极值竞技场
中/EN
P93 · 装箱与覆盖 · 经典问题 · 难

正三角形内圆的半径之和 · n = 46

在边长为 1 的正三角形内放置 n 个互不重叠的圆，每个圆的半径各自随意，使所有半径之和尽可能大。

子题n = 46
目标最大化 半径之和

纪录对比

半径之和 · 越大越好
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2.332746296zzzcy纪录详情 ↓

严格定义

  • 容器顶点 (0, 0)、(1, 0)、(1/2, √3/2) 的闭正三角形；圆可以与边相切
  • 提交恰好 n 个圆，每个是 [x, y, r]，r > 0，全部数字最多九位小数
  • 约束每个圆完整落在三角形内：y ≥ r，√3·x − y ≥ 2r，√3·(1 − x) − y ≥ 2r；任意两圆内部不相交：圆心距 ≥ 半径之和。斜边条件化为整数平方比较，全部精确判定，相切允许
  • 分数半径之和，精确整数求和后按九位小数显示；越大越好S=Σr
  • 目标在所有合法放置中最大化半径之和maxSΣr
放大来摆，然后提交 ↗

当前展示：本站纪录

0.866025404y0
0x1
已验证构造计分是全部半径之和

帮助理解

哪里有优化空间

等圆三角阵是最差的策略之一：一个大圆坐在中间，几个中等的圆塞进三个角，再让小圆钻进它们和三条边之间的缝隙，半径之和才会上去。角落既是浪费也是机会：一个刚好卡住角的小圆，几乎不占别人的位置。

前沿在哪里

Erich Friedman 的“最大化总周长”表列到 n = 30。其中 26 行来自 David W. Cantrell 2011 年的搜索，只公布到三位小数；n = 27、29、30 是 Haowei Lin 与 Jonathan Viquerat 2026 年的新行。除 n = 1 的内切圆外，没有一行有最优性证明；本站展示按图重建并精确验算的构型（n = 29 只引用数值）。n = 31–50 超出该表，目前没有公布的纪录，是完全开放的子题。

查看来源
正三角形内圆的半径之和 n = 46 的当前纪录构型，2.332746296
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 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。
纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

历史纪录（3 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    2.332746288→2.332746296
  2. zzzcyAI · OpenAI: GPT-6 Astra
    1.800669652→2.332746288
  3. NUE_13人工
    1.714490564→1.800669652
ANSWER FORMAT

答案怎么写

容器是边长 1 的正三角形：底边从 (0, 0) 到 (1, 0)，顶点在 (1/2, √3/2)。坐标与半径共用一套单位，直接写成小数，最多九位。

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成最多九位小数的十进制字符串，例如 "0.25"。容器是底边从 (0, 0) 到 (1, 0)、顶点在 (1/2, √3/2) 的正三角形。

当前第一名的答案

{
  "circles": [
    [
      "0.500000000",
      "0.762006203",
      "0.052009600"
    ],
    [
      "0.552012893",
      "0.671935216",
      "0.052000607"
    ],
    [
      "0.604025964",
      "0.581914054",
      "0.051966547"
    ],
    [
      "0.656027203",
      "0.492034944",
      "0.051871708"
    ],
    [
      "0.707978802",
      "0.402489296",
      "0.051653127"
    ],
    [
      "0.759769549",
      "0.313538944",
      "0.051276201"
    ],
    [
      "0.811206603",
      "0.225328869",
      "0.050835443"
    ],
    [
      "0.862159709",
      "0.137719817",
      "0.050513285"
    ],
    [
      "0.912717045",
      "0.050392837",
      "0.050392837"
    ],
    [
      "0.447993692",
      "0.671910644",
      "0.052018596"
    ],
    [
      "0.500042564",
      "0.581831255",
      "0.052016885"
    ],
    [
      "0.552156319",
      "0.491811549",
      "0.051999416"
    ],
    [
      "0.604412855",
      "0.401993657",
      "0.051914005"
    ],
    [
      "0.656884555",
      "0.312750643",
      "0.051611811"
    ],
    [
      "0.709340369",
      "0.224520549",
      "0.051033997"
    ],
    [
      "0.761167009",
      "0.137232459",
      "0.050480589"
    ],
    [
      "0.812051550",
      "0.050272729",
      "0.050272729"
    ],
    [
      "0.395983672",
      "0.581779799",
      "0.052042018"
    ],
    [
      "0.448082849",
      "0.491636231",
      "0.052074201"
    ],
    [
      "0.500346510",
      "0.401458807",
      "0.052153713"
    ],
    [
      "0.553083927",
      "0.311414805",
      "0.052197412"
    ],
    [
      "0.606720498",
      "0.222543886",
      "0.051604909"
    ],
    [
      "0.660287741",
      "0.135724641",
      "0.050409946"
    ],
    [
      "0.711952309",
      "0.049827502",
      "0.049827502"
    ],
    [
      "0.343948346",
      "0.491615402",
      "0.052060304"
    ],
    [
      "0.396019750",
      "0.401320444",
      "0.052173137"
    ],
    [
      "0.448323073",
      "0.310574539",
      "0.052566811"
    ],
    [
      "0.501813033",
      "0.219148154",
      "0.053357499"
    ],
    [
      "0.558882728",
      "0.131688016",
      "0.051075378"
    ],
    [
      "0.613628097",
      "0.048505595",
      "0.048505595"
    ],
    [
      "0.291860857",
      "0.401545836",
      "0.051985998"
    ],
    [
      "0.343557452",
      "0.311090318",
      "0.052200079"
    ],
    [
      "0.395118104",
      "0.219005115",
      "0.053337523"
    ],
    [
      "0.448578343",
      "0.119135079",
      "0.059940988"
    ],
    [
      "0.520985396",
      "0.044235464",
      "0.044235464"
    ],
    [
      "0.239741264",
      "0.312003794",
      "0.051620127"
    ],
    [
      "0.290284158",
      "0.222072199",
      "0.051541279"
    ],
    [
      "0.338323078",
      "0.131476606",
      "0.051002854"
    ],
    [
      "0.376291479",
      "0.044186908",
      "0.044186908"
    ],
    [
      "0.187857551",
      "0.223585802",
      "0.050896510"
    ],
    [
      "0.237137954",
      "0.135253412",
      "0.050252731"
    ],
    [
      "0.283824262",
      "0.048375109",
      "0.048375109"
    ],
    [
      "0.136703166",
      "0.136399634",
      "0.050188597"
    ],
    [
      "0.185872576",
      "0.049584038",
      "0.049584037"
    ],
    [
      "0.086403760",
      "0.049885234",
      "0.049885234"
    ],
    [
      "0.448618573",
      "0.029597050",
      "0.029597050"
    ]
  ]
}
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": 46
}

