文献中也称packing circles with radii 1 to ngraduated circle packing
将半径分别为 1, 2, …, n 的 n 个圆互不重叠地放入一个圆内，使外围容器圆的半径尽可能小。
| 纪录 | 数值 / 区间 | 作者 / 持有人 | 来源 |
|---|---|---|---|
| 外部已知最好 | [1102.85560543245, 1102.85560543255] | 见来源署名 | packomania.com ↗ |
| 本站纪录 | 1102.855605448 | 邱仲普 | 纪录详情 ↓ |
本站纪录尚未确认达到或超越此参考纪录。
当前展示：本站纪录
半径最大的几个圆决定了容器半径的下界与主体骨架，而半径较小的圆可以填入大圆之间及靠近边界的空隙。每当 n 增加 1 时，新增的最大圆往往会改变大圆之间的接触关系，使整体排布发生重排。
将半径为 1..n 的圆装入最小圆曾是 2005 年 Al Zimmermann 编程竞赛的赛题。本站已导入 Packomania 收录的 n = 1..200 公开已知最优构型；n ≥ 5 的最优性尚未给出数学证明，较大规模的子题合并为同一张数据表供算法挑战。
查看来源
查看构型、求解笔记与纪录历史。
从实时页面的 148 圆九位整数证书出发，将容器半径目标降至 1102.855605448，在每个圆心 ±32 个整数单位窗口内建立线性化修复模型并由 HiGHS 求解；独立整数检查确认全部 148 个包含约束与 10878 个成对分离约束。该解严格领先 1e-9；更低目标可扩大窗口或进行全局重启。
在讨论区查看与回复 →单位就是最小那个圆的半径：第 i 个圆的半径正好是 i。容器是你自己给出的圆，圆心在 (radius, radius)，所以坐标范围是 0 到 2·radius，radius 越小越好。坐标和半径用同一个单位，直接写成小数，例如 "7.5"，最多九位小数。
提交容器圆半径 radius 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。容器圆的圆心固定在 (radius, radius)。所有数值均以十进制字符串表示（如 "7.5"）。
{
"centers": [
[
"1153.839731459",
"1160.316991143"
],
[
"1521.726770658",
"1403.921088819"
],
[
"1647.115435634",
"783.272245315"
],
[
"226.271541186",
"1645.732750265"
],
[
"86.813864244",
"693.598611804"
],
[
"134.289384412",
"679.548291506"
],
[
"441.331505111",
"229.241422158"
],
[
"1789.641783771",
"635.365148787"
],
[
"1789.085550427",
"951.312094533"
],
[
"1169.990239435",
"1683.628894526"
],
[
"2016.730504737",
"505.486045089"
],
[
"995.795294486",
"2182.762724583"
],
[
"1383.854840251",
"792.367728075"
],
[
"359.846244219",
"563.797446341"
],
[
"1689.331371210",
"332.481602539"
],
[
"937.527482116",
"28.736368447"
],
[
"1937.015558995",
"1331.285893532"
],
[
"364.691849860",
"1479.410893407"
],
[
"1530.493435729",
"863.853949277"
],
[
"1370.834093522",
"645.334616798"
],
[
"2182.104731254",
"1083.850607335"
],
[
"1989.326352180",
"484.665326453"
],
[
"1718.429130824",
"215.676324081"
],
[
"1262.102640716",
"37.305380265"
],
[
"764.719567051",
"79.555570417"
],
[
"1590.766840937",
"2057.222994049"
],
[
"329.877372063",
"1851.096400501"
],
[
"77.937754163",
"1426.504354388"
],
[
"500.362428979",
"1991.578472813"
],
[
"189.913449596",
"1666.065063035"
],
[
"2103.195990162",
"718.234407192"
],
[
"1450.238797282",
"377.223673415"
],
[
"1488.463233445",
"104.908826439"
],
[
"698.505000694",
"1807.832493787"
],
[
"1353.720890128",
"2140.825837694"
],
[
"1803.085144473",
"1907.752788666"
],
[
"752.721848698",
"2109.553264881"
],
[
"1046.005020549",
"2165.985113283"
],
[
"2001.107592437",
"1672.687680404"
],
[
"2128.604465182",
"1380.834517884"
],
[
"270.021587324",
"843.012527796"
],
[
"1017.722213482",
"942.250439647"
],
[
"1518.374962261",
"526.286870590"
],
[
"534.746769047",
"1081.554240349"
],
[
"1669.900685749",
"1634.005939372"
],
[
"243.531371617",
"1330.758169190"
],
[
"969.496323360",
"1228.751082717"
],
[
"749.162560807",
"1296.223960706"
],
[
"1335.307361485",
"1920.778480900"
],
[
"1847.034425360",
"358.071871794"
],
[
"521.057206309",
"226.550386093"
],
[
"1352.942943541",
"1686.235636325"
],
[
"1516.713693528",
"430.221001368"
],
[
"71.896070637",
"909.929405414"
],
[
"557.205159229",
"326.196387046"
],
[
"339.767209625",
"386.194765997"
],
[
"481.504100352",
"1714.703664453"
],
[
