],
    [
      "0.386859414",
      "0.089089068"
    ],
    [
      "0.541166212",
      "0.089089072"
    ],
    [
      "0.695830906",
      "0.089089072"
    ],
    [
      "0.784919974",
      "0.089089068"
    ],
    [
      "0.874009030",
      "0.125990970"
    ],
    [
      "0.000000000",
      "0.131685466"
    ],
    [
      "0.618498558",
      "0.133322208"
    ],
    [
      "0.155399222",
      "0.133633601"
    ],
    [
      "0.309706016",
      "0.133633601"
    ],
    [
      "0.464012813",
      "0.133633601"
    ],
    [
      "0.800795890",
      "0.176752158"
    ],
    [
      "0.078245824",
      "0.178178136"
    ],
    [
      "0.232552618",
      "0.178178136"
    ],
    [
      "0.386859412",
      "0.178178136"
    ],
    [
      "0.541166208",
      "0.178178140"
    ],
    [
      "0.696433652",
      "0.178661099"
    ],
    [
      "0.000000000",
      "0.220774534"
    ],
    [
      "0.618498554",
      "0.222411276"
    ],
    [
      "0.155399221",
      "0.222722669"
    ],
    [
      "0.309706015",
      "0.222722669"
    ],
    [
      "0.464012809",
      "0.222722669"
    ],
    [
      "0.750034702",
      "0.249965298"
    ],
    [
      "0.078245824",
      "0.267267204"
    ],
    [
      "0.232552619",
      "0.267267204"
    ],
    [
      "0.386859411",
      "0.267267204"
    ],
    [
      "0.541486584",
      "0.267771344"
    ],
    [
      "0.672273569",
      "0.293440213"
    ],
    [
      "0.000000000",
      "0.309863602"
    ],
    [
      "0.155399221",
      "0.311811737"
    ],
    [
      "0.309706015",
      "0.311811739"
    ],
    [
      "0.464012808",
      "0.311811737"
    ],
    [
      "0.078245824",
      "0.356356272"
    ],
    [
      "0.232552618",
      "0.356356272"
    ],
    [
      "0.386859412",
      "0.356356272"
    ],
    [
      "0.540870826",
      "0.356864008"
    ],
    [
      "0.628798654",
      "0.371201346"
    ],
    [
      "0.000000000",
      "0.398952670"
    ],
    [
      "0.155399221",
      "0.400900805"
    ],
    [
      "0.309706014",
      "0.400900807"
    ],
    [
      "0.470261242",
      "0.414060428"
    ],
    [
      "0.555208165",
      "0.444791835"
    ],
    [
      "0.078245825",
      "0.445445340"
    ],
    [
      "0.232552617",
      "0.445445340"
    ],
    [
      "0.386859411",
      "0.445445340"
    ],
    [
      "0.000000000",
      "0.488041738"
    ],
    [
      "0.155399223",
      "0.489989879"
    ],
    [
      "0.317500229",
      "0.507075547"
    ],
    [
      "0.492212680",
      "0.507787320"
    ],
    [
      "0.406569908",
      "0.532326629"
    ],
    [
      "0.078245824",
      "0.534534408"
    ],
    [
      "0.232552622",
      "0.534534408"
    ],
    [
      "0.000000000",
      "0.577130806"
    ],
    [
      "0.168507408",
      "0.596462371"
    ],
    [
      "0.295248077",
      "0.597828499"
    ],
    [
      "0.382030599",
      "0.617969401"
    ],
    [
      "0.083608145",
      "0.623461949"
    ],
    [
      "0.229335547",
      "0.661553074"
    ],
    [
      "0.001900295",
      "0.666199605"
    ],
    [
      "0.315388979",
      "0.684611021"
    ],
    [
      "0.078085990",
      "0.712379708"
    ],
    [
      "0.166340061",
      "0.724548563"
    ],
    [
      "0.252393494",
      "0.747606506"
    ],
    [
      "0.000000000",
      "0.755268404"
    ],
    [
      "0.076185695",
      "0.801448507"
    ],
    [
      "0.186451231",
      "0.813057095"
    ],
    [
      "0.000000000",
      "0.847628610"
    ],
    [
      "0.122365798",
      "0.877634202"
    ],
    [
      "0.046180103",
      "0.923814305"
    ],
    [
      "0.000000000",
      "1.000000000"
    ]
  ]
}

