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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 = 29

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 = 29
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[1.83673, 1.83674)见来源署名Erich Friedman ↗
本站纪录1.83671165zzzcy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 = 29 的当前纪录构型，1.83671165
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（5 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    1.624026196→1.83671165
  2. NUE_13人工
    1.606018151→1.624026196
  3. 今天人工
    1.592136851→1.606018151
  4. NUE_13人工
    1.400133372→1.592136851
  5. NUE_13人工
    1.328439337→1.400133372
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",
      "0.740343825",
      "0.062840789"
    ],
    [
      "0.437016863",
      "0.632033290",
      "0.062451060"
    ],
    [
      "0.562696779",
      "0.630964600",
      "0.063233398"
    ],
    [
      "0.374001455",
      "0.525997840",
      "0.060895841"
    ],
    [
      "0.498141604",
      "0.522132330",
      "0.063304475"
    ],
    [
      "0.625589817",
      "0.520180081",
      "0.064158689"
    ],
    [
      "0.311214454",
      "0.427914919",
      "0.055562163"
    ],
    [
      "0.430200634",
      "0.414226543",
      "0.064208795"
    ],
    [
      "0.559290707",
      "0.409367906",
      "0.064972678"
    ],
    [
      "0.689388949",
      "0.407722176",
      "0.065135972"
    ],
    [
      "0.249838101",
      "0.358418190",
      "0.037157046"
    ],
    [
      "0.347190784",
      "0.302063893",
      "0.075330071"
    ],
    [
      "0.489645640",
      "0.297016080",
      "0.067214189"
    ],
    [
      "0.622737986",
      "0.294909411",
      "0.065894828"
    ],
    [
      "0.754321293",
      "0.294146623",
      "0.065690689"
    ],
    [
      "0.220269898",
      "0.270702726",
      "0.055407964"
    ],
    [
      "0.422099768",
      "0.180817466",
      "0.067190285"
    ],
    [
      "0.555687980",
      "0.180866997",
      "0.066397936"
    ],
    [
      "0.688072182",
      "0.180347926",
      "0.065987282"
    ],
    [
      "0.819941715",
      "0.180105149",
      "0.065882474"
    ],
    [
      "0.291059415",
      "0.174540454",
      "0.064000319"
    ],
    [
      "0.166768759",
      "0.167918765",
      "0.060466599"
    ],
    [
      "0.489724277",
      "0.066033210",
      "0.066033210"
    ],
    [
      "0.621841362",
      "0.066083884",
      "0.066083883"
    ],
    [
      "0.753901450",
      "0.065976248",
      "0.065976248"
    ],
    [
      "0.885807031",
      "0.065929342",
      "0.065929341"
    ],
    [
      "0.358805869",
      "0.064890188",
      "0.064890187"
    ],
    [
      "0.106726537",
      "0.061618595",
      "0.061618594"
    ],
    [
      "0.231136200",
      "0.062796646",
      "0.062796645"
    ]
  ]
}
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": 29
}

当前第一名的答案

{
  "circles": [
    [
      "0.500000000",
      "0.740343825",
      "0.062840789"
    ],
    [
      "0.437016863",
      "0.632033290",
      "0.062451060"
    ],
    [
      "0.562696779",
      "0.630964600",
      "0.063233398"
    ],
    [
      "0.374001455",
      "0.525997840",
      "0.060895841"
    ],
    [
      "0.498141604",
      "0.522132330",
      "0.063304475"
    ],
    [
      "0.625589817",
      "0.520180081",
      "0.064158689"
    ],
    [
      "0.311214454",
      "0.427914919",
      "0.055562163"
    ],
    [
      "0.430200634",
      "0.414226543",
      "0.064208795"
    ],
    [
      "0.559290707",
      "0.409367906",
      "0.064972678"
    ],
    [
      "0.689388949",
      "0.407722176",
      "0.065135972"
    ],
    [
      "0.249838101",
      "0.358418190",
      "0.037157046"
    ],
    [
      "0.347190784",
      "0.302063893",
      "0.075330071"
    ],
    [
      "0.489645640",
      "0.297016080",
      "0.067214189"
    ],
    [
      "0.622737986",
      "0.294909411",
      "0.065894828"
    ],
    [
      "0.754321293",
      "0.294146623",
      "0.065690689"
    ],
    [
      "0.220269898",
      "0.270702726",
      "0.055407964"
    ],
    [
      "0.422099768",
      "0.180817466",
      "0.067190285"
    ],
    [
      "0.555687980",
      "0.180866997",
      "0.066397936"
    ],
    [
      "0.688072182",
      "0.180347926",
      "0.065987282"
    ],
    [
      "0.819941715",
      "0.180105149",
      "0.065882474"
    ],
    [
      "0.291059415",
      "0.174540454",
      "0.064000319"
    ],
    [
      "0.166768759",
      "0.167918765",
      "0.060466599"
    ],
    [
      "0.489724277",
      "0.066033210",
      "0.066033210"
    ],
    [
      "0.621841362",
      "0.066083884",
      "0.066083883"
    ],
    [
      "0.753901450",
      "0.065976248",
      "0.065976248"
    ],
    [
      "0.885807031",
      "0.065929342",
      "0.065929341"
    ],
    [
      "0.358805869",
      "0.064890188",
      "0.064890187"
    ],
    [
      "0.106726537",
      "0.061618595",
      "0.061618594"
    ],
    [
      "0.231136200",
      "0.062796646",
      "0.062796645"
    ]
  ]
}

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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