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 | [1427.19848366325, 1427.19848366335] | 见来源署名 | packomania.com ↗ |
| 本站纪录 | 1427.198483682 | 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 176 圆九位整数证书出发，将容器半径目标降至 1427.198483682，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 176 个包含约束与 15400 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": [
[
"444.421286387",
"2421.264826579"
],
[
"2361.843382442",
"1239.796462862"
],
[
"2805.996739787",
"1070.541103338"
],
[
"102.999100411",
"1033.303163477"
],
[
"1109.741268425",
"2481.847913388"
],
[
"830.146128530",
"426.269271536"
],
[
"961.710132342",
"1080.606474305"
],
[
"28.659900656",
"1598.789352257"
],
[
"2519.101998515",
"1684.974381890"
],
[
"2786.575885813",
"1824.795787919"
],
[
"1363.315769487",
"571.787990466"
],
[
"1172.959715688",
"914.461440110"
],
[
"2512.812279089",
"522.276635400"
],
[
"2257.864471973",
"2570.387988536"
],
[
"727.442606768",
"1122.227810926"
],
[
"1178.172684263",
"1678.985230353"
],
[
"903.439560097",
"693.116982016"
],
[
"1460.698925195",
"1009.319400105"
],
[
"238.604576204",
"2182.248919830"
],
[
"1401.691703114",
"2067.043697170"
],
[
"774.127896593",
"181.880735995"
],
[
"1502.566252928",
"837.041881157"
],
[
"1001.931355325",
"944.770293309"
],
[
"2129.067800850",
"212.318242646"
],
[
"1377.624809707",
"28.031812403"
],
[
"1864.620412384",
"2758.203423852"
],
[
"2035.572287259",
"2688.284891004"
],
[
"577.863712454",
"2539.090043052"
],
[
"2439.716466601",
"2391.436941732"
],
[
"1579.069423862",
"38.280307643"
],
[
"2281.519629878",
"2531.430225531"
],
[
"792.443151067",
"2669.639235716"
],
[
"419.320071150",
"2390.507830055"
],
[
"1562.548702505",
"2813.806695542"
],
[
"2603.430045525",
"2171.977599185"
],
[
"226.101084438",
"2128.485407420"
],
[
"1874.696201421",
"111.205284854"
],
[
"460.495266314",
"429.554850424"
],
[
"41.875325685",
"1338.048547798"
],
[
"274.212912197",
"655.872414454"
],
[
"2294.403735518",
"518.628367023"
],
[
"1849.052677239",
"2231.929417677"
],
[
"600.488595984",
"628.733614805"
],
[
"1076.163081973",
"89.344180809"
],
[
"731.876210341",
"232.695253573"
],
[
"2569.953626496",
"1656.234048105"
],
[
"120.870104814",
"981.905822064"
],
[
"1888.191026511",
"2560.511960229"
],
[
"2060.977015898",
"1176.407816321"
],
[
"2281.179077675",
"2131.783359038"
],
[
"1829.558694081",
"958.249820136"
],
[
"1179.841692993",
"2438.560628210"
],
[
"939.922034439",
"1158.143898088"
],
[
"1488.164932804",
"1802.664242328"
],
[
"2799.346394647",
"1434.023417003"
],
[
"2792.842062084",
"1549.687185222"
],
[
"404.407750048",
"1297.678922221"
],
[
"1171.198764392",
"347.679664187"
],
[
"1683.904469942",
"1603.625628317"
],
[
"1181.266229723",
"1258.999952245"
],
[
"2626.314774541",
"772.508925335"
],
[
"1872.514286886",
"1374.633710682"
],
[
"2514.053523706",
"825.172115146"
],
[
"1450.346897670",
"1427.854542036"
],
[
"1640.912943133",
"1176.911143606"
],
[
"2055.324093713",
"1061.433605294"
],
[
"1084.238722002",
"2743.407878805"
],
[
"2388.326329429",
"788.379824926"
],
[
"1950.142389386",
"957.271841161"
],
[
"714.412357688",
"633.843411484"
],
[
"92.255100811",
"1666.364726415"
],
[
"125.833117237",
"1805.366585444"
],
[
"2704.646575729",
"1876.620229179"
],
[
"1934.336564386",
"2681.772954703"
],
[
"2743.704547916",
"1118.568410453"
],
[
"2176.919084464",
"303.073773939"
],
[
"2339.514913101",
"1782.955465864"
],
[
"2413.547779568",
"506.696969310"
],
[
"2298.895480964",
"398.712462227"
],
[
"2201.154838147",
"1857.154452795"
],
[
"1468.744382835",
"250.069255485"
],
[
"902.739789162",
"1774.004911048"
],
[
"833.663257126",
"1923.879856912"
],
[
"1475.156907126",
"85.182890320"
],
[
"817.705358267",
"749.409652372"
],
[
"1645.854254141",
"2057.872653835"
],
[
"326.761815766",
"1419.056234262"
],
[
"1081.393661042",
"2151.825191072"
],
[
"2324.953692575",
"2419.041330470"
],
[
"2316.202359707",
"647.801971079"
],
[
