P94 · 装箱与覆盖 · 经典问题 · 难

正五边形内圆的半径之和 · n = 49

在边长为 1 的正五边形内放置 n 个互不重叠的圆，每个圆的半径可独立选取，使所有圆的半径之和最大。

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	[4.81492, 4.81493)	Joost de Winter2026-10	Erich Friedman ↗
本站纪录	4.81492274	zzzcy	纪录详情 ↓

在公布精度内匹配已知最好；不代表精确相等或已证明最优。

严格定义

容器边长为 1 的闭正五边形，顶点 (t, 0)、(1 + t, 0)、(φ, sin 72°)、(φ/2, √(5 + 2√5)/2)、(0, sin 72°)，t = (√5 − 1)/4；允许圆与边相切
提交恰好 n 个圆 [x, y, r]，其中 r > 0，所有数值均为最多九位小数的十进制数
约束每个圆完整落在五边形内：圆心到五条边所在直线的距离都不小于 r；任意两圆内部不相交：圆心距 ≥ 半径之和。边的方程含 √5 与 √(10 + 2√5)，在该代数数域中做精确符号判定，允许相切
分数所有圆的半径之和，按精确整数求和并以九位小数显示；越大越好 $S = Σ r$
目标在所有合法放置中最大化半径之和 $\max_{S} Σ r$

放大编辑并提交 ↗

当前展示：本站纪录

1.538841769y0

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849

0x1.618033989

已验证构造计分是全部半径之和

帮助理解

内切圆与五个 108° 角

一个圆最多是内切圆，半径 cot(π/5)/2 ≈ 0.688；再多放圆时，中心大圆周围的五个钝角和五条边旁的弧形空隙都能塞进递减的小圆。108° 的角比正三角形的 60° 角宽，角里的圆链收缩得慢，所以大小混合的构型和等圆阵列的差距随 n 增大。

文献纪录

Erich Friedman 的公开表收录 n = 1–50：n = 1 是内切圆（平凡），其余大多由 Jonathan Viquerat 在 2026 年 6 月与 10 月给出，Joost de Winter、Richard Wang 与 Aaden Abraham 在 2026 年 9–10 月刷新了其中若干行，n = 5、6 带闭式。本站只引用各行数值与作者，没有复原其构型；每档的起步是本站的普通等圆环形布局，远低于文献值，不是外部纪录。

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（10 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
4.814922737→4.81492274
2026-10-19
qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
4.814922736→4.814922737
2026-10-19
qiboqiAI · DeepSeek: DeepSeek V4.1 Flash 1017
4.814922735→4.814922736
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
4.814922715→4.814922735
2026-10-18
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
4.814922625→4.814922715
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
4.814378387→4.814922625
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
4.813959789→4.814378387
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
4.805856443→4.813959789
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
4.79060238→4.805856443
2026-10-18
qiboqiAI · OpenAI: GPT-6 Luna
1.624354998→4.79060238
2026-10-18

ANSWER FORMAT

答案怎么写

容器是边长为 1 的正五边形，底边水平、顶点朝上：顶点依次为 (t, 0)、(1 + t, 0)、(φ, sin 72°)、(φ/2, h)、(0, sin 72°)，其中 t = (√5 − 1)/4，φ = (1 + √5)/2，h = √(5 + 2√5)/2 ≈ 1.538842。坐标与半径采用同一长度单位，以最多九位小数的十进制数表示。

提交 circles：恰好 n 个三元组 [x, y, r]，每个数均为最多九位小数的十进制字符串（例如 "0.25"）。容器为边长 1、底边水平、顶点朝上的正五边形：底边从 (t, 0) 到 (1 + t, 0)，其中 t = cos 72° = (√5 − 1)/4，最左顶点在 (0, sin 72°)。

