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P93 · 装箱与覆盖 · 经典问题 · 难

正三角形内圆的半径之和 · n = 43

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 = 43
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2.247076374zzzcy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 = 43 的当前纪录构型，2.247076374
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（2 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    1.789644351→2.247076374
  2. NUE_13人工
    1.767356158→1.789644351
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": [
    [
      "0.096228095",
      "0.055557317",
      "0.055557316"
    ],
    [
      "0.207784692",
      "0.056000159",
      "0.056000159"
    ],
    [
      "0.321272836",
      "0.057497867",
      "0.057497867"
    ],
    [
      "0.439364514",
      "0.060635486",
      "0.060635485"
    ],
    [
      "0.560635486",
      "0.060635486",
      "0.060635485"
    ],
    [
      "0.678727164",
      "0.057497867",
      "0.057497867"
    ],
    [
      "0.792215308",
      "0.056000159",
      "0.056000159"
    ],
    [
      "0.903771905",
      "0.055557317",
      "0.055557316"
    ],
    [
      "0.151185694",
      "0.151624082",
      "0.055118610"
    ],
    [
      "0.261770289",
      "0.153539665",
      "0.055482574"
    ],
    [
      "0.374558174",
      "0.159354942",
      "0.057455127"
    ],
    [
      "0.500000000",
      "0.175196859",
      "0.068983071"
    ],
    [
      "0.625441826",
      "0.159354942",
      "0.057455127"
    ],
    [
      "0.738229711",
      "0.153539665",
      "0.055482574"
    ],
    [
      "0.848814306",
      "0.151624082",
      "0.055118610"
    ],
    [
      "0.204490315",
      "0.246600359",
      "0.053793627"
    ],
    [
      "0.311412052",
      "0.250210309",
      "0.053189032"
    ],
    [
      "0.415092004",
      "0.259887806",
      "0.050941589"
    ],
    [
      "0.500000000",
      "0.280647287",
      "0.036467356"
    ],
    [
      "0.584907996",
      "0.259887806",
      "0.050941589"
    ],
    [
      "0.688587948",
      "0.250210309",
      "0.053189032"
    ],
    [
      "0.795509685",
      "0.246600359",
      "0.053793627"
    ],
    [
      "0.255487258",
      "0.339009851",
      "0.051753529"
    ],
    [
      "0.357279364",
      "0.342765327",
      "0.050107828"
    ],
    [
      "0.453582344",
      "0.349315522",
      "0.046417655"
    ],
    [
      "0.546417656",
      "0.349315522",
      "0.046417655"
    ],
    [
      "0.642720636",
      "0.342765327",
      "0.050107828"
    ],
    [
      "0.744512742",
      "0.339009851",
      "0.051753529"
    ],
    [
      "0.304801337",
      "0.427984543",
      "0.049973429"
    ],
    [
      "0.403427478",
      "0.430106642",
      "0.048675539"
    ],
    [
      "0.500000000",
      "0.431427371",
      "0.047906013"
    ],
    [
      "0.596572522",
      "0.430106642",
      "0.048675539"
    ],
    [
      "0.695198663",
      "0.427984543",
      "0.049973429"
    ],
    [
      "0.353577013",
      "0.514217293",
      "0.049098028"
    ],
    [
      "0.451336532",
      "0.514839191",
      "0.048663468"
    ],
    [
      "0.548663468",
      "0.514839191",
      "0.048663468"
    ],
    [
      "0.646422987",
      "0.514217293",
      "0.049098028"
    ],
    [
      "0.402352803",
      "0.599171022",
      "0.048862237"
    ],
    [
      "0.500000000",
      "0.599267102",
      "0.048785006"
    ],
    [
      "0.597647197",
      "0.599171022",
      "0.048862237"
    ],
    [
      "0.451170423",
      "0.683790941",
      "0.048829577"
    ],
    [
      "0.548829577",
      "0.683790941",
      "0.048829577"
    ],
    [
      "0.500000000",
      "0.768366250",
      "0.048829576"
    ]
  ]
}
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": 43
}