"1010.420522945",
"62.096751652"
],
[
"573.039344373",
"1643.448495501"
],
[
"62.492140569",
"1174.908882650"
],
[
"1912.781116221",
"447.505493218"
],
[
"1756.179298248",
"292.578773855"
],
[
"651.076793256",
"397.695452251"
],
[
"179.809994757",
"626.192443927"
],
[
"874.479783714",
"259.380368980"
],
[
"1378.349895496",
"310.540794361"
],
[
"436.290799512",
"309.957002332"
],
[
"446.083666359",
"1149.985623662"
],
[
"2131.986962882",
"1005.866643283"
],
[
"1483.648313764",
"1336.054754355"
],
[
"1991.380859869",
"997.686690130"
],
[
"1614.329603512",
"1736.966392068"
],
[
"596.653480456",
"1775.041217121"
],
[
"795.354895782",
"679.738709179"
],
[
"786.257970678",
"828.460751775"
],
[
"77.938523284",
"1039.788901665"
],
[
"1625.697603812",
"1373.883304380"
],
[
"707.106281340",
"1671.645240217"
],
[
"202.427698229",
"947.444733858"
],
[
"1344.457958339",
"1391.965799391"
],
[
"1215.811989432",
"1099.246978627"
],
[
"1032.993480616",
"1505.151975226"
],
[
"1011.358573738",
"203.093631267"
],
[
"872.449191051",
"110.394206221"
],
[
"274.598310879",
"511.230819343"
],
[
"1045.936032659",
"1876.882826023"
],
[
"800.984377307",
"392.430830241"
],
[
"110.407608264",
"1314.937836644"
],
[
"1740.398556587",
"728.930165661"
],
[
"880.382282748",
"963.980543235"
],
[
"704.069386644",
"542.290697838"
],
[
"981.077478901",
"375.694793939"
],
[
"1157.902515260",
"94.501410948"
],
[
"443.522271929",
"1561.997111560"
],
[
"148.373693403",
"782.053789223"
],
[
"900.396532213",
"546.073999491"
],
[
"1156.555565396",
"931.399797504"
],
[
"454.307332280",
"1870.397086520"
],
[
"1739.641186268",
"1508.020771246"
],
[
"2072.575012109",
"847.192472161"
],
[
"1871.407405152",
"868.486019613"
],
[
"1042.250922314",
"1688.918946615"
],
[
"865.369178646",
"1583.427727471"
],
[
"1364.679797569",
"1209.080378379"
],
[
"1030.098878526",
"1088.920292509"
],
[
"318.540251760",
"1718.136455208"
],
[
"1352.788529656",
"139.439606237"
],
[
"1451.015979160",
"1812.654304593"
],
[
"826.814418442",
"1157.150386076"
],
[
"614.781344398",
"1213.123487321"
],
[
"1450.661664731",
"2031.654017981"
],
[
"1826.356895269",
"1076.782145801"
],
[
"892.692743527",
"1369.162884805"
],
[
"1229.279594151",
"1796.975764635"
],
[
"1804.006228453",
"1302.679130494"
],
[
"1112.244736023",
"538.052868460"
],
[
"1650.789443583",
"1922.416331572"
],
[
"345.527863747",
"701.435967887"
],
[
"1657.136909284",
"919.538397616"
],
[
"852.494751273",
"1806.055778534"
],
[
"670.772592797",
"988.162724279"
],
[
"693.477507512",
"212.399777123"
],
[
"1230.525738470",
"1559.979040765"
],
[
"1188.349922678",
"310.439159373"
],
[
"1254.838916912",
"732.341055382"
],
[
"1351.333784175",
"500.630588220"
],
[
"416.237047886",
"1342.687931293"
],
[
"250.085049543",
"1149.249275779"
],
[
"488.275336022",
"499.477992615"
],
[
"217.042089332",
"1505.077489135"
],
[
"661.201007935",
"1968.560251580"
],
[
"1566.892132298",
"250.077376150"
],
[
"1597.668231095",
"1164.828422759"
],
[
"914.161653153",
"2053.083556142"
],
[
"595.455925051",
"740.742000160"
],
[
"417.414373376",
"945.051094129"
],
[
"1515.538003837",
"706.264512724"
],
[
"1950.466635904",
"642.904576139"
],
[
"991.543901917",
"762.677685982"
],
[
"656.250615455",
"1459.660096285"
