P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 97

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 = 97

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	[0.083053857, 0.083053861]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.083057101	matrixfang	纪录详情 ↓

已超越外部纪录 · 审核通过。审核依据 ↗

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

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485868788899091929394959697

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（7 次易主）

matrixfangAI · OpenAI: GPT-6.1 Sol
0.08305710100000002→0.08305710100000005
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.08305710100000000→0.08305710100000002
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.083057101000000000→0.083057101000000006
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.083056845→0.083057101
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.083053859→0.083056845
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.0830538594→0.0830538596
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.071428571→0.083053859
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"
    ],
    [
      "0.431577370",
      "0.000000000"
    ],
    [
      "0.578114469",
      "0.000000004"
    ],
    [
      "0.661171570",
      "0.000000001"
    ],
    [
      "0.753896858",
      "0.000000000"
    ],
    [
      "0.836953960",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.143859122",
      "0.000000000"
    ],
    [
      "0.287718244",
      "0.000000000"
    ],
    [
      "0.918476980",
      "0.015889596"
    ],
    [
      "0.359647805",
      "0.041528548"
    ],
    [
      "0.215788683",
      "0.041528548"
    ],
    [
      "0.071929561",
      "0.041528548"
    ],
    [
      "0.504845914",
      "0.039117810"
    ],
    [
      "0.707534215",
      "0.068912897"
    ],
    [
      "0.812300804",
      "0.079313958"
    ],
    [
      "0.431577363",
      "0.083057101"
    ],
    [
      "0.287718247",
      "0.083057101"
    ],
    [
      "0.143859125",
      "0.083057101"
    ],
    [
      "0.000000003",
      "0.083057101"
    ],
    [
      "0.578114474",
      "0.083057105"
    ],
    [
      "0.902587384",
      "0.097412616"
    ],
    [
      "0.504845905",
      "0.122174911"
    ],
    [
      "0.215788686",
      "0.124585649"
    ],
    [
      "0.359647805",
      "0.124585654"
    ],
    [
      "0.071929564",
      "0.124585649"
    ],
    [
      "0.648803977",
      "0.127643138"
    ],
    [
      "0.753570564",
      "0.138044197"
    ],
    [
      "0.843857143",
      "0.156142856"
    ],
    [
      "0.575255306",
      "0.166231673"
    ],
    [
      "0.431577367",
      "0.166114204"
    ],
    [
      "0.287718244",
      "0.166114202"
    ],
    [
      "0.143859122",
      "0.166114202"
    ],
    [
      "0.000000000",
      "0.166114202"
    ],
    [
      "0.694840326",
      "0.196774438"
    ],
    [
      "0.505531404",
      "0.211365469"
    ],
    [
      "0.359647802",
      "0.207642755"
    ],
    [
      "0.071929561",
      "0.207642750"
    ],
    [
      "0.215788683",
      "0.207642750"
    ],
    [
      "0.785126904",
      "0.214873096"
    ],
    [
      "0.621291655",
      "0.235362974"
    ],
    [
      "0.431577360",
      "0.249171308"
    ],
    [
      "0.287718241",
      "0.249171303"
    ],
    [
      "0.000000003",
      "0.249171303"
    ],
    [
      "0.143859119",
      "0.249171304"
    ],
    [
      "0.726396664",
      "0.273603336"
    ],
    [
      "0.505531405",
      "0.294422571"
    ],
    [
      "0.359647799",
      "0.290699856"
    ],
    [
      "0.071929561",
      "0.290699857"
    ],
    [
      "0.215788680",
      "0.290699852"
    ],
    [
      "0.586922358",
      "0.310975364"
    ],
    [
      "0.667287007",
      "0.331951691"