当前第一名的答案

{
  "circles": [
    [
      "0.096228095",
      "0.055557317",
      "0.055557316"
    ],
    [
      "0.207784692",
      "0.056000159",
      "0.056000159"
    ],
    [
      "0.321272836",
      "0.057497867",
      "0.057497867"
    ],
    [
      "0.439364514",
      "0.060635486",
      "0.060635485"
    ],
    [
      "0.560635486",
      "0.060635486",
      "0.060635485"
    ],
    [
      "0.678727164",
      "0.057497867",
      "0.057497867"
    ],
    [
      "0.792215308",
      "0.056000159",
      "0.056000159"
    ],
    [
      "0.903771905",
      "0.055557317",
      "0.055557316"
    ],
    [
      "0.151185694",
      "0.151624082",
      "0.055118610"
    ],
    [
      "0.261770289",
      "0.153539665",
      "0.055482574"
    ],
    [
      "0.374558174",
      "0.159354942",
      "0.057455127"
    ],
    [
      "0.500000000",
      "0.175196859",
      "0.068983071"
    ],
    [
      "0.625441826",
      "0.159354942",
      "0.057455127"
    ],
    [
      "0.738229711",
      "0.153539665",
      "0.055482574"
    ],
    [
      "0.848814306",
      "0.151624082",
      "0.055118610"
    ],
    [
      "0.204490315",
      "0.246600359",
      "0.053793627"
    ],
    [
      "0.311412052",
      "0.250210309",
      "0.053189032"
    ],
    [
      "0.415092004",
      "0.259887806",
      "0.050941589"
    ],
    [
      "0.500000000",
      "0.280647287",
      "0.036467356"
    ],
    [
      "0.584907996",
      "0.259887806",
      "0.050941589"
    ],
    [
      "0.688587948",
      "0.250210309",
      "0.053189032"
    ],
    [
      "0.795509685",
      "0.246600359",
      "0.053793627"
    ],
    [
      "0.255487258",
      "0.339009851",
      "0.051753529"
    ],
    [
      "0.357279364",
      "0.342765327",
      "0.050107828"
    ],
    [
      "0.453582344",
      "0.349315522",
      "0.046417655"
    ],
    [
      "0.546417656",
      "0.349315522",
      "0.046417655"
    ],
    [
      "0.642720636",
      "0.342765327",
      "0.050107828"
    ],
    [
      "0.744512742",
      "0.339009851",
      "0.051753529"
    ],
    [
      "0.304801337",
      "0.427984543",
      "0.049973429"
    ],
    [
      "0.403427478",
      "0.430106642",
      "0.048675539"
    ],
    [
      "0.500000000",
      "0.431427371",
      "0.047906013"
    ],
    [
      "0.596572522",
      "0.430106642",
      "0.048675539"
    ],
    [
      "0.695198663",
      "0.427984543",
      "0.049973429"
    ],
    [
      "0.353577013",
      "0.514217293",
      "0.049098028"
    ],
    [
      "0.451336532",
      "0.514839191",
      "0.048663468"
    ],
    [
      "0.548663468",
      "0.514839191",
      "0.048663468"
    ],
    [
      "0.646422987",
      "0.514217293",
      "0.049098028"
    ],
    [
      "0.402352803",
      "0.599171022",
      "0.048862237"
    ],
    [
      "0.500000000",
      "0.599267102",
      "0.048785006"
    ],
    [
      "0.597647197",
      "0.599171022",
      "0.048862237"
    ],
    [
      "0.451170423",
      "0.683790941",
      "0.048829577"
    ],
    [
      "0.548829577",
      "0.683790941",
      "0.048829577"
    ],
    [
      "0.500000000",
      "0.768366250",
      "0.048829576"
    ]
  ]
}

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

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