P83 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入正方形 · n = 26

把 n 个边长为 1 的等边三角形装入正方形，允许接触但内部不能重叠，最小化容器正方形边长。

子题n = 26

目标最小化容器正方形边长

纪录对比

容器正方形边长 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[3.575, 3.576)	26. Found by Maurizio Morandi	Erich Friedman ↗
本站纪录	3.577046309	zzzcy	纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

容器编辑器与证书将容器统一归一化为单位正方形 [0,1]²；公开得分按文献的单位边长图形规范折算。允许接触边界。
提交提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。
约束恰好 n 个全等图形完全位于容器内，任意两图形内部互不重叠。多边形允许任意角度独立旋转。
精度输入的十进制小数按精确有理数解析，正多边形顶点由精确代数数定义，可行性检验不设浮点容差。
目标容器正方形边长 = 1/(2 sin(π/3) r)，越小越好。页面、排行榜与文献纪录统一使用该单位并自动换算。

放大编辑并提交 ↗

当前展示：本站纪录

已验证构造已验证构造

帮助理解

文献来源与坐标证书

文献目标值按原始发表精度换算，九位小数坐标证书单独存储并与文献目标对照展示。

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zzzcy

解题方式AI · OpenAI: GPT-6 Astra

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（19 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
3.593136397→3.577046309
2026-10-03
NUE_13人工
3.593239689→3.593136397
2026-10-03
hadsajnc人工
3.614045539→3.593239689
2026-10-02
NUE_13AI · MoonshotAI: Kimi K3
3.634602917→3.614045539
2026-10-01
hadsajnc人工
3.636382441→3.634602917
2026-10-01
NUE_13人工
3.659000765→3.636382441
2026-10-01
NUE_13人工
3.841963063→3.659000765
2026-10-01
今天人工
3.849001692→3.841963063
2026-09-19
NUE_13人工
3.997781693→3.849001692
2026-09-19
今天人工
3.998021434→3.997781693
2026-09-19
今天人工
3.998021517→3.998021434
2026-09-19
ALG人工
4.088470518→3.998021517
2026-09-19
SegmaTree人工
4.088470547→4.088470518
2026-09-19
SegmaTree人工
4.088470605→4.088470547
2026-09-19
ALG人工
4.930467623→4.088470605
2026-09-19
ALG人工
5.428651616→4.930467623
2026-09-18
ALG人工
5.428651718→5.428651616
2026-09-18
ALG人工
5.432341284→5.428651718
2026-09-18
CJerry人工
9.52627938→5.432341284
2026-09-17

ANSWER FORMAT

答案怎么写

单位正方形 [0,1]²

提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.268627388",
      "x": "0.301609277",
      "y": "0.919121102"
    },
    {
      "turn": "0.268627388",
      "x": "0.580965685",
      "y": "0.919121102"
    },
    {
      "turn": "0.268627388",
      "x": "0.860322094",
      "y": "0.919121102"
    },
    {
      "turn": "0.000632658",
      "x": "0.080878898",
      "y": "0.860322094"
    },
    {
      "turn": "-0.267271227",
      "x": "0.441287779",
      "y": "0.838242204"
    },
    {
      "turn": "-0.267271227",
      "x": "0.720644188",
      "y": "0.838242204"
    },
    {
      "turn": "-0.576507034",
      "x": "0.161757796",
      "y": "0.720644188"
    },
    {
      "turn": "-0.576507034",
      "x": "0.919121102",
      "y": "0.698390723"
    },
    {
      "turn": "0.228942858",
      "x": "0.375887130",
      "y": "0.670975878"
    },
    {
      "turn": "0.268627388",
      "x": "0.651787488",
      "y": "0.676750757"
    },
    {
      "turn": "-0.307731406",
      "x": "0.509366278",
      "y": "0.580231013"
    },
    {
      "turn": "0.000632658",
      "x": "0.080878898",
      "y": "0.580965685"
    },
    {
      "turn": "0.000632658",
      "x": "0.838242204",
      "y": "0.558712221"
    },
    {
      "turn": "-0.576507034",
      "x": "0.161757796",
      "y": "0.441287779"
    },
    {
      "turn": "-0.576507034",
      "x": "0.919121102",
      "y": "0.419034315"
    },
    {
      "turn": "0.228942858",
      "x": "0.490633722",
      "y": "0.419768987"
    },
    {
      "turn": "-0.307731406",
      "x": "0.624112870",
      "y": "0.329024122"
    },
    {
      "turn": "-0.267271227",
      "x": "0.348212512",
      "y": "0.323249243"
    },
    {
      "turn": "0.000632658",
      "x": "0.080878898",
      "y": "0.301609277"
    },
    {
      "turn": "0.000632658",
      "x": "0.838242204",
      "y": "0.279355812"
    },
    {
      "turn": "-0.576507034",
      "x": "0.919121102",
      "y": "0.139677906"
    },
    {
      "turn": "0.268627388",
      "x": "0.279355812",
      "y": "0.161757796"
    },
    {
      "turn": "0.268627388",
      "x": "0.558712221",
      "y": "0.161757796"
    },
    {
      "turn": "-0.267271227",
      "x": "0.139677906",
      "y": "0.080878898"
    },
    {
      "turn": "-0.267271227",
      "x": "0.419034315",
      "y": "0.080878898"
    },
    {
      "turn": "-0.267271227",
      "x": "0.698390723",
      "y": "0.080878898"
    }
  ],
  "radius": "0.161404192"
}