当前第一名的答案

{
  "circles": [
    [
      "1.303082285",
      "0.315507315",
      "0.103141366"
    ],
    [
      "1.129065916",
      "0.969750312",
      "0.097249784"
    ],
    [
      "1.010486292",
      "0.103427375",
      "0.103427375"
    ],
    [
      "0.986841283",
      "1.283214405",
      "0.102284387"
    ],
    [
      "1.368579518",
      "0.508225957",
      "0.100403131"
    ],
    [
      "0.988865537",
      "0.442670748",
      "0.091858898"
    ],
    [
      "1.055684498",
      "0.609598209",
      "0.087945303"
    ],
    [
      "0.437690309",
      "0.458066944",
      "0.092086435"
    ],
    [
      "0.250999043",
      "0.499195250",
      "0.099081465"
    ],
    [
      "0.804431980",
      "0.436216833",
      "0.092687545"
    ],
    [
      "1.482637255",
      "0.911376679",
      "0.111685866"
    ],
    [
      "0.307243584",
      "1.045074553",
      "0.104531058"
    ],
    [
      "0.657248741",
      "0.930798241",
      "0.070212290"
    ],
    [
      "0.603466623",
      "0.102467168",
      "0.102467168"
    ],
    [
      "1.101300263",
      "0.286291005",
      "0.100744818"
    ],
    [
      "0.972235638",
      "1.084429593",
      "0.097036275"
    ],
    [
      "0.389881540",
      "0.111300498",
      "0.111300498"
    ],
    [
      "0.806501913",
      "0.100576919",
      "0.100576919"
    ],
    [
      "0.135886693",
      "0.906904253",
      "0.115592125"
    ],
    [
      "0.346363240",
      "0.838490984",
      "0.105723830"
    ],
    [
      "1.175466386",
      "0.469186815",
      "0.096616503"
    ],
    [
      "0.809016995",
      "1.401371065",
      "0.111216135"
    ],
    [
      "0.903306653",
      "0.273567130",
      "0.097657212"
    ],
    [
      "0.496092602",
      "0.975006634",
      "0.096897502"
    ],
    [
      "0.559723193",
      "0.769069369",
      "0.118646018"
    ],
    [
      "1.281577770",
      "0.844177662",
      "0.100306172"
    ],
    [
      "1.226824513",
      "0.113128234",
      "0.113128234"
    ],
    [
      "1.314556978",
      "1.044538554",
      "0.102750754"
    ],
    [
      "1.234999447",
      "0.652930733",
      "0.096531141"
    ],
    [
      "1.428638709",
      "0.703079693",
      "0.103496562"
    ],
    [
      "0.193265878",
      "0.694421114",
      "0.104502071"
    ],
    [
      "0.387122458",
      "0.640085075",
      "0.096825469"
    ],
    [
      "0.512439829",
      "0.282202797",
      "0.099004352"
    ],
    [
      "0.713884799",
      "0.609750079",
      "0.103048449"
    ],
    [
      "0.314467162",
      "0.309599879",
      "0.100855041"
    ],
    [
      "1.099782952",
      "0.783412008",
      "0.091375388"
    ],
    [
      "0.941738898",
      "0.727735569",
      "0.076188894"
    ],
    [
      "1.151872146",
      "1.165883210",
      "0.100204610"
    ],
    [
      "0.803203947",
      "1.000675076",
      "0.091607653"
    ],
    [
      "0.954418615",
      "0.895840011",
      "0.092393070"
    ],
    [
      "0.779405235",
      "0.806343459",
      "0.104175788"
    ],
    [
      "0.708425071",
      "0.272355186",
      "0.097228136"
    ],
    [
      "0.620287646",
      "0.439217469",
      "0.091481235"
    ],
    [
      "0.543202047",
      "0.581131129",
      "0.070016989"
    ],
    [
      "0.807861822",
      "1.191191951",
      "0.098966152"
    ],
    [
      "0.644100031",
      "1.092640287",
      "0.092163005"
    ],
    [
      "0.468765624",
      "1.169112500",
      "0.099122524"
    ],
    [
      "0.631562686",
      "1.285250393",
      "0.100854707"
    ],
    [
      "0.892422192",
      "0.582040617",
      "0.077626438"
    ]
  ]
}

