P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 50

在半径为 1 的圆内放置 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 = 50

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	[6.534855, 6.534865]	见来源署名	Erich Friedman ↗
本站纪录	暂无选手纪录	—	—

严格定义

容器圆心 (1, 1)、半径 1 的闭圆盘；圆可以与容器边界相切
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圆盘内：(x − 1)² + (y − 1)² ≤ (1 − r)²；任意两圆内部不相交：圆心距 ≥ 半径之和。全部按整数精确判定，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 $S = Σ r$
目标在所有合法放置中最大化半径之和 $\max_{S} Σ r$

放大来摆，然后提交 ↗

当前展示：起步布局

2y0

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950

0x2

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

和正方形里一样，等圆是最差的策略之一：一个大圆压在圆心附近，或者几个大圆围成一圈，再让小圆钻进它们和圆周之间的缝隙。圆盘没有角，所以“靠墙的大圆”换成了“沿圆周的一圈”，大小的取舍换了一种几何。

前沿在哪里

Erich Friedman 的“最大化总周长”表列到 n = 50：n ≤ 4 和 6–10 是 Elser 与 Cantrell 的闭式构型，n = 5 是 Haowei Lin 2026 年 7 月的改进，n = 33–50 是 Jonathan Viquerat 2026 年 9 月新加的行。表还在被改写，没有一行有最优性证明；本站展示的是按图重建并精确验算的构型。

查看来源

构型与历史

起步布局

查看构型、求解笔记与纪录历史。

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。

ANSWER FORMAT

答案怎么写

容器是圆心 (1, 1)、半径 1 的圆盘，所以坐标都在 [0, 2] 之内，不会出现负数。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"。容器是圆心 (1, 1)、半径 1 的闭圆盘。

起步布局答案

{
  "circles": [
    [
      "0.655503953",
      "0.881279390",
      "0.181224401"
    ],
    [
      "1.395095977",
      "0.520690051",
      "0.168360364"
    ],
    [
      "1.154047332",
      "1.827299484",
      "0.158480530"
    ],
    [
      "1.713027896",
      "0.537310531",
      "0.150005691"
    ],
    [
      "0.981937567",
      "0.845289069",
      "0.147187240"
    ],
    [
      "1.271381596",
      "0.808187187",
      "0.144625020"
    ],
    [
      "0.879050457",
      "1.118385203",
      "0.144647054"
    ],
    [
      "1.558073072",
      "0.785908191",
      "0.142930810"
    ],
    [
      "0.170778635",
      "1.218704583",
      "0.142422151"
    ],
    [
      "0.822113648",
      "0.605174439",
      "0.141254643"
    ],
    [
      "1.160529047",
      "1.071167820",
      "0.140764364"
    ],
    [
      "0.856323190",
      "1.847954641",
      "0.139959245"
    ],
    [
      "1.092252545",
      "0.144215369",
      "0.139257374"
    ],
    [
      "1.438733649",
      "1.740750399",
      "0.139071217"
    ],
    [
      "0.542358487",
      "0.581142735",
      "0.139530814"
    ],
    [
      "0.283078502",
      "1.476770998",
      "0.139019733"
    ],
    [
      "1.242983173",
      "1.543922612",
      "0.138524579"
    ],
    [
      "0.970362070",
      "1.593787094",
      "0.138619303"
    ],
    [
      "1.063620943",
      "1.332954874",
      "0.138383700"
    ],
    [
      "1.359772135",
      "0.216514223",
      "0.137859667"
    ],
    [
      "1.838175572",
      "0.795986071",
      "0.137352926"
    ],
    [
      "0.138692357",
      "0.941429562",
      "0.136703207"
    ],
    [
      "1.434955527",
      "1.036081082",
      "0.135896026"
    ],
    [
      "1.098491754",
      "0.587545185",
      "0.135685148"
    ],
    [
      "1.332299633",
      "1.286074986",
      "0.134354199"
    ],
    [
      "0.387706687",
      "1.047466340",
      "0.133947738"
    ],
    [
      "0.274345275",
      "0.527130832",
      "0.133870661"
    ],
    [
      "0.796342097",
      "1.384233279",
      "0.133769669"
    ],
    [
      "0.356792936",
      "0.781645306",
      "0.133664820"
    ],
    [
      "0.611694772",
      "1.192122570",
      "0.132690757"
    ],
    [
      "0.448281204",
      "0.327998696",
      "0.130529251"
    ],
    [
      "0.459098474",
      "1.680823901",
      "0.130462395"
    ],
    [
      "0.823504991",
      "0.147591683",
      "0.129511386"
    ],
    [
      "1.592705160",
      "1.248559688",
      "0.128739769"
    ],
    [
      "0.951215993",
      "0.370040022",
      "0.126990784"
    ],
    [
      "1.748152122",
      "1.450945266",
      "0.126453647"
    ],
    [
      "1.846000918",
      "1.217769443",
      "0.126420533"
    ],
    [
      "1.695301568",
      "1.016264717",
      "0.125203090"
    ],
    [
      "0.708968198",
      "1.627779790",
      "0.124975574"
    ],
    [
      "0.544160317",
      "1.440377729",
      "0.124586381"
    ],
    [
      "1.499253332",
      "1.484076905",
      "0.124640583"
    ],
    [
      "0.699830876",
      "0.369342261",
      "0.124395300"
    ],
    [
      "1.580307111",
      "0.319123181",
      "0.105377405"
    ],
    [
      "1.176911140",
      "0.366635980",
      "0.098730031"
    ],
    [
      "0.400564206",
      "1.275276306",
      "0.094224776"
    ],
    [
      "0.140783507",
      "0.711192123",
      "0.093543728"
    ],
    [
      "0.625656449",
      "1.827686068",
      "0.091596278"
    ],
    [
      "1.645292615",
      "1.641503903",
      "0.090093511"
    ],
    [
      "1.910188895",
      "1.011358735",
      "0.089740231"
    ],
    [
      "0.607466948",
      "0.177439044",
      "0.088578734"
    ]
  ]
}

