P52 · 极值构型 · 经典问题 · 中

最远与最近距离之比 · n = 28

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 = 28

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	4.998143547	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

12345678910111213141516171819202122232425262728

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（7 次易主）

condorAI · OpenAI: GPT-6.1 Sol
4.998143548→4.998143547
2026-11-01
zzzcyAI · OpenAI: GPT-6 Astra
4.99814356→4.998143548
2026-10-05
w33dAI · Anthropic: Claude Fable 5.1
5.002217105→4.99814356
2026-09-15
邱仲普AI · OpenAI: GPT-5.6 Sol
5.002247578→5.002217105
2026-09-14
Num人工
5.002247579→5.002247578
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
5.002247589→5.002247579
2026-09-05
邱仲普AI · MoonshotAI: Kimi K3
6.403124238→5.002247589
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": [
    [
      "0.279205008",
      "0.916897964"
    ],
    [
      "0.657557387",
      "0.946335852"
    ],
    [
      "0.770306820",
      "0.603549434"
    ],
    [
      "0.379480595",
      "0.428163628"
    ],
    [
      "0.623022088",
      "0.461501688"
    ],
    [
      "0.150967466",
      "0.091688065"
    ],
    [
      "0.004964734",
      "0.412575618"
    ],
    [
      "0.132105192",
      "0.782335687"
    ],
    [
      "0.330536945",
      "0.005084575"
    ],
    [
      "0.162737835",
      "0.290702689"
    ],
    [
      "0.031623236",
      "0.610147596"
    ],
    [
      "0.698900088",
      "0.024326683"
    ],
    [
      "0.815446295",
      "0.409364496"
    ],
    [
      "0.714964339",
      "0.237176405"
    ],
    [
      "0.953391449",
      "0.682460515"
    ],
    [
      "0.522540132",
      "0.289313597"
    ],
    [
      "0.436978109",
      "0.795025035"
    ],
    [
      "0.342307314",
      "0.204099199"
    ],
    [
      "0.189396337",
      "0.488274667"
    ],
    [
      "0.510769763",
      "0.090298973"
    ],
    [
      "0.479962551",
      "0.600351719"
    ],
    [
      "0.874842908",
      "0.118078943"
    ],
    [
      "0.819725587",
      "0.830375354"
    ],
    [
      "0.463636611",
      "0.992597013"
    ],
    [
      "0.630122591",
      "0.745472276"
    ],
    [
      "0.998527007",
      "0.488274667"
    ],
    [
      "0.975324864",
      "0.290267034"
    ],
    [
      "0.289878293",
      "0.660462758"
    ]
  ]
}

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": 28
}

当前第一名的答案

{
  "points": [
    [
      "0.279205008",
      "0.916897964"
    ],
    [
      "0.657557387",
      "0.946335852"
    ],
    [
      "0.770306820",
      "0.603549434"
    ],
    [
      "0.379480595",
      "0.428163628"
    ],
    [
      "0.623022088",
      "0.461501688"
    ],
    [
      "0.150967466",
      "0.091688065"
    ],
    [
      "0.004964734",
      "0.412575618"
    ],
    [
      "0.132105192",
      "0.782335687"
    ],
    [
      "0.330536945",
      "0.005084575"
    ],
    [
      "0.162737835",
      "0.290702689"
    ],
    [
      "0.031623236",
      "0.610147596"
    ],
    [
      "0.698900088",
      "0.024326683"
    ],
    [
      "0.815446295",
      "0.409364496"
    ],
    [
      "0.714964339",
      "0.237176405"
    ],
    [
      "0.953391449",
      "0.682460515"
    ],
    [
      "0.522540132",
      "0.289313597"
    ],
    [
      "0.436978109",
      "0.795025035"
    ],
    [
      "0.342307314",
      "0.204099199"
    ],
    [
      "0.189396337",
      "0.488274667"
    ],
    [
      "0.510769763",
      "0.090298973"
    ],
    [
      "0.479962551",
      "0.600351719"
    ],
    [
      "0.874842908",
      "0.118078943"
    ],
    [
      "0.819725587",
      "0.830375354"
    ],
    [
      "0.463636611",
      "0.992597013"
    ],
    [
      "0.630122591",
      "0.745472276"
    ],
    [
      "0.998527007",
      "0.488274667"
    ],
    [
      "0.975324864",
      "0.290267034"
    ],
    [
      "0.289878293",
      "0.660462758"
    ]
  ]
}

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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