],
[
"1139.523911625",
"1309.243151358"
],
[
"1829.351579465",
"1731.701448447"
],
[
"1980.998548706",
"1490.396254304"
],
[
"1498.312746887",
"1554.768453059"
],
[
"1396.595956801",
"962.134313861"
],
[
"1194.178635982",
"2055.343271389"
],
[
"1705.859362496",
"495.471295467"
],
[
"2051.886524986",
"1208.162495127"
]
],
"radius": "1102.855605448"
}{
"n": 148
}{
"centers": [
[
"1153.839731459",
"1160.316991143"
],
[
"1521.726770658",
"1403.921088819"
],
[
"1647.115435634",
"783.272245315"
],
[
"226.271541186",
"1645.732750265"
],
[
"86.813864244",
"693.598611804"
],
[
"134.289384412",
"679.548291506"
],
[
"441.331505111",
"229.241422158"
],
[
"1789.641783771",
"635.365148787"
],
[
"1789.085550427",
"951.312094533"
],
[
"1169.990239435",
"1683.628894526"
],
[
"2016.730504737",
"505.486045089"
],
[
"995.795294486",
"2182.762724583"
],
[
"1383.854840251",
"792.367728075"
],
[
"359.846244219",
"563.797446341"
],
[
"1689.331371210",
"332.481602539"
],
[
"937.527482116",
"28.736368447"
],
[
"1937.015558995",
"1331.285893532"
],
[
"364.691849860",
"1479.410893407"
],
[
"1530.493435729",
"863.853949277"
],
[
"1370.834093522",
"645.334616798"
],
[
"2182.104731254",
"1083.850607335"
],
[
"1989.326352180",
"484.665326453"
],
[
"1718.429130824",
"215.676324081"
],
[
"1262.102640716",
"37.305380265"
],
[
"764.719567051",
"79.555570417"
],
[
"1590.766840937",
"2057.222994049"
],
[
"329.877372063",
"1851.096400501"
],
[
"77.937754163",
"1426.504354388"
],
[
"500.362428979",
"1991.578472813"
],
[
"189.913449596",
"1666.065063035"
],
[
"2103.195990162",
"718.234407192"
],
[
"1450.238797282",
"377.223673415"
],
[
"1488.463233445",
"104.908826439"
],
[
"698.505000694",
"1807.832493787"
],
[
"1353.720890128",
"2140.825837694"
],
[
"1803.085144473",
"1907.752788666"
],
[
"752.721848698",
"2109.553264881"
],
[
"1046.005020549",
"2165.985113283"
],
[
"2001.107592437",
"1672.687680404"
],
[
"2128.604465182",
"1380.834517884"
],
[
"270.021587324",
"843.012527796"
],
[
"1017.722213482",
"942.250439647"
],
[
"1518.374962261",
"526.286870590"
],
[
"534.746769047",
"1081.554240349"
],
[
"1669.900685749",
"1634.005939372"
],
[
"243.531371617",
"1330.758169190"
],
[
"969.496323360",
"1228.751082717"
],
[
"749.162560807",
"1296.223960706"
],
[
"1335.307361485",
"1920.778480900"
],
[
"1847.034425360",
"358.071871794"
],
[
"521.057206309",
"226.550386093"
],
[
"1352.942943541",
"1686.235636325"
],
[
"1516.713693528",
"430.221001368"
],
[
"71.896070637",
"909.929405414"
],
[
"557.205159229",
"326.196387046"
],
[
"339.767209625",
"386.194765997"
],
[
"481.504100352",
"1714.703664453"
],
[
"1010.420522945",
"62.096751652"
],
[
"573.039344373",
"1643.448495501"
],
[
"62.492140569",
"1174.908882650"
],
[
"1912.781116221",
"447.505493218"
],
[
"1756.179298248",
"292.578773855"
],
[
"651.076793256",
"397.695452251"
],
[
"179.809994757",
"626.192443927"
],
[
"874.479783714",
"259.380368980"
],
[
"1378.349895496",
"310.540794361"
],
[
"436.290799512",
"309.957002332"
],
[
"446.083666359",
"1149.985623662"
],
[
"2131.986962882",
"1005.866643283"
],
[
"1483.648313764",
"1336.054754355"
],
[
"1991.380859869",
"997.686690130"
],
[
"1614.329603512",
"1736.966392068"
],
[
"596.653480456",
"1775.041217121"