    ],
    [
      "0.431577361",
      "0.332228410"
    ],
    [
      "0.000000000",
      "0.332228405"
    ],
    [
      "0.143859122",
      "0.332228405"
    ],
    [
      "0.287718238",
      "0.332228405"
    ],
    [
      "0.359647800",
      "0.373756958"
    ],
    [
      "0.071929558",
      "0.373756958"
    ],
    [
      "0.215788680",
      "0.373756958"
    ],
    [
      "0.505531418",
      "0.377479687"
    ],
    [
      "0.608938652",
      "0.391061348"
    ],
    [
      "0.431577358",
      "0.415285511"
    ],
    [
      "0.000000000",
      "0.415285511"
    ],
    [
      "0.143859119",
      "0.415285506"
    ],
    [
      "0.287718240",
      "0.415285508"
    ],
    [
      "0.550208412",
      "0.449791588"
    ],
    [
      "0.071929561",
      "0.456814059"
    ],
    [
      "0.215788677",
      "0.456814059"
    ],
    [
      "0.359775683",
      "0.457035562"
    ],
    [
      "0.475386456",
      "0.485849277"
    ],
    [
      "0.000000003",
      "0.498342612"
    ],
    [
      "0.143859119",
      "0.498342612"
    ],
    [
      "0.287718237",
      "0.498342609"
    ],
    [
      "0.071929561",
      "0.539871165"
    ],
    [
      "0.215788679",
      "0.539871162"
    ],
    [
      "0.358410412",
      "0.541944197"
    ],
    [
      "0.439328767",
      "0.560671233"
    ],
    [
      "0.000000000",
      "0.581399713"
    ],
    [
      "0.143859121",
      "0.581399715"
    ],
    [
      "0.293606101",
      "0.593894975"
    ],
    [
      "0.071929562",
      "0.622928267"
    ],
    [
      "0.215788682",
      "0.622928263"
    ],
    [
      "0.377137448",
      "0.622862552"
    ],
    [
      "0.000000000",
      "0.664456815"
    ],
    [
      "0.143859123",
      "0.664456819"
    ],
    [
      "0.318407208",
      "0.681592791"
    ],
    [
      "0.238135661",
      "0.702922604"
    ],
    [
      "0.071929558",
      "0.705985368"
    ],
    [
      "0.000000000",
      "0.747513930"
    ],
    [
      "0.138402022",
      "0.755783898"
    ],
    [
      "0.216805849",
      "0.783194151"
    ],
    [
      "0.079671782",
      "0.814514137"
    ],
    [
      "0.000000000",
      "0.837985166"
    ],
    [
      "0.158075609",
      "0.841924390"
    ],
    [
      "0.099345369",
      "0.900654630"
    ],
    [
      "0.018337952",
      "0.918992583"
    ],
    [
      "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": 97
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.431577370",
      "0.000000000"
    ],
    [
      "0.578114469",
      "0.000000004"
    ],
    [
      "0.661171570",
      "0.000000001"
    ],
    [
      "0.753896858",
      "0.000000000"
    ],
    [
      "0.836953960",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.143859122",
      "0.000000000"
    ],
    [
      "0.287718244",
      "0.000000000"
    ],
    [
      "0.918476980",
      "0.015889596"
    ],
    [
      "0.359647805",
      "0.041528548"
    ],
    [
      "0.215788683",
      "0.041528548"
    ],
    [
      "0.071929561",
      "0.041528548"
    ],
    [
      "0.504845914",
      "0.039117810"
    ],
    [
      "0.707534215",
      "0.068912897"
    ],
    [
      "0.812300804",
      "0.079313958"
    ],
    [
      "0.431577363",
      "0.083057101"
    ],
    [
      "0.287718247",
      "0.083057101"
    ],
    [
      "0.143859125",
      "0.083057101"
    ],
    [
      "0.000000003",
      "0.083057101"
    ],
    [
      "0.578114474",
      "0.083057105"
    ],
    [
      "0.902587384",
      "0.097412616"
    ],
    [
      "0.504845905",
      "0.122174911"
    ],
    [
      "0.215788686",
      "0.124585649"
    ],
    [
      "0.359647805",
      "0.124585654"
    ],
    [
      "0.071929564",
      "0.124585649"
    ],
    [
      "0.648803977",
