P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 51

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 = 51

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	[0.118297038, 0.118297042]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.118297041	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

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051

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
0.1→0.118297041
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": [
    [
      "0.203860519",
      "0.000000000"
    ],
    [
      "0.408639823",
      "0.000000000"
    ],
    [
      "0.526936865",
      "0.000000000"
    ],
    [
      "0.656364575",
      "0.000000000"
    ],
    [
      "0.851597291",
      "0.000000000"
    ],
    [
      "0.000013072",
      "0.000013516"
    ],
    [
      "0.995348527",
      "0.001872339"
    ],
    [
      "0.306250171",
      "0.059249888"
    ],
    [
      "0.102002504",
      "0.060159246"
    ],
    [
      "0.753980933",
      "0.066822427"
    ],
    [
      "0.591650720",
      "0.099026800"
    ],
    [
      "0.887181887",
      "0.112818113"
    ],
    [
      "0.408756904",
      "0.118296986"
    ],
    [
      "0.204392156",
      "0.119409134"
    ],
    [
      "0.000000000",
      "0.120073178"
    ],
    [
      "0.689267078",
      "0.165849227"
    ],
    [
      "0.306898889",
      "0.178456232"
    ],
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      "0.102389652",
      "0.179323066"
    ],
    [
      "0.508002079",
      "0.182675440"
    ],
    [
      "0.803533246",
      "0.196466754"
    ],
    [
      "0.204896385",
      "0.238370164"
    ],
    [
      "0.000000000",
      "0.238572955"
    ],
    [
      "0.406144064",
      "0.242834686"
    ],
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      "0.605618437",
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      "0.719884605",
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      "0.297620072"
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      "0.521969796",
      "0.333146508"
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    [
      "0.000000000",
      "0.356869998"
    ],
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      "0.407029798",
      "0.361128415"
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      "0.201751909",
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    ],
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      "0.099307073",
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      "0.308322900",
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    ],
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      "0.201758723",
      "0.480295550"
    ],
    [
      "0.000000000",
      "0.485435862"
    ],
    [
      "0.102450382",
      "0.544580995"
    ],
    [
      "0.299781185",
      "0.546520825"
    ],
    [
      "0.409665419",
      "0.590334581"
    ],
    [
      "0.000000000",
      "0.603732904"
    ],
    [
      "0.216132543",
      "0.630169465"
    ],
    [
      "0.102448250",
      "0.662881414"
    ],
    [
      "0.326016778",
      "0.673983221"
    ],
    [
      "0.000000000",
      "0.722029946"
    ],
    [
      "0.242368138",
      "0.757631862"
    ],
    [
      "0.125982927",
      "0.778813761"
    ],
    [
      "0.000000000",
      "0.840326988"
    ],
    [
      "0.104801028",
      "0.895198972"
    ],
    [
      "0.001309309",
      "0.996242022"
    ]
  ]
}

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": 51
}

当前第一名的答案

{
  "points": [
    [
      "0.203860519",
      "0.000000000"
    ],
    [
      "0.408639823",
      "0.000000000"
    ],
    [
      "0.526936865",
      "0.000000000"
    ],
    [
      "0.656364575",
      "0.000000000"
    ],
    [
      "0.851597291",
      "0.000000000"
    ],
    [
      "0.000013072",
      "0.000013516"
    ],
    [
      "0.995348527",
      "0.001872339"
    ],
    [
      "0.306250171",
      "0.059249888"
    ],
    [
      "0.102002504",
      "0.060159246"
    ],
    [
      "0.753980933",
      "0.066822427"
    ],
    [
      "0.591650720",
      "0.099026800"
    ],
    [
      "0.887181887",
      "0.112818113"
    ],
    [
      "0.408756904",
      "0.118296986"
    ],
    [
      "0.204392156",
      "0.119409134"
    ],
    [
      "0.000000000",
      "0.120073178"
    ],
    [
      "0.689267078",
      "0.165849227"
    ],
    [
      "0.306898889",
      "0.178456232"
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      "0.102389652",
      "0.179323066"
    ],
    [
      "0.508002079",
      "0.182675440"
    ],
    [
      "0.803533246",
      "0.196466754"
    ],
    [
      "0.204896385",
      "0.238370164"
    ],
    [
      "0.000000000",
      "0.238572955"
    ],
    [
      "0.406144064",
      "0.242834686"
    ],
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      "0.605618437",
      "0.249497867"
    ],
    [
      "0.719884605",
      "0.280115395"
    ],
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      "0.102506722",
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    ],
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    ],
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    ],
    [
      "0.000000000",
      "0.356869998"
    ],
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      "0.407029798",
      "0.361128415"
    ],
    [
      "0.201751909",
      "0.361998508"
    ],
    [
      "0.636235964",
      "0.363764035"
    ],
    [
      "0.099307073",
      "0.421152930"
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      "0.427029958"
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      "0.447412676"
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    ],
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      "0.480295550"
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    ],
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      "0.102450382",
      "0.544580995"
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      "0.299781185",
      "0.546520825"
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      "0.590334581"
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    ],
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      "0.630169465"
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      "0.102448250",
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      "0.757631862"
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      "0.000000000",
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      "0.104801028",
      "0.895198972"
    ],
    [
      "0.001309309",
      "0.996242022"
    ]
  ]
}

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

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