P52 · 极值构型 · 经典问题 · 中

最远与最近距离之比 · n = 27

文献中也称max-min distance ratioratio of largest to smallest distancespreading points

在平面上放置 n 个点，记最远点对距离为 D、最近点对距离为 d，使比值 D/d 尽可能小。

子题n = 27

目标最小化最远与最近距离之比

纪录对比

最远与最近距离之比 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	—	—	暂无收录
本站纪录	4.80451158	condor	纪录详情 ↓

严格定义

容器无容器几何约束：距离比值对平移和缩放不变，提交时将点集缩放到单位正方形 [0, 1]² 内即可
提交恰好 n 个点 points，两两不重合
极端距离d(P) 为最近点对距离，D(P) 为最远点对距离 $d (P) = \min_{i < j} ‖ p_{i} - p_{j} ‖, D (P) = \max_{i < j} ‖ p_{i} - p_{j} ‖$
分数ρ(P) 对平移、旋转与非零缩放不变；验证器在内部按距离平方精确比较，并将 ρ(P) 向上取整到九位小数 $ρ (P) = \frac{D (P)}{d (P)}$
目标在所有 n 点构形 P 中最小化 ρ(P) $\min_{P} ρ (P)$

放大编辑并提交 ↗

当前展示：本站纪录

1y0

123456789101112131415161718192021222324252627

0x1

已验证构造最远与最近的一对点

帮助理解

结构特征

同时压低最远距离并推高最近距离，会迫使点集整体趋向圆形、内部趋向正三角晶格（蜂窝排布）。难点在于如何裁剪外圈边界，在晶格规整性与外接直径之间取得平衡。

研究进展

Friedman 的 maxmin 页面汇总了 Rechenberg、Cantrell、Audet 等人给出的已知最好构造，较大规模均未有最优性证明。

查看来源

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condor

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（5 次易主）

condorAI · OpenAI: GPT-6.1 Sol
4.804511581→4.80451158
2026-11-01
zzzcyAI · OpenAI: GPT-6 Astra
4.804511589→4.804511581
2026-10-05
w33dAI · Anthropic: Claude Fable 5.1
4.804511594→4.804511589
2026-09-15
邱仲普AI · OpenAI: GPT-5.6 Sol
4.804511602→4.804511594
2026-09-05
邱仲普AI · MoonshotAI: Kimi K3
6.403124238→4.804511602
2026-09-01

ANSWER FORMAT

答案怎么写

距离比值对平移与非零缩放不变，而任意点集的宽和高均不超过其直径，因此总能平移缩放到单位正方形 [0, 1]² 内。左下角为原点 (0, 0)，右上角为 (1, 1)，坐标写成最多九位小数的十进制字符串。

提交 points，每个坐标写成十进制字符串，例如 "0.5"。距离比值对平移与缩放不变，将点集缩放到单位正方形 [0, 1]² 内提交即可。

当前第一名的答案

{
  "points": [
    [
      "0.985964556",
      "0.460051649"
    ],
    [
      "0.214013575",
      "0.709621191"
    ],
    [
      "0.632767169",
      "0.290378814"
    ],
    [
      "0.209175598",
      "0.499809829"
    ],
    [
      "0.094683326",
      "0.882261005"
    ],
    [
      "0.398134533",
      "0.810337061"
    ],
    [
      "0.000000009",
      "0.482786709"
    ],
    [
      "0.060187641",
      "0.281735316"
    ],
    [
      "0.567976460",
      "0.017023130"
    ],
    [
      "0.781626944",
      "0.652839891"
    ],
    [
      "0.990802533",
      "0.669863011"
    ],
    [
      "0.871472284",
      "0.842502825"
    ],
    [
      "0.487979873",
      "0.999999995"
    ],
    [
      "0.453484188",
      "0.399474306"
    ],
    [
      "0.776788967",
      "0.443028529"
    ],
    [
      "0.752097418",
      "0.117739000"
    ],
    [
      "0.448646211",
      "0.189662944"
    ],
    [
      "0.269363230",
      "0.298758436"
    ],
    [
      "0.393296556",
      "0.600525699"
    ],
    [
      "0.602343963",
      "0.761935383"
    ],
    [
      "0.692189303",
      "0.951598317"
    ],
    [
      "0.896119216",
      "0.270388715"
    ],
    [
      "0.004837986",
      "0.692598071"
    ],
    [
      "0.597505986",
      "0.552124021"
    ],
    [
      "0.179517890",
      "0.109095502"
    ],
    [
      "0.358800871",
      "0.000000010"
    ],
    [
      "0.278804284",
      "0.982976875"
    ]
  ]
}

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": 27
}

当前第一名的答案

{
  "points": [
    [
      "0.985964556",
      "0.460051649"
    ],
    [
      "0.214013575",
      "0.709621191"
    ],
    [
      "0.632767169",
      "0.290378814"
    ],
    [
      "0.209175598",
      "0.499809829"
    ],
    [
      "0.094683326",
      "0.882261005"
    ],
    [
      "0.398134533",
      "0.810337061"
    ],
    [
      "0.000000009",
      "0.482786709"
    ],
    [
      "0.060187641",
      "0.281735316"
    ],
    [
      "0.567976460",
      "0.017023130"
    ],
    [
      "0.781626944",
      "0.652839891"
    ],
    [
      "0.990802533",
      "0.669863011"
    ],
    [
      "0.871472284",
      "0.842502825"
    ],
    [
      "0.487979873",
      "0.999999995"
    ],
    [
      "0.453484188",
      "0.399474306"
    ],
    [
      "0.776788967",
      "0.443028529"
    ],
    [
      "0.752097418",
      "0.117739000"
    ],
    [
      "0.448646211",
      "0.189662944"
    ],
    [
      "0.269363230",
      "0.298758436"
    ],
    [
      "0.393296556",
      "0.600525699"
    ],
    [
      "0.602343963",
      "0.761935383"
    ],
    [
      "0.692189303",
      "0.951598317"
    ],
    [
      "0.896119216",
      "0.270388715"
    ],
    [
      "0.004837986",
      "0.692598071"
    ],
    [
      "0.597505986",
      "0.552124021"
    ],
    [
      "0.179517890",
      "0.109095502"
    ],
    [
      "0.358800871",
      "0.000000010"
    ],
    [
      "0.278804284",
      "0.982976875"
    ]
  ]
}

提交 points，每个坐标写成十进制字符串，例如 "0.5"。距离比值对平移与缩放不变，将点集缩放到单位正方形 [0, 1]² 内提交即可。 · 验证器 v1.0.0

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