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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 = 49

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 = 49
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2.40336169zzzcy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 = 49 的当前纪录构型，2.40336169
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
    2.403361683→2.40336169
  2. zzzcyAI · OpenAI: GPT-6 Astra
    1.878950552→2.403361683
  3. NUE_13人工
    1.826305166→1.878950552
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": [
    [
      "0.081952887",
      "0.047315521",
      "0.047315521"
    ],
    [
      "0.500000000",
      "0.771394361",
      "0.047315521"
    ],
    [
      "0.904310752",
      "0.055246213",
      "0.055246213"
    ],
    [
      "0.478086936",
      "0.301326647",
      "0.033847512"
    ],
    [
      "0.175659336",
      "0.210762748",
      "0.046744073"
    ],
    [
      "0.405303774",
      "0.608518582",
      "0.046744073"
    ],
    [
      "0.266623450",
      "0.370973139",
      "0.045416111"
    ],
    [
      "0.312039562",
      "0.449636152",
      "0.045416111"
    ],
    [
      "0.358302050",
      "0.528524410",
      "0.046036472"
    ],
    [
      "0.221435458",
      "0.291464521",
      "0.046036471"
    ],
    [
      "0.272136867",
      "0.047950882",
      "0.047950882"
    ],
    [
      "0.594541751",
      "0.606372522",
      "0.047950882"
    ],
    [
      "0.683504520",
      "0.055067737",
      "0.055067737"
    ],
    [
      "0.794062200",
      "0.246559257",
      "0.055067737"
    ],
    [
      "0.369184521",
      "0.049103617",
      "0.049103617"
    ],
    [
      "0.642067280",
      "0.521750421",
      "0.049103617"
    ],
    [
      "0.469565761",
      "0.051301685",
      "0.051301685"
    ],
    [
      "0.690354318",
      "0.433718683",
      "0.051301685"
    ],
    [
      "0.574640899",
      "0.053803226",
      "0.053803226"
    ],
    [
      "0.740725489",
      "0.341470173",
      "0.053803226"
    ],
    [
      "0.452632031",
      "0.689637510",
      "0.047172082"
    ],
    [
      "0.129072412",
      "0.129215811",
      "0.047172082"
    ],
    [
      "0.793818326",
      "0.055246213",
      "0.055246213"
    ],
    [
      "0.849064539",
      "0.150935461",
      "0.055246213"
    ],
    [
      "0.176727774",
      "0.047459475",
      "0.047459475"
    ],
    [
      "0.547262776",
      "0.689244924",
      "0.047459475"
    ],
    [
      "0.362048172",
      "0.215253545",
      "0.046498635"
    ],
    [
      "0.494609047",
      "0.444855717",
      "0.046498635"
    ],
    [
      "0.455425875",
      "0.223470114",
      "0.047239869"
    ],
    [
      "0.534182141",
      "0.359879969",
      "0.047239869"
    ],
    [
      "0.444434634",
      "0.370572276",
      "0.043142321"
    ],
    [
      "0.401292312",
      "0.295847582",
      "0.043142321"
    ],
    [
      "0.682836702",
      "0.247970041",
      "0.056166706"
    ],
    [
      "0.626669996",
      "0.150686451",
      "0.056166707"
    ],
    [
      "0.633719337",
      "0.345787481",
      "0.053289984"
    ],
    [
      "0.517398925",
      "0.144314618",
      "0.053289984"
    ],
    [
      "0.356667449",
      "0.371428221",
      "0.044629037"
    ],
    [
      "0.545248948",
      "0.524343443",
      "0.047749432"
    ],
    [
      "0.318529732",
      "0.131654241",
      "0.047749432"
    ],
    [
      "0.269001635",
      "0.211965578",
      "0.046605975"
    ],
    [
      "0.450933242",
      "0.527080364",
      "0.046605975"
    ],
    [
      "0.402820654",
      "0.448691980",
      "0.045369889"
    ],
    [
      "0.312831673",
      "0.292826493",
      "0.045369890"
    ],
    [
      "0.590027197",
      "0.438426823",
      "0.049135846"
    ],
    [
      "0.415324832",
      "0.135833450",
      "0.049135846"
    ],
    [
      "0.499320324",
      "0.607534212",
      "0.047277630"
    ],
    [
      "0.223520100",
      "0.129834212",
      "0.047277630"
    ],
    [
      "0.738327218",
      "0.151076851",
      "0.055491197"
    ],
    [
      "0.562791565",
      "0.252422407",
      "0.063960968"
    ]
  ]
}
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": 49
}