],
[
"795.354895782",
"679.738709179"
],
[
"786.257970678",
"828.460751775"
],
[
"77.938523284",
"1039.788901665"
],
[
"1625.697603812",
"1373.883304380"
],
[
"707.106281340",
"1671.645240217"
],
[
"202.427698229",
"947.444733858"
],
[
"1344.457958339",
"1391.965799391"
],
[
"1215.811989432",
"1099.246978627"
],
[
"1032.993480616",
"1505.151975226"
],
[
"1011.358573738",
"203.093631267"
],
[
"872.449191051",
"110.394206221"
],
[
"274.598310879",
"511.230819343"
],
[
"1045.936032659",
"1876.882826023"
],
[
"800.984377307",
"392.430830241"
],
[
"110.407608264",
"1314.937836644"
],
[
"1740.398556587",
"728.930165661"
],
[
"880.382282748",
"963.980543235"
],
[
"704.069386644",
"542.290697838"
],
[
"981.077478901",
"375.694793939"
],
[
"1157.902515260",
"94.501410948"
],
[
"443.522271929",
"1561.997111560"
],
[
"148.373693403",
"782.053789223"
],
[
"900.396532213",
"546.073999491"
],
[
"1156.555565396",
"931.399797504"
],
[
"454.307332280",
"1870.397086520"
],
[
"1739.641186268",
"1508.020771246"
],
[
"2072.575012109",
"847.192472161"
],
[
"1871.407405152",
"868.486019613"
],
[
"1042.250922314",
"1688.918946615"
],
[
"865.369178646",
"1583.427727471"
],
[
"1364.679797569",
"1209.080378379"
],
[
"1030.098878526",
"1088.920292509"
],
[
"318.540251760",
"1718.136455208"
],
[
"1352.788529656",
"139.439606237"
],
[
"1451.015979160",
"1812.654304593"
],
[
"826.814418442",
"1157.150386076"
],
[
"614.781344398",
"1213.123487321"
],
[
"1450.661664731",
"2031.654017981"
],
[
"1826.356895269",
"1076.782145801"
],
[
"892.692743527",
"1369.162884805"
],
[
"1229.279594151",
"1796.975764635"
],
[
"1804.006228453",
"1302.679130494"
],
[
"1112.244736023",
"538.052868460"
],
[
"1650.789443583",
"1922.416331572"
],
[
"345.527863747",
"701.435967887"
],
[
"1657.136909284",
"919.538397616"
],
[
"852.494751273",
"1806.055778534"
],
[
"670.772592797",
"988.162724279"
],
[
"693.477507512",
"212.399777123"
],
[
"1230.525738470",
"1559.979040765"
],
[
"1188.349922678",
"310.439159373"
],
[
"1254.838916912",
"732.341055382"
],
[
"1351.333784175",
"500.630588220"
],
[
"416.237047886",
"1342.687931293"
],
[
"250.085049543",
"1149.249275779"
],
[
"488.275336022",
"499.477992615"
],
[
"217.042089332",
"1505.077489135"
],
[
"661.201007935",
"1968.560251580"
],
[
"1566.892132298",
"250.077376150"
],
[
"1597.668231095",
"1164.828422759"
],
[
"914.161653153",
"2053.083556142"
],
[
"595.455925051",
"740.742000160"
],
[
"417.414373376",
"945.051094129"
],
[
"1515.538003837",
"706.264512724"
],
[
"1950.466635904",
"642.904576139"
],
[
"991.543901917",
"762.677685982"
],
[
"656.250615455",
"1459.660096285"
],
[
"1139.523911625",
"1309.243151358"
],
[
"1829.351579465",
"1731.701448447"
],
[
"1980.998548706",
"1490.396254304"
],
[
"1498.312746887",
"1554.768453059"
],
[
"1396.595956801",
"962.134313861"
],
[
"1194.178635982",
"2055.343271389"
],
[
"1705.859362496",
"495.471295467"
],
[
"2051.886524986",
"1208.162495127"
]
],
"radius": "1102.855605448"
}提交容器圆半径 radius 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。容器圆的圆心固定在 (radius, radius)。所有数值均以十进制字符串表示（如 "7.5"）。 · 验证器 v1.0.0
欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。
按选手与子题归档，默认展示最新一篇。历史版本与回复完整保留，点击即可展开查看或回复。求解笔记可在提交破纪录答案时一并发布。