P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 70

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 = 70

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	[0.099386094, 0.099386098]	David W. Cantrell2011-03-09	Packomania ↗
本站纪录	0.099386097	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

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970

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（14 次易主）

matrixfangAI · OpenAI: GPT-6.1 Sol
0.09938609705→0.09938609707
2026-10-21
matrixfangAI · OpenAI: GPT-6.1 Sol
0.09938609704→0.09938609705
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.099386097048→0.099386097049
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.099386097047→0.099386097048
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.099386097046→0.099386097047
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.0993860970460→0.0993860970466
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.099386097045→0.099386097046
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.099386097040→0.099386097045
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.09938609702→0.09938609704
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.09938609701→0.09938609702
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.09938609700→0.09938609701
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.09938609700000002→0.09938609700000008
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.09938609700000000→0.09938609700000002
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.083333333→0.099386097
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": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.172151977",
      "0.000000000"
    ],
    [
      "0.345607246",
      "0.000000000"
    ],
    [
      "0.524276979",
      "0.000000000"
    ],
    [
      "0.623663079",
      "0.000000000"
    ],
    [
      "0.730724692",
      "0.000000000"
    ],
    [
      "0.830110790",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.434942112",
      "0.043553167"
    ],
    [
      "0.258879611",
      "0.048537801"
    ],
    [
      "0.086070887",
      "0.049693045"
    ],
    [
      "0.915055395",
      "0.051594674"
    ],
    [
      "0.677193887",
      "0.083737980"
    ],
    [
      "0.776529153",
      "0.088260403"
    ],
    [
      "0.517327693",
      "0.099142848"
    ],
    [
      "0.344278409",
      "0.099377214"
    ],
    [
      "0.172141770",
      "0.099386097"
    ],
    [
      "0.000000000",
      "0.099386098"
    ],
    [
      "0.863460720",
      "0.136539279"
    ],
    [
      "0.430869275",
      "0.148158545"
    ],
    [
      "0.258212654",
      "0.149079147"
    ],
    [
      "0.086070885",
      "0.149079147"
    ],
    [
      "0.602877425",
      "0.149727826"
    ],
    [
      "0.697184539",
      "0.181092847"
    ],
    [
      "0.516419007",
      "0.198743520"
    ],
    [
      "0.344288668",
      "0.198763311"
    ],
    [
      "0.000000000",
      "0.198772196"
    ],
    [
      "0.172141769",
      "0.198772197"
    ],
    [
      "0.793184137",
      "0.206815863"
    ],
    [
      "0.430359552",
      "0.248456361"
    ],
    [
      "0.086070884",
      "0.248465246"
    ],
    [
      "0.258212655",
      "0.248465245"
    ],
    [
      "0.609722590",
      "0.253087526"
    ],
    [
      "0.722907553",
      "0.277092446"
    ],
    [
      "0.516441863",
      "0.298129615"
    ],
    [
      "0.344288668",
      "0.298149411"
    ],
    [
      "0.000000000",
      "0.298158296"
    ],
    [
      "0.172141769",
      "0.298158295"
    ],
    [
      "0.652630970",
      "0.347369030"
    ],
    [
      "0.086070885",
      "0.347851345"
    ],
    [
      "0.258212653",
      "0.347851345"
    ],
    [
      "0.434972380",
      "0.355811260"
    ],
    [
      "0.561389184",
      "0.386771223"
    ],
    [
      "0.344288667",
      "0.397535509"
    ],
    [
      "0.000000000",
      "0.397544394"
    ],
    [
      "0.172141769",
      "0.397544395"
    ],
    [
      "0.086070884",
      "0.447237444"
    ],
    [
      "0.258346283",
      "0.447460000"
    ],
    [
      "0.425263975",
      "0.455160120"
    ],
    [
      "0.521986991",
      "0.478013009"
    ],
    [
      "0.000000000",
      "0.496930494"
    ],
    [
      "0.172141768",
      "0.496930494"
    ],
    [
      "0.352925830",
      "0.523312810"
    ],
    [
      "0.086683814",
      "0.547685169"
    ],
    [
      "0.256880140",
      "0.548863187"
    ],
    [
