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 | [1308.66609660615, 1308.66609660625] | 见来源署名 | packomania.com ↗ |
| 本站纪录 | 1308.666096622 | 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 166 圆九位整数证书出发，将容器半径目标降至 1308.666096622，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 166 个包含约束与 13695 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": [
[
"1062.856408351",
"2482.600283896"
],
[
"1449.987978345",
"100.714978696"
],
[
"1729.150568770",
"2350.697895853"
],
[
"1043.049025939",
"1951.500927793"
],
[
"783.361010813",
"1969.339648029"
],
[
"22.989779989",
"1518.149748448"
],
[
"1102.939072565",
"2242.211997970"
],
[
"34.966328309",
"1560.438453179"
],
[
"1896.001455686",
"472.381496176"
],
[
"1242.465733775",
"2121.419054973"
],
[
"516.790035337",
"2065.400523466"
],
[
"2571.249578958",
"1013.635877398"
],
[
"1417.276586913",
"1661.159237435"
],
[
"1428.656880427",
"239.134993014"
],
[
"1045.074046870",
"337.513482937"
],
[
"1254.597009443",
"674.911450704"
],
[
"1287.033744244",
"1205.212954783"
],
[
"1891.315732350",
"1964.868107717"
],
[
"477.189367847",
"788.476792180"
],
[
"1702.807189231",
"1472.384529005"
],
[
"41.573527232",
"1085.069810409"
],
[
"273.669566462",
"1582.525534008"
],
[
"1427.823603183",
"2588.662530821"
],
[
"1730.648559851",
"1523.905167361"
],
[
"2577.266784474",
"1503.790028257"
],
[
"2288.942160897",
"2135.802771928"
],
[
"1655.129898451",
"2542.615373234"
],
[
"549.216273521",
"2339.850666533"
],
[
"2120.308387424",
"2297.779573878"
],
[
"30.252875605",
"1286.527773931"
],
[
"2541.960600886",
"975.047013653"
],
[
"264.430767514",
"579.710098722"
],
[
"2582.630894505",
"1245.584497232"
],
[
"1922.508532822",
"1005.036032701"
],
[
"115.751061349",
"862.575124761"
],
[
"181.076932114",
"1893.965478664"
],
[
"365.518503639",
"2161.658057943"
],
[
"838.981610456",
"128.076498769"
],
[
"1140.323583507",
"50.259237421"
],
[
"1457.636906410",
"48.878644882"
],
[
"77.057905988",
"1608.631150774"
],
[
"1779.928853468",
"132.930405942"
],
[
"1545.502891994",
"380.738450415"
],
[
"1700.915406354",
"1920.581862925"
],
[
"2169.113342178",
"2000.065901880"
],
[
"679.032701351",
"590.471959899"
],
[
"1512.342698646",
"2310.869632043"
],
[
"904.959065478",
"571.169326673"
],
[
"534.290584855",
"1154.541842641"
],
[
"1988.141988732",
"2136.866360712"
],
[
"917.838753822",
"2504.064774346"
],
[
"2013.618549512",
"1821.553165780"
],
[
"1018.465514438",
"2530.337479729"
],
[
"723.011882386",
"1958.350236653"
],
[
"462.398877406",
"383.728003861"
],
[
"1865.459358947",
"2430.770154430"
],
[
"1247.244648679",
"1561.907986169"
],
[
"399.554608152",
"1755.404026162"
],
[
"552.428265918",
"313.794398133"
],
[
"1398.368340486",
"1022.592046197"
],
[
"2399.610002734",
"1914.061114651"
],
[
"2196.356767408",
"433.718833153"
],
[
"426.994102830",
"1122.425414674"
],
[
"1799.241966708",
"2329.269466529"
],
[
"1650.917790344",
"572.265756306"
],
[
"2404.943305125",
"723.520550790"
],
[
"2090.214040213",
"343.825432386"
],
[
"2018.860134579",
"963.190368128"
],
[
"1823.693046477",
"1038.814310829"
],
[
"922.518902933",
"1104.511885948"
],
[
"1637.288852142",
"1550.340678861"
],
[
"1578.721995299",
"1995.906379951"
],
[
"1061.162913260",
"1139.545408970"
],
[
"2286.670502779",
"798.457857174"
],
[
"651.719347599",
"2349.956100941"
],
[
"957.417076980",
"1246.493066516"
],
[
"336.929925003",
"1010.862347623"