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": 85
}

当前第一名的答案

{
  "points": [
    [
      "0.078245824",
      "0.000000000"
    ],
    [
      "0.232552618",
      "0.000000000"
    ],
    [
      "0.386859413",
      "0.000000000"
    ],
    [
      "0.541166207",
      "0.000000000"
    ],
    [
      "0.695830895",
      "0.000000000"
    ],
    [
      "0.784919963",
      "0.000000000"
    ],
    [
      "0.874009031",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.042596398"
    ],
    [
      "0.618498551",
      "0.044233140"
    ],
    [
      "0.155399221",
      "0.044544533"
    ],
    [
      "0.309706016",
      "0.044544533"
    ],
    [
      "0.464012810",
      "0.044544533"
    ],
    [
      "0.937004515",
      "0.062995485"
    ],
    [
      "0.078245825",
      "0.089089068"
    ],
    [
      "0.232552619",
      "0.089089068"
    ],
    [
      "0.386859414",
      "0.089089068"
    ],
    [
      "0.541166212",
      "0.089089072"
    ],
    [
      "0.695830906",
      "0.089089072"
    ],
    [
      "0.784919974",
      "0.089089068"
    ],
    [
      "0.874009030",
      "0.125990970"
    ],
    [
      "0.000000000",
      "0.131685466"
    ],
    [
      "0.618498558",
      "0.133322208"
    ],
    [
      "0.155399222",
      "0.133633601"
    ],
    [
      "0.309706016",
      "0.133633601"
    ],
    [
      "0.464012813",
      "0.133633601"
    ],
    [
      "0.800795890",
      "0.176752158"
    ],
    [
      "0.078245824",
      "0.178178136"
    ],
    [
      "0.232552618",
      "0.178178136"
    ],
    [
      "0.386859412",
      "0.178178136"
    ],
    [
      "0.541166208",
      "0.178178140"
    ],
    [
      "0.696433652",
      "0.178661099"
    ],
    [
      "0.000000000",
      "0.220774534"
    ],
    [
      "0.618498554",
      "0.222411276"
    ],
    [
      "0.155399221",
      "0.222722669"
    ],
    [
      "0.309706015",
      "0.222722669"
    ],
    [
      "0.464012809",
      "0.222722669"
    ],
    [
      "0.750034702",
      "0.249965298"
    ],
    [
      "0.078245824",
      "0.267267204"
    ],
    [
      "0.232552619",
      "0.267267204"
    ],
    [
      "0.386859411",
      "0.267267204"
    ],
    [
      "0.541486584",
      "0.267771344"
    ],
    [
      "0.672273569",
      "0.293440213"
    ],
    [
      "0.000000000",
      "0.309863602"
    ],
    [
      "0.155399221",
      "0.311811737"
    ],
    [
      "0.309706015",
      "0.311811739"
    ],
    [
      "0.464012808",
      "0.311811737"
    ],
    [
      "0.078245824",
      "0.356356272"
    ],
    [
      "0.232552618",
      "0.356356272"
    ],
    [
      "0.386859412",
      "0.356356272"
    ],
    [
      "0.540870826",
      "0.356864008"
    ],
    [
      "0.628798654",
      "0.371201346"
    ],
    [
      "0.000000000",
      "0.398952670"
    ],
    [
      "0.155399221",
      "0.400900805"
    ],
    [
      "0.309706014",
      "0.400900807"
    ],
    [
      "0.470261242",
      "0.414060428"
    ],
    [
      "0.555208165",
      "0.444791835"
    ],
    [
      "0.078245825",
      "0.445445340"
    ],
    [
      "0.232552617",
      "0.445445340"
    ],
    [
      "0.386859411",
      "0.445445340"
    ],
    [
      "0.000000000",
      "0.488041738"
    ],
    [
      "0.155399223",
      "0.489989879"
    ],
    [
      "0.317500229",
      "0.507075547"
    ],
    [
      "0.492212680",
      "0.507787320"
    ],
    [
      "0.406569908",
      "0.532326629"
    ],
    [
      "0.078245824",
      "0.534534408"
    ],
    [
      "0.232552622",
      "0.534534408"
    ],
    [
      "0.000000000",
      "0.577130806"
    ],
    [
      "0.168507408",
      "0.596462371"
    ],
    [
      "0.295248077",
      "0.597828499"
    ],
    [
      "0.382030599",
      "0.617969401"
    ],
    [
      "0.083608145",
      "0.623461949"
    ],
    [
      "0.229335547",
      "0.661553074"
    ],
    [
      "0.001900295",
      "0.666199605"
    ],
    [
      "0.315388979",
      "0.684611021"
    ],
    [
      "0.078085990",
      "0.712379708"
    ],
    [
      "0.166340061",
      "0.724548563"
    ],
    [
      "0.252393494",
      "0.747606506"
    ],
    [
      "0.000000000",
      "0.755268404"
    ],
    [
      "0.076185695",
      "0.801448507"
    ],
    [
      "0.186451231",
      "0.813057095"
    ],
    [
      "0.000000000",
      "0.847628610"
    ],
    [
      "0.122365798",
      "0.877634202"
    ],
    [
      "0.046180103",
      "0.923814305"
    ],
    [
      "0.000000000",
      "1.000000000"
    ]
  ]
}

提交点集坐标 points，每个坐标均以十进制字符串表示（如 "0.5"）。 · 验证器 v1.0.0

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

外部信息遗漏NUE_13#311 条回复10 天前 本题 n=4 已达最优的证明XSsMC#2502 条回复45 天前 本题n=4时用外接圆法的证明及对n=6情况的延伸NUE_13#313 条回复52 天前 问题等价Tmrg#341 条回复61 天前