当前第一名的答案

{
  "circles": [
    [
      "0.500000000",
      "0.762006203",
      "0.052009600"
    ],
    [
      "0.552012893",
      "0.671935216",
      "0.052000607"
    ],
    [
      "0.604025964",
      "0.581914054",
      "0.051966547"
    ],
    [
      "0.656027203",
      "0.492034944",
      "0.051871708"
    ],
    [
      "0.707978802",
      "0.402489296",
      "0.051653127"
    ],
    [
      "0.759769549",
      "0.313538944",
      "0.051276201"
    ],
    [
      "0.811206603",
      "0.225328869",
      "0.050835443"
    ],
    [
      "0.862159709",
      "0.137719817",
      "0.050513285"
    ],
    [
      "0.912717045",
      "0.050392837",
      "0.050392837"
    ],
    [
      "0.447993692",
      "0.671910644",
      "0.052018596"
    ],
    [
      "0.500042564",
      "0.581831255",
      "0.052016885"
    ],
    [
      "0.552156319",
      "0.491811549",
      "0.051999416"
    ],
    [
      "0.604412855",
      "0.401993657",
      "0.051914005"
    ],
    [
      "0.656884555",
      "0.312750643",
      "0.051611811"
    ],
    [
      "0.709340369",
      "0.224520549",
      "0.051033997"
    ],
    [
      "0.761167009",
      "0.137232459",
      "0.050480589"
    ],
    [
      "0.812051550",
      "0.050272729",
      "0.050272729"
    ],
    [
      "0.395983672",
      "0.581779799",
      "0.052042018"
    ],
    [
      "0.448082849",
      "0.491636231",
      "0.052074201"
    ],
    [
      "0.500346510",
      "0.401458807",
      "0.052153713"
    ],
    [
      "0.553083927",
      "0.311414805",
      "0.052197412"
    ],
    [
      "0.606720498",
      "0.222543886",
      "0.051604909"
    ],
    [
      "0.660287741",
      "0.135724641",
      "0.050409946"
    ],
    [
      "0.711952309",
      "0.049827502",
      "0.049827502"
    ],
    [
      "0.343948346",
      "0.491615402",
      "0.052060304"
    ],
    [
      "0.396019750",
      "0.401320444",
      "0.052173137"
    ],
    [
      "0.448323073",
      "0.310574539",
      "0.052566811"
    ],
    [
      "0.501813033",
      "0.219148154",
      "0.053357499"
    ],
    [
      "0.558882728",
      "0.131688016",
      "0.051075378"
    ],
    [
      "0.613628097",
      "0.048505595",
      "0.048505595"
    ],
    [
      "0.291860857",
      "0.401545836",
      "0.051985998"
    ],
    [
      "0.343557452",
      "0.311090318",
      "0.052200079"
    ],
    [
      "0.395118104",
      "0.219005115",
      "0.053337523"
    ],
    [
      "0.448578343",
      "0.119135079",
      "0.059940988"
    ],
    [
      "0.520985396",
      "0.044235464",
      "0.044235464"
    ],
    [
      "0.239741264",
      "0.312003794",
      "0.051620127"
    ],
    [
      "0.290284158",
      "0.222072199",
      "0.051541279"
    ],
    [
      "0.338323078",
      "0.131476606",
      "0.051002854"
    ],
    [
      "0.376291479",
      "0.044186908",
      "0.044186908"
    ],
    [
      "0.187857551",
      "0.223585802",
      "0.050896510"
    ],
    [
      "0.237137954",
      "0.135253412",
      "0.050252731"
    ],
    [
      "0.283824262",
      "0.048375109",
      "0.048375109"
    ],
    [
      "0.136703166",
      "0.136399634",
      "0.050188597"
    ],
    [
      "0.185872576",
      "0.049584038",
      "0.049584037"
    ],
    [
      "0.086403760",
      "0.049885234",
      "0.049885234"
    ],
    [
      "0.448618573",
      "0.029597050",
      "0.029597050"
    ]
  ]
}

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成最多九位小数的十进制字符串，例如 "0.25"。容器是底边从 (0, 0) 到 (1, 0)、顶点在 (1/2, √3/2) 的正三角形。 · 验证器 v1.0.0

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

ai证明zzzcy#3083 条回复对本周挑战的评价*2NUE_13#312 条回复