],
[
"1903.058366739",
"682.913053333"
],
[
"1333.776095087",
"2538.075791212"
],
[
"2505.951366112",
"1584.579779590"
],
[
"1522.607338409",
"502.606385618"
],
[
"599.323789852",
"2015.525820183"
],
[
"88.788999430",
"1189.681861753"
],
[
"1728.338646966",
"2459.179894323"
],
[
"778.985730941",
"505.793793432"
],
[
"1645.342688070",
"2310.816587515"
],
[
"2299.591853576",
"2023.161742637"
],
[
"1670.322434939",
"420.501268468"
],
[
"1628.099011872",
"1390.604813018"
],
[
"812.852903784",
"988.002603586"
],
[
"1777.714904716",
"1631.115203063"
],
[
"805.021400876",
"2416.194159949"
],
[
"1163.513415146",
"2513.690963561"
],
[
"1811.273737905",
"536.941841777"
],
[
"1142.782772230",
"1859.138022467"
],
[
"831.109376694",
"1642.832159476"
],
[
"2076.221009513",
"1411.507296461"
],
[
"1729.973135427",
"714.811371510"
],
[
"1530.115715996",
"2165.955451042"
],
[
"1403.495467621",
"363.919073987"
],
[
"272.354003817",
"1177.037133012"
],
[
"594.007519694",
"469.332707447"
],
[
"2438.959407970",
"889.061798663"
],
[
"2428.916645076",
"1751.677510739"
],
[
"392.705110989",
"527.751539866"
],
[
"1525.372119161",
"689.751609761"
],
[
"952.525761967",
"423.648448855"
],
[
"2016.967687346",
"1981.518109677"
],
[
"1870.268559415",
"867.015819570"
],
[
"148.852450005",
"1005.957133575"
],
[
"548.364412594",
"677.505277164"
],
[
"1237.212092358",
"1082.697484299"
],
[
"2151.057064899",
"2157.191211852"
],
[
"1974.873495574",
"2300.328732803"
],
[
"1304.467891408",
"1724.110022200"
],
[
"1337.673456599",
"570.492020027"
],
[
"1725.071941032",
"1203.479940355"
],
[
"516.489923991",
"2197.565790153"
],
[
"2265.590520461",
"601.841458886"
],
[
"1523.334742018",
"2477.507487836"
],
[
"1841.027438941",
"1833.440493155"
],
[
"899.301049539",
"1435.772033532"
],
[
"418.611499486",
"1574.886564544"
],
[
"1180.880376249",
"388.789158313"
],
[
"2033.289860616",
"527.192926173"
],
[
"2489.223416049",
"1379.260088165"
],
[
"2087.155918492",
"773.545261687"
],
[
"655.871053869",
"1492.935435166"
],
[
"1015.354656998",
"2348.364068027"
],
[
"2256.224963482",
"1273.204814358"
],
[
"1846.136817967",
"1419.922301216"
],
[
"2470.422831003",
"1121.946003820"
],
[
"208.526690888",
"1723.195805964"
],
[
"458.138057357",
"1320.944344294"
],
[
"2200.616681166",
"1822.389083195"
],
[
"2006.094552205",
"1633.703786050"
],
[
"1433.569079499",
"1507.434974373"
],
[
"1084.062277323",
"2090.227547133"
],
[
"1066.026185515",
"1636.579725209"
],
[
"748.999882686",
"282.800858772"
],
[
"2223.807068383",
"1004.150767598"
],
[
"1037.416557975",
"925.860803507"
],
[
"1013.092960113",
"181.096156536"
],
[
"1297.692166625",
"144.051701436"
],
[
"902.749831393",
"1872.937885727"
],
[
"1085.597895167",
"641.919685052"
],
[
"2276.696541201",
"1550.667954445"
],
[
"1312.236082118",
"833.263669906"
],
[
"1459.744028045",
"1222.376993020"
],
[
"1991.550510502",
"1179.473017533"
],
[
"1876.679377567",
"299.928418167"
],
[
"1354.808503487",
"1986.321430756"
],
[
"804.263381705",
"745.127308008"
],
[
"1592.227694839",
"189.359787028"
],
[
"728.188131994",
"1217.943271758"
],
[
"1571.934451322",
"1767.729276295"
],
[
"364.661564995",
"1967.630985725"
],
[
"614.834105873",
"1776.027531886"
],
[
"1781.287919657",
"2106.993394504"
],
[
"565.860266918",
"947.939918218"
],
[
"793.659406940",
"2161.411624603"
],
[
"293.631668524",
"775.689945524"
],
[
"1302.801313285",
"2296.998517409"
],
[