当前第一名的答案

{
  "circles": [
    [
      "0.081952887",
      "0.047315521",
      "0.047315521"
    ],
    [
      "0.500000000",
      "0.771394361",
      "0.047315521"
    ],
    [
      "0.904310752",
      "0.055246213",
      "0.055246213"
    ],
    [
      "0.478086936",
      "0.301326647",
      "0.033847512"
    ],
    [
      "0.175659336",
      "0.210762748",
      "0.046744073"
    ],
    [
      "0.405303774",
      "0.608518582",
      "0.046744073"
    ],
    [
      "0.266623450",
      "0.370973139",
      "0.045416111"
    ],
    [
      "0.312039562",
      "0.449636152",
      "0.045416111"
    ],
    [
      "0.358302050",
      "0.528524410",
      "0.046036472"
    ],
    [
      "0.221435458",
      "0.291464521",
      "0.046036471"
    ],
    [
      "0.272136867",
      "0.047950882",
      "0.047950882"
    ],
    [
      "0.594541751",
      "0.606372522",
      "0.047950882"
    ],
    [
      "0.683504520",
      "0.055067737",
      "0.055067737"
    ],
    [
      "0.794062200",
      "0.246559257",
      "0.055067737"
    ],
    [
      "0.369184521",
      "0.049103617",
      "0.049103617"
    ],
    [
      "0.642067280",
      "0.521750421",
      "0.049103617"
    ],
    [
      "0.469565761",
      "0.051301685",
      "0.051301685"
    ],
    [
      "0.690354318",
      "0.433718683",
      "0.051301685"
    ],
    [
      "0.574640899",
      "0.053803226",
      "0.053803226"
    ],
    [
      "0.740725489",
      "0.341470173",
      "0.053803226"
    ],
    [
      "0.452632031",
      "0.689637510",
      "0.047172082"
    ],
    [
      "0.129072412",
      "0.129215811",
      "0.047172082"
    ],
    [
      "0.793818326",
      "0.055246213",
      "0.055246213"
    ],
    [
      "0.849064539",
      "0.150935461",
      "0.055246213"
    ],
    [
      "0.176727774",
      "0.047459475",
      "0.047459475"
    ],
    [
      "0.547262776",
      "0.689244924",
      "0.047459475"
    ],
    [
      "0.362048172",
      "0.215253545",
      "0.046498635"
    ],
    [
      "0.494609047",
      "0.444855717",
      "0.046498635"
    ],
    [
      "0.455425875",
      "0.223470114",
      "0.047239869"
    ],
    [
      "0.534182141",
      "0.359879969",
      "0.047239869"
    ],
    [
      "0.444434634",
      "0.370572276",
      "0.043142321"
    ],
    [
      "0.401292312",
      "0.295847582",
      "0.043142321"
    ],
    [
      "0.682836702",
      "0.247970041",
      "0.056166706"
    ],
    [
      "0.626669996",
      "0.150686451",
      "0.056166707"
    ],
    [
      "0.633719337",
      "0.345787481",
      "0.053289984"
    ],
    [
      "0.517398925",
      "0.144314618",
      "0.053289984"
    ],
    [
      "0.356667449",
      "0.371428221",
      "0.044629037"
    ],
    [
      "0.545248948",
      "0.524343443",
      "0.047749432"
    ],
    [
      "0.318529732",
      "0.131654241",
      "0.047749432"
    ],
    [
      "0.269001635",
      "0.211965578",
      "0.046605975"
    ],
    [
      "0.450933242",
      "0.527080364",
      "0.046605975"
    ],
    [
      "0.402820654",
      "0.448691980",
      "0.045369889"
    ],
    [
      "0.312831673",
      "0.292826493",
      "0.045369890"
    ],
    [
      "0.590027197",
      "0.438426823",
      "0.049135846"
    ],
    [
      "0.415324832",
      "0.135833450",
      "0.049135846"
    ],
    [
      "0.499320324",
      "0.607534212",
      "0.047277630"
    ],
    [
      "0.223520100",
      "0.129834212",
      "0.047277630"
    ],
    [
      "0.738327218",
      "0.151076851",
      "0.055491197"
    ],
    [
      "0.562791565",
      "0.252422407",
      "0.063960968"
    ]
  ]
}

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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