      "0.127643138"
    ],
    [
      "0.753570564",
      "0.138044197"
    ],
    [
      "0.843857143",
      "0.156142856"
    ],
    [
      "0.575255306",
      "0.166231673"
    ],
    [
      "0.431577367",
      "0.166114204"
    ],
    [
      "0.287718244",
      "0.166114202"
    ],
    [
      "0.143859122",
      "0.166114202"
    ],
    [
      "0.000000000",
      "0.166114202"
    ],
    [
      "0.694840326",
      "0.196774438"
    ],
    [
      "0.505531404",
      "0.211365469"
    ],
    [
      "0.359647802",
      "0.207642755"
    ],
    [
      "0.071929561",
      "0.207642750"
    ],
    [
      "0.215788683",
      "0.207642750"
    ],
    [
      "0.785126904",
      "0.214873096"
    ],
    [
      "0.621291655",
      "0.235362974"
    ],
    [
      "0.431577360",
      "0.249171308"
    ],
    [
      "0.287718241",
      "0.249171303"
    ],
    [
      "0.000000003",
      "0.249171303"
    ],
    [
      "0.143859119",
      "0.249171304"
    ],
    [
      "0.726396664",
      "0.273603336"
    ],
    [
      "0.505531405",
      "0.294422571"
    ],
    [
      "0.359647799",
      "0.290699856"
    ],
    [
      "0.071929561",
      "0.290699857"
    ],
    [
      "0.215788680",
      "0.290699852"
    ],
    [
      "0.586922358",
      "0.310975364"
    ],
    [
      "0.667287007",
      "0.331951691"
    ],
    [
      "0.431577361",
      "0.332228410"
    ],
    [
      "0.000000000",
      "0.332228405"
    ],
    [
      "0.143859122",
      "0.332228405"
    ],
    [
      "0.287718238",
      "0.332228405"
    ],
    [
      "0.359647800",
      "0.373756958"
    ],
    [
      "0.071929558",
      "0.373756958"
    ],
    [
      "0.215788680",
      "0.373756958"
    ],
    [
      "0.505531418",
      "0.377479687"
    ],
    [
      "0.608938652",
      "0.391061348"
    ],
    [
      "0.431577358",
      "0.415285511"
    ],
    [
      "0.000000000",
      "0.415285511"
    ],
    [
      "0.143859119",
      "0.415285506"
    ],
    [
      "0.287718240",
      "0.415285508"
    ],
    [
      "0.550208412",
      "0.449791588"
    ],
    [
      "0.071929561",
      "0.456814059"
    ],
    [
      "0.215788677",
      "0.456814059"
    ],
    [
      "0.359775683",
      "0.457035562"
    ],
    [
      "0.475386456",
      "0.485849277"
    ],
    [
      "0.000000003",
      "0.498342612"
    ],
    [
      "0.143859119",
      "0.498342612"
    ],
    [
      "0.287718237",
      "0.498342609"
    ],
    [
      "0.071929561",
      "0.539871165"
    ],
    [
      "0.215788679",
      "0.539871162"
    ],
    [
      "0.358410412",
      "0.541944197"
    ],
    [
      "0.439328767",
      "0.560671233"
    ],
    [
      "0.000000000",
      "0.581399713"
    ],
    [
      "0.143859121",
      "0.581399715"
    ],
    [
      "0.293606101",
      "0.593894975"
    ],
    [
      "0.071929562",
      "0.622928267"
    ],
    [
      "0.215788682",
      "0.622928263"
    ],
    [
      "0.377137448",
      "0.622862552"
    ],
    [
      "0.000000000",
      "0.664456815"
    ],
    [
      "0.143859123",
      "0.664456819"
    ],
    [
      "0.318407208",
      "0.681592791"
    ],
    [
      "0.238135661",
      "0.702922604"
    ],
    [
      "0.071929558",
      "0.705985368"
    ],
    [
      "0.000000000",
      "0.747513930"
    ],
    [
      "0.138402022",
      "0.755783898"
    ],
    [
      "0.216805849",
      "0.783194151"
    ],
    [
      "0.079671782",
      "0.814514137"
    ],
    [
      "0.000000000",
      "0.837985166"
    ],
    [
      "0.158075609",
      "0.841924390"
    ],
    [
      "0.099345369",
      "0.900654630"
    ],
    [
      "0.018337952",
      "0.918992583"
    ],
    [
      "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