"1609.372653401",
"947.404414690"
],
[
"1173.915069450",
"1352.368555962"
],
[
"171.877142241",
"1424.410131313"
]
],
"radius": "1308.666096622"
}{
"n": 166
}{
"centers": [
[
"1062.856408351",
"2482.600283896"
],
[
"1449.987978345",
"100.714978696"
],
[
"1729.150568770",
"2350.697895853"
],
[
"1043.049025939",
"1951.500927793"
],
[
"783.361010813",
"1969.339648029"
],
[
"22.989779989",
"1518.149748448"
],
[
"1102.939072565",
"2242.211997970"
],
[
"34.966328309",
"1560.438453179"
],
[
"1896.001455686",
"472.381496176"
],
[
"1242.465733775",
"2121.419054973"
],
[
"516.790035337",
"2065.400523466"
],
[
"2571.249578958",
"1013.635877398"
],
[
"1417.276586913",
"1661.159237435"
],
[
"1428.656880427",
"239.134993014"
],
[
"1045.074046870",
"337.513482937"
],
[
"1254.597009443",
"674.911450704"
],
[
"1287.033744244",
"1205.212954783"
],
[
"1891.315732350",
"1964.868107717"
],
[
"477.189367847",
"788.476792180"
],
[
"1702.807189231",
"1472.384529005"
],
[
"41.573527232",
"1085.069810409"
],
[
"273.669566462",
"1582.525534008"
],
[
"1427.823603183",
"2588.662530821"
],
[
"1730.648559851",
"1523.905167361"
],
[
"2577.266784474",
"1503.790028257"
],
[
"2288.942160897",
"2135.802771928"
],
[
"1655.129898451",
"2542.615373234"
],
[
"549.216273521",
"2339.850666533"
],
[
"2120.308387424",
"2297.779573878"
],
[
"30.252875605",
"1286.527773931"
],
[
"2541.960600886",
"975.047013653"
],
[
"264.430767514",
"579.710098722"
],
[
"2582.630894505",
"1245.584497232"
],
[
"1922.508532822",
"1005.036032701"
],
[
"115.751061349",
"862.575124761"
],
[
"181.076932114",
"1893.965478664"
],
[
"365.518503639",
"2161.658057943"
],
[
"838.981610456",
"128.076498769"
],
[
"1140.323583507",
"50.259237421"
],
[
"1457.636906410",
"48.878644882"
],
[
"77.057905988",
"1608.631150774"
],
[
"1779.928853468",
"132.930405942"
],
[
"1545.502891994",
"380.738450415"
],
[
"1700.915406354",
"1920.581862925"
],
[
"2169.113342178",
"2000.065901880"
],
[
"679.032701351",
"590.471959899"
],
[
"1512.342698646",
"2310.869632043"
],
[
"904.959065478",
"571.169326673"
],
[
"534.290584855",
"1154.541842641"
],
[
"1988.141988732",
"2136.866360712"
],
[
"917.838753822",
"2504.064774346"
],
[
"2013.618549512",
"1821.553165780"
],
[
"1018.465514438",
"2530.337479729"
],
[
"723.011882386",
"1958.350236653"
],
[
"462.398877406",
"383.728003861"
],
[
"1865.459358947",
"2430.770154430"
],
[
"1247.244648679",
"1561.907986169"
],
[
"399.554608152",
"1755.404026162"
],
[
"552.428265918",
"313.794398133"
],
[
"1398.368340486",
"1022.592046197"
],
[
"2399.610002734",
"1914.061114651"
],
[
"2196.356767408",
"433.718833153"
],
[
"426.994102830",
"1122.425414674"
],
[
"1799.241966708",
"2329.269466529"
],
[
"1650.917790344",
"572.265756306"
],
[
"2404.943305125",
"723.520550790"
],
[
"2090.214040213",
"343.825432386"
],
[
"2018.860134579",
"963.190368128"
],
[
"1823.693046477",
"1038.814310829"
],
[
"922.518902933",
"1104.511885948"
],
[
"1637.288852142",
"1550.340678861"
],
[
"1578.721995299",
"1995.906379951"
],
[
"1061.162913260",
"1139.545408970"
],
[
"2286.670502779",
"798.457857174"
],
[
"651.719347599",
"2349.956100941"
],
[
"957.417076980",
"1246.493066516"
],
[
"336.929925003",
"1010.862347623"
],
[
"1903.058366739",
"682.913053333"
],
[
"1333.776095087",
"2538.075791212"
],
[
"2505.951366112",
"1584.579779590"
],
[
"1522.607338409",
"502.606385618"
],
[
"599.323789852",
"2015.525820183"