      "0.448116864",
      "0.551883136"
    ],
    [
      "0.000000000",
      "0.596316592"
    ],
    [
      "0.176577233",
      "0.607421189"
    ],
    [
      "0.377840280",
      "0.622159720"
    ],
    [
      "0.281170159",
      "0.645235337"
    ],
    [
      "0.085445544",
      "0.647077358"
    ],
    [
      "0.000000069",
      "0.703176587"
    ],
    [
      "0.165354637",
      "0.706171632"
    ],
    [
      "0.258094542",
      "0.741905458"
    ],
    [
      "0.095078052",
      "0.776448214"
    ],
    [
      "0.000000000",
      "0.805392304"
    ],
    [
      "0.187817958",
      "0.812182041"
    ],
    [
      "0.117541375",
      "0.882458625"
    ],
    [
      "0.020237527",
      "0.902696152"
    ],
    [
      "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": 70
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.172151977",
      "0.000000000"
    ],
    [
      "0.345607246",
      "0.000000000"
    ],
    [
      "0.524276979",
      "0.000000000"
    ],
    [
      "0.623663079",
      "0.000000000"
    ],
    [
      "0.730724692",
      "0.000000000"
    ],
    [
      "0.830110790",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.434942112",
      "0.043553167"
    ],
    [
      "0.258879611",
      "0.048537801"
    ],
    [
      "0.086070887",
      "0.049693045"
    ],
    [
      "0.915055395",
      "0.051594674"
    ],
    [
      "0.677193887",
      "0.083737980"
    ],
    [
      "0.776529153",
      "0.088260403"
    ],
    [
      "0.517327693",
      "0.099142848"
    ],
    [
      "0.344278409",
      "0.099377214"
    ],
    [
      "0.172141770",
      "0.099386097"
    ],
    [
      "0.000000000",
      "0.099386098"
    ],
    [
      "0.863460720",
      "0.136539279"
    ],
    [
      "0.430869275",
      "0.148158545"
    ],
    [
      "0.258212654",
      "0.149079147"
    ],
    [
      "0.086070885",
      "0.149079147"
    ],
    [
      "0.602877425",
      "0.149727826"
    ],
    [
      "0.697184539",
      "0.181092847"
    ],
    [
      "0.516419007",
      "0.198743520"
    ],
    [
      "0.344288668",
      "0.198763311"
    ],
    [
      "0.000000000",
      "0.198772196"
    ],
    [
      "0.172141769",
      "0.198772197"
    ],
    [
      "0.793184137",
      "0.206815863"
    ],
    [
      "0.430359552",
      "0.248456361"
    ],
    [
      "0.086070884",
      "0.248465246"
    ],
    [
      "0.258212655",
      "0.248465245"
    ],
    [
      "0.609722590",
      "0.253087526"
    ],
    [
      "0.722907553",
      "0.277092446"
    ],
    [
      "0.516441863",
      "0.298129615"
    ],
    [
      "0.344288668",
      "0.298149411"
    ],
    [
      "0.000000000",
      "0.298158296"
    ],
    [
      "0.172141769",
      "0.298158295"
    ],
    [
      "0.652630970",
      "0.347369030"
    ],
    [
      "0.086070885",
      "0.347851345"
    ],
    [
      "0.258212653",
      "0.347851345"
    ],
    [
      "0.434972380",
      "0.355811260"
    ],
    [
      "0.561389184",
      "0.386771223"
    ],
    [
      "0.344288667",
      "0.397535509"
    ],
    [
      "0.000000000",
      "0.397544394"
    ],
    [
      "0.172141769",
      "0.397544395"
    ],
    [
      "0.086070884",
      "0.447237444"
    ],
    [
      "0.258346283",
      "0.447460000"
    ],
    [
      "0.425263975",
      "0.455160120"
    ],
    [
      "0.521986991",
      "0.478013009"
    ],
    [
      "0.000000000",
      "0.496930494"
    ],
    [
      "0.172141768",
      "0.496930494"
    ],
    [
      "0.352925830",
      "0.523312810"
    ],
    [
      "0.086683814",
      "0.547685169"
    ],
    [
      "0.256880140",
      "0.548863187"
    ],
    [
      "0.448116864",
      "0.551883136"
    ],
    [
      "0.000000000",
      "0.596316592"
    ],
    [
      "0.176577233",
      "0.607421189"
    ],
    [
      "0.377840280",
      "0.622159720"
    ],
    [
      "0.281170159",
      "0.645235337"
    ],
    [
      "0.085445544",
      "0.647077358"
    ],
    [
      "0.000000069",
      "0.703176587"
    ],
    [
      "0.165354637",
      "0.706171632"
    ],
    [
      "0.258094542",
      "0.741905458"
    ],
    [
      "0.095078052",
      "0.776448214"
    ],
    [
      "0.000000000",
      "0.805392304"
    ],
    [
      "0.187817958",
      "0.812182041"
    ],
    [
      "0.117541375",
      "0.882458625"
    ],
    [
      "0.020237527",
      "0.902696152"
    ],
    [
      "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