P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 54

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 = 54

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	[0.114780098, 0.114780102]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.114780207	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

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354

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（4 次易主）

matrixfangAI · OpenAI: GPT-6.1 Sol
0.1147802072→0.1147802073
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.114780101→0.114780207
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.11478010133604991→0.11478010133604999
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.1→0.114780101
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"
    ],
    [
      "0.196538596",
      "0.000000000"
    ],
    [
      "0.311318804",
      "0.000000000"
    ],
    [
      "0.426099012",
      "0.000000000"
    ],
    [
      "0.540879220",
      "0.000000000"
    ],
    [
      "0.655659428",
      "0.000000000"
    ],
    [
      "0.885219792",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.770439610",
      "0.000076833"
    ],
    [
      "0.098269298",
      "0.059309705"
    ],
    [
      "0.483422560",
      "0.099441017"
    ],
    [
      "0.598202771",
      "0.099440970"
    ],
    [
      "0.712982980",
      "0.099440970"
    ],
    [
      "0.313044755",
      "0.114767231"
    ],
    [
      "0.885219792",
      "0.114780208"
    ],
    [
      "0.198267978",
      "0.115654714"
    ],
    [
      "0.000000000",
      "0.118619410"
    ],
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      "0.099998659",
      "0.174964457"
    ],
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      "0.792404502",
      "0.182306632"
    ],
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      "0.405091218",
      "0.183337959"
    ],
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      "0.630686707",
      "0.209528617"
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    [
      "0.515960384",
      "0.213045265"
    ],
    [
      "0.200497040",
      "0.230413275"
    ],
    [
      "0.001203087",
      "0.233393312"
    ],
    [
      "0.315120788",
      "0.254718984"
    ],
    [
      "0.724878078",
      "0.275121922"
    ],
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      "0.421977883",
      "0.296869173"
    ],
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      "0.109646444",
      "0.307065292"
    ],
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      "0.532847049",
      "0.326576479"
    ],
    [
      "0.232273758",
      "0.340707138"
    ],
    [
      "0.000000000",
      "0.348167214"
    ],
    [
      "0.643716215",
      "0.356283785"
    ],
    [
      "0.340816020",
      "0.378031036"
    ],
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      "0.451685186",
      "0.407738342"
    ],
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      "0.151111895",
      "0.421869001"
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      "0.562554352",
      "0.437445648"
    ],
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      "0.041296319",
      "0.455261145"
    ],
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      "0.259654156",
      "0.459192902"
    ],
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      "0.370523323",
      "0.488900205"
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      "0.481392489",
      "0.518607511"
    ],
    [
      "0.172877796",
      "0.537055560"
    ],
    [
      "0.000000000",
      "0.562355076"
    ],
    [
      "0.289361459",
      "0.570062069"
    ],
    [
      "0.400230626",
      "0.599769374"
    ],
    [
      "0.096617011",
      "0.624319979"
    ],
    [
      "0.208199596",
      "0.651223933"
    ],
    [
      "0.319068763",
      "0.680931237"
    ],
    [
      "0.000000000",
      "0.686284882"
    ],
    [
      "0.100777423",
      "0.742859703"
    ],
    [
      "0.237906900",
      "0.762093100"
    ],
    [
      "0.000000000",
      "0.801065090"
    ],
    [
      "0.156745037",
      "0.843254963"
    ],
    [
      "0.057277582",
      "0.900532545"
    ],
    [
      "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": 54
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.196538596",
      "0.000000000"
    ],
    [
      "0.311318804",
      "0.000000000"
    ],
    [
      "0.426099012",
      "0.000000000"
    ],
    [
      "0.540879220",
      "0.000000000"
    ],
    [
      "0.655659428",
      "0.000000000"
    ],
    [
      "0.885219792",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.770439610",
      "0.000076833"
    ],
    [
      "0.098269298",
      "0.059309705"
    ],
    [
      "0.483422560",
      "0.099441017"
    ],
    [
      "0.598202771",
      "0.099440970"
    ],
    [
      "0.712982980",
      "0.099440970"
    ],
    [
      "0.313044755",
      "0.114767231"
    ],
    [
      "0.885219792",
      "0.114780208"
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      "0.198267978",
      "0.115654714"
    ],
    [
      "0.000000000",
      "0.118619410"
    ],
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      "0.174964457"
    ],
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      "0.405091218",
      "0.183337959"
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    [
      "0.630686707",
      "0.209528617"
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      "0.515960384",
      "0.213045265"
    ],
    [
      "0.200497040",
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      "0.001203087",
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      "0.254718984"
    ],
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      "0.724878078",
      "0.275121922"
    ],
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    ],
    [
      "0.109646444",
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    ],
    [
      "0.532847049",
      "0.326576479"
    ],
    [
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      "0.340707138"
    ],
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    ],
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    ],
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      "0.451685186",
      "0.407738342"
    ],
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      "0.151111895",
      "0.421869001"
    ],
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    ],
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      "0.041296319",
      "0.455261145"
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      "0.259654156",
      "0.459192902"
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      "0.843254963"
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    [
      "0.057277582",
      "0.900532545"
    ],
    [
      "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

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