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": 50
}

起步布局答案

{
  "circles": [
    [
      "0.655503953",
      "0.881279390",
      "0.181224401"
    ],
    [
      "1.395095977",
      "0.520690051",
      "0.168360364"
    ],
    [
      "1.154047332",
      "1.827299484",
      "0.158480530"
    ],
    [
      "1.713027896",
      "0.537310531",
      "0.150005691"
    ],
    [
      "0.981937567",
      "0.845289069",
      "0.147187240"
    ],
    [
      "1.271381596",
      "0.808187187",
      "0.144625020"
    ],
    [
      "0.879050457",
      "1.118385203",
      "0.144647054"
    ],
    [
      "1.558073072",
      "0.785908191",
      "0.142930810"
    ],
    [
      "0.170778635",
      "1.218704583",
      "0.142422151"
    ],
    [
      "0.822113648",
      "0.605174439",
      "0.141254643"
    ],
    [
      "1.160529047",
      "1.071167820",
      "0.140764364"
    ],
    [
      "0.856323190",
      "1.847954641",
      "0.139959245"
    ],
    [
      "1.092252545",
      "0.144215369",
      "0.139257374"
    ],
    [
      "1.438733649",
      "1.740750399",
      "0.139071217"
    ],
    [
      "0.542358487",
      "0.581142735",
      "0.139530814"
    ],
    [
      "0.283078502",
      "1.476770998",
      "0.139019733"
    ],
    [
      "1.242983173",
      "1.543922612",
      "0.138524579"
    ],
    [
      "0.970362070",
      "1.593787094",
      "0.138619303"
    ],
    [
      "1.063620943",
      "1.332954874",
      "0.138383700"
    ],
    [
      "1.359772135",
      "0.216514223",
      "0.137859667"
    ],
    [
      "1.838175572",
      "0.795986071",
      "0.137352926"
    ],
    [
      "0.138692357",
      "0.941429562",
      "0.136703207"
    ],
    [
      "1.434955527",
      "1.036081082",
      "0.135896026"
    ],
    [
      "1.098491754",
      "0.587545185",
      "0.135685148"
    ],
    [
      "1.332299633",
      "1.286074986",
      "0.134354199"
    ],
    [
      "0.387706687",
      "1.047466340",
      "0.133947738"
    ],
    [
      "0.274345275",
      "0.527130832",
      "0.133870661"
    ],
    [
      "0.796342097",
      "1.384233279",
      "0.133769669"
    ],
    [
      "0.356792936",
      "0.781645306",
      "0.133664820"
    ],
    [
      "0.611694772",
      "1.192122570",
      "0.132690757"
    ],
    [
      "0.448281204",
      "0.327998696",
      "0.130529251"
    ],
    [
      "0.459098474",
      "1.680823901",
      "0.130462395"
    ],
    [
      "0.823504991",
      "0.147591683",
      "0.129511386"
    ],
    [
      "1.592705160",
      "1.248559688",
      "0.128739769"
    ],
    [
      "0.951215993",
      "0.370040022",
      "0.126990784"
    ],
    [
      "1.748152122",
      "1.450945266",
      "0.126453647"
    ],
    [
      "1.846000918",
      "1.217769443",
      "0.126420533"
    ],
    [
      "1.695301568",
      "1.016264717",
      "0.125203090"
    ],
    [
      "0.708968198",
      "1.627779790",
      "0.124975574"
    ],
    [
      "0.544160317",
      "1.440377729",
      "0.124586381"
    ],
    [
      "1.499253332",
      "1.484076905",
      "0.124640583"
    ],
    [
      "0.699830876",
      "0.369342261",
      "0.124395300"
    ],
    [
      "1.580307111",
      "0.319123181",
      "0.105377405"
    ],
    [
      "1.176911140",
      "0.366635980",
      "0.098730031"
    ],
    [
      "0.400564206",
      "1.275276306",
      "0.094224776"
    ],
    [
      "0.140783507",
      "0.711192123",
      "0.093543728"
    ],
    [
      "0.625656449",
      "1.827686068",
      "0.091596278"
    ],
    [
      "1.645292615",
      "1.641503903",
      "0.090093511"
    ],
    [
      "1.910188895",
      "1.011358735",
      "0.089740231"
    ],
    [
      "0.607466948",
      "0.177439044",
      "0.088578734"
    ]
  ]
}

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"。容器是圆心 (1, 1)、半径 1 的闭圆盘。 · 验证器 v1.0.0

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