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": 26,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.268627388",
      "x": "0.301609277",
      "y": "0.919121102"
    },
    {
      "turn": "0.268627388",
      "x": "0.580965685",
      "y": "0.919121102"
    },
    {
      "turn": "0.268627388",
      "x": "0.860322094",
      "y": "0.919121102"
    },
    {
      "turn": "0.000632658",
      "x": "0.080878898",
      "y": "0.860322094"
    },
    {
      "turn": "-0.267271227",
      "x": "0.441287779",
      "y": "0.838242204"
    },
    {
      "turn": "-0.267271227",
      "x": "0.720644188",
      "y": "0.838242204"
    },
    {
      "turn": "-0.576507034",
      "x": "0.161757796",
      "y": "0.720644188"
    },
    {
      "turn": "-0.576507034",
      "x": "0.919121102",
      "y": "0.698390723"
    },
    {
      "turn": "0.228942858",
      "x": "0.375887130",
      "y": "0.670975878"
    },
    {
      "turn": "0.268627388",
      "x": "0.651787488",
      "y": "0.676750757"
    },
    {
      "turn": "-0.307731406",
      "x": "0.509366278",
      "y": "0.580231013"
    },
    {
      "turn": "0.000632658",
      "x": "0.080878898",
      "y": "0.580965685"
    },
    {
      "turn": "0.000632658",
      "x": "0.838242204",
      "y": "0.558712221"
    },
    {
      "turn": "-0.576507034",
      "x": "0.161757796",
      "y": "0.441287779"
    },
    {
      "turn": "-0.576507034",
      "x": "0.919121102",
      "y": "0.419034315"
    },
    {
      "turn": "0.228942858",
      "x": "0.490633722",
      "y": "0.419768987"
    },
    {
      "turn": "-0.307731406",
      "x": "0.624112870",
      "y": "0.329024122"
    },
    {
      "turn": "-0.267271227",
      "x": "0.348212512",
      "y": "0.323249243"
    },
    {
      "turn": "0.000632658",
      "x": "0.080878898",
      "y": "0.301609277"
    },
    {
      "turn": "0.000632658",
      "x": "0.838242204",
      "y": "0.279355812"
    },
    {
      "turn": "-0.576507034",
      "x": "0.919121102",
      "y": "0.139677906"
    },
    {
      "turn": "0.268627388",
      "x": "0.279355812",
      "y": "0.161757796"
    },
    {
      "turn": "0.268627388",
      "x": "0.558712221",
      "y": "0.161757796"
    },
    {
      "turn": "-0.267271227",
      "x": "0.139677906",
      "y": "0.080878898"
    },
    {
      "turn": "-0.267271227",
      "x": "0.419034315",
      "y": "0.080878898"
    },
    {
      "turn": "-0.267271227",
      "x": "0.698390723",
      "y": "0.080878898"
    }
  ],
  "radius": "0.161404192"
}

提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。 · 验证器 v1.0.0

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