],
[
"88.788999430",
"1189.681861753"
],
[
"1728.338646966",
"2459.179894323"
],
[
"778.985730941",
"505.793793432"
],
[
"1645.342688070",
"2310.816587515"
],
[
"2299.591853576",
"2023.161742637"
],
[
"1670.322434939",
"420.501268468"
],
[
"1628.099011872",
"1390.604813018"
],
[
"812.852903784",
"988.002603586"
],
[
"1777.714904716",
"1631.115203063"
],
[
"805.021400876",
"2416.194159949"
],
[
"1163.513415146",
"2513.690963561"
],
[
"1811.273737905",
"536.941841777"
],
[
"1142.782772230",
"1859.138022467"
],
[
"831.109376694",
"1642.832159476"
],
[
"2076.221009513",
"1411.507296461"
],
[
"1729.973135427",
"714.811371510"
],
[
"1530.115715996",
"2165.955451042"
],
[
"1403.495467621",
"363.919073987"
],
[
"272.354003817",
"1177.037133012"
],
[
"594.007519694",
"469.332707447"
],
[
"2438.959407970",
"889.061798663"
],
[
"2428.916645076",
"1751.677510739"
],
[
"392.705110989",
"527.751539866"
],
[
"1525.372119161",
"689.751609761"
],
[
"952.525761967",
"423.648448855"
],
[
"2016.967687346",
"1981.518109677"
],
[
"1870.268559415",
"867.015819570"
],
[
"148.852450005",
"1005.957133575"
],
[
"548.364412594",
"677.505277164"
],
[
"1237.212092358",
"1082.697484299"
],
[
"2151.057064899",
"2157.191211852"
],
[
"1974.873495574",
"2300.328732803"
],
[
"1304.467891408",
"1724.110022200"
],
[
"1337.673456599",
"570.492020027"
],
[
"1725.071941032",
"1203.479940355"
],
[
"516.489923991",
"2197.565790153"
],
[
"2265.590520461",
"601.841458886"
],
[
"1523.334742018",
"2477.507487836"
],
[
"1841.027438941",
"1833.440493155"
],
[
"899.301049539",
"1435.772033532"
],
[
"418.611499486",
"1574.886564544"
],
[
"1180.880376249",
"388.789158313"
],
[
"2033.289860616",
"527.192926173"
],
[
"2489.223416049",
"1379.260088165"
],
[
"2087.155918492",
"773.545261687"
],
[
"655.871053869",
"1492.935435166"
],
[
"1015.354656998",
"2348.364068027"
],
[
"2256.224963482",
"1273.204814358"
],
[
"1846.136817967",
"1419.922301216"
],
[
"2470.422831003",
"1121.946003820"
],
[
"208.526690888",
"1723.195805964"
],
[
"458.138057357",
"1320.944344294"
],
[
"2200.616681166",
"1822.389083195"
],
[
"2006.094552205",
"1633.703786050"
],
[
"1433.569079499",
"1507.434974373"
],
[
"1084.062277323",
"2090.227547133"
],
[
"1066.026185515",
"1636.579725209"
],
[
"748.999882686",
"282.800858772"
],
[
"2223.807068383",
"1004.150767598"
],
[
"1037.416557975",
"925.860803507"
],
[
"1013.092960113",
"181.096156536"
],
[
"1297.692166625",
"144.051701436"
],
[
"902.749831393",
"1872.937885727"
],
[
"1085.597895167",
"641.919685052"
],
[
"2276.696541201",
"1550.667954445"
],
[
"1312.236082118",
"833.263669906"
],
[
"1459.744028045",
"1222.376993020"
],
[
"1991.550510502",
"1179.473017533"
],
[
"1876.679377567",
"299.928418167"
],
[
"1354.808503487",
"1986.321430756"
],
[
"804.263381705",
"745.127308008"
],
[
"1592.227694839",
"189.359787028"
],
[
"728.188131994",
"1217.943271758"
],
[
"1571.934451322",
"1767.729276295"
],
[
"364.661564995",
"1967.630985725"
],
[
"614.834105873",
"1776.027531886"
],
[
"1781.287919657",
"2106.993394504"
],
[
"565.860266918",
"947.939918218"
],
[
"793.659406940",
"2161.411624603"
],
[
"293.631668524",
"775.689945524"
],
[
"1302.801313285",
"2296.998517409"
],
[
"1609.372653401",
"947.404414690"
],
[
"1173.915069450",
"1352.368555962"
],
[
"171.877142241",
"1424.410131313"
]
],
"radius": "1308.666096622"
}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。