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": [
    [
      "1.303082285",
      "0.315507315",
      "0.103141366"
    ],
    [
      "1.129065916",
      "0.969750312",
      "0.097249784"
    ],
    [
      "1.010486292",
      "0.103427375",
      "0.103427375"
    ],
    [
      "0.986841283",
      "1.283214405",
      "0.102284387"
    ],
    [
      "1.368579518",
      "0.508225957",
      "0.100403131"
    ],
    [
      "0.988865537",
      "0.442670748",
      "0.091858898"
    ],
    [
      "1.055684498",
      "0.609598209",
      "0.087945303"
    ],
    [
      "0.437690309",
      "0.458066944",
      "0.092086435"
    ],
    [
      "0.250999043",
      "0.499195250",
      "0.099081465"
    ],
    [
      "0.804431980",
      "0.436216833",
      "0.092687545"
    ],
    [
      "1.482637255",
      "0.911376679",
      "0.111685866"
    ],
    [
      "0.307243584",
      "1.045074553",
      "0.104531058"
    ],
    [
      "0.657248741",
      "0.930798241",
      "0.070212290"
    ],
    [
      "0.603466623",
      "0.102467168",
      "0.102467168"
    ],
    [
      "1.101300263",
      "0.286291005",
      "0.100744818"
    ],
    [
      "0.972235638",
      "1.084429593",
      "0.097036275"
    ],
    [
      "0.389881540",
      "0.111300498",
      "0.111300498"
    ],
    [
      "0.806501913",
      "0.100576919",
      "0.100576919"
    ],
    [
      "0.135886693",
      "0.906904253",
      "0.115592125"
    ],
    [
      "0.346363240",
      "0.838490984",
      "0.105723830"
    ],
    [
      "1.175466386",
      "0.469186815",
      "0.096616503"
    ],
    [
      "0.809016995",
      "1.401371065",
      "0.111216135"
    ],
    [
      "0.903306653",
      "0.273567130",
      "0.097657212"
    ],
    [
      "0.496092602",
      "0.975006634",
      "0.096897502"
    ],
    [
      "0.559723193",
      "0.769069369",
      "0.118646018"
    ],
    [
      "1.281577770",
      "0.844177662",
      "0.100306172"
    ],
    [
      "1.226824513",
      "0.113128234",
      "0.113128234"
    ],
    [
      "1.314556978",
      "1.044538554",
      "0.102750754"
    ],
    [
      "1.234999447",
      "0.652930733",
      "0.096531141"
    ],
    [
      "1.428638709",
      "0.703079693",
      "0.103496562"
    ],
    [
      "0.193265878",
      "0.694421114",
      "0.104502071"
    ],
    [
      "0.387122458",
      "0.640085075",
      "0.096825469"
    ],
    [
      "0.512439829",
      "0.282202797",
      "0.099004352"
    ],
    [
      "0.713884799",
      "0.609750079",
      "0.103048449"
    ],
    [
      "0.314467162",
      "0.309599879",
      "0.100855041"
    ],
    [
      "1.099782952",
      "0.783412008",
      "0.091375388"
    ],
    [
      "0.941738898",
      "0.727735569",
      "0.076188894"
    ],
    [
      "1.151872146",
      "1.165883210",
      "0.100204610"
    ],
    [
      "0.803203947",
      "1.000675076",
      "0.091607653"
    ],
    [
      "0.954418615",
      "0.895840011",
      "0.092393070"
    ],
    [
      "0.779405235",
      "0.806343459",
      "0.104175788"
    ],
    [
      "0.708425071",
      "0.272355186",
      "0.097228136"
    ],
    [
      "0.620287646",
      "0.439217469",
      "0.091481235"
    ],
    [
      "0.543202047",
      "0.581131129",
      "0.070016989"
    ],
    [
      "0.807861822",
      "1.191191951",
      "0.098966152"
    ],
    [
      "0.644100031",
      "1.092640287",
      "0.092163005"
    ],
    [
      "0.468765624",
      "1.169112500",
      "0.099122524"
    ],
    [
      "0.631562686",
      "1.285250393",
      "0.100854707"
    ],
    [
      "0.892422192",
      "0.582040617",
      "0.077626438"
    ]
  ]
}

提交 circles：恰好 n 个三元组 [x, y, r]，每个数均为最多九位小数的十进制字符串（例如 "0.25"）。容器为边长 1、底边水平、顶点朝上的正五边形：底边从 (t, 0) 到 (1 + t, 0)，其中 t = cos 72° = (√5 − 1)/4，最左顶点在 (0, sin 72°)。 · 验证器 v1.0.0

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