"1567.605283229",
"392.213018434"
],
[
"539.961919049",
"2423.256164809"
],
[
"2754.595050697",
"1292.837163522"
],
[
"1446.479490953",
"2759.844318555"
],
[
"1067.598784872",
"1708.124275453"
],
[
"1619.792459453",
"2399.812025032"
],
[
"1710.078842581",
"2229.232515647"
],
[
"1132.552135467",
"2582.888142536"
],
[
"1278.986651237",
"647.602088257"
],
[
"2421.573988554",
"1626.126552277"
],
[
"1252.314704745",
"2741.815569370"
],
[
"2507.910919455",
"660.259673081"
],
[
"465.818818508",
"1925.608592649"
],
[
"2204.775388723",
"2266.959990764"
],
[
"1019.750914041",
"1297.017043797"
],
[
"1995.052208567",
"2256.180950664"
],
[
"1476.866582937",
"1964.640862123"
],
[
"2718.686516887",
"1696.165580981"
],
[
"2414.948613621",
"965.750267066"
],
[
"1229.430549789",
"2283.313135655"
],
[
"933.758165252",
"2647.401376890"
],
[
"2594.490803595",
"2025.249653810"
],
[
"2529.214994874",
"1809.926480891"
],
[
"1113.944659953",
"509.870809216"
],
[
"881.183634841",
"2116.092795349"
],
[
"1831.686943039",
"2406.499911608"
],
[
"1264.473769158",
"2058.075511779"
],
[
"887.130731916",
"558.650334131"
],
[
"2645.038508327",
"951.532448355"
],
[
"661.606860058",
"2032.360842210"
],
[
"121.937300949",
"1476.672958022"
],
[
"2363.114181984",
"1980.551200912"
],
[
"793.899617256",
"1256.530638734"
],
[
"729.495002025",
"2526.892944550"
],
[
"241.250197341",
"1965.015881158"
],
[
"702.809196172",
"2278.305930929"
],
[
"1987.685341165",
"1833.693475588"
],
[
"2122.919504534",
"2049.880249139"
],
[
"431.591299898",
"594.284728221"
],
[
"1495.417540181",
"1617.542427928"
],
[
"1862.848298561",
"2059.480350307"
],
[
"1048.410026102",
"1934.311789559"
],
[
"1346.095374222",
"425.520077929"
],
[
"1972.583185496",
"254.629575166"
],
[
"2341.935913368",
"1405.032054424"
],
[
"2441.325130746",
"2226.411013598"
],
[
"603.800303456",
"808.811904250"
],
[
"2072.179821189",
"1363.071945594"
],
[
"2246.828690079",
"1148.067823296"
],
[
"2608.098518803",
"1474.187440885"
],
[
"1790.686916039",
"460.878810712"
],
[
"1471.591621350",
"2213.584981920"
],
[
"1113.795949196",
"1067.540397127"
],
[
"1288.963855266",
"1798.227025754"
],
[
"1886.662527139",
"1581.149635730"
],
[
"1338.137764767",
"885.354366415"
],
[
"2174.243946115",
"1631.755242740"
],
[
"1255.254139076",
"159.608682916"
],
[
"2533.172687783",
"1195.068971687"
],
[
"1661.144944228",
"1390.973875448"
],
[
"831.620601904",
"984.999053658"
],
[
"1626.568696311",
"960.264964682"
],
[
"1859.642920777",
"1160.019340392"
],
[
"1693.831316051",
"182.232093954"
],
[
"653.628108523",
"417.209445842"
],
[
"1057.458191140",
"773.895957340"
],
[
"1855.129284447",
"751.827571072"
],
[
"1700.236531337",
"1820.010155890"
],
[
"662.633793821",
"1752.669685827"
],
[
"888.100223346",
"1527.002033829"
],
[
"2090.228541686",
"2505.924464764"
],
[
"973.394707130",
"2377.294093484"
],
[
"576.390381638",
"1442.349954215"
],
[
"1384.066087937",
"2509.507411318"
],
[
"2088.729485695",
"530.208650436"
],
[
"423.970190697",
"2191.498560480"
],
[
"2162.638351068",
"853.877431172"
],
[
"1546.380748544",
"650.341899282"
],
[
"302.624302348",
"862.932292054"
],
[
"1691.711646436",
"2655.905036541"
],
[
"535.251461371",
"1110.893185933"
],
[
"1404.639473512",
"1196.323050908"
],
[
"1221.249068342",
"1488.543788182"
],
[
"337.548929801",
"1680.097216838"
],
[
"197.574621535",
"1190.499940887"
],
[
"950.312113390",
"270.445685807"
]
],
"radius": "1427.198483682"
}{
"n": 176
}{
"centers": [
[
"444.421286387",
"2421.264826579"
],
[
"2361.843382442",
"1239.796462862"
],
[
"2805.996739787",
"1070.541103338"
],
[
"102.999100411",
"1033.303163477"
],
[
"1109.741268425",
"2481.847913388"
],
[
"830.146128530",
"426.269271536"
],
[
"961.710132342",
"1080.606474305"
],
[
"28.659900656",
"1598.789352257"
],
[
"2519.101998515",
"1684.974381890"
],
[
"2786.575885813",
"1824.795787919"
],
[
"1363.315769487",
"571.787990466"
],
[
"1172.959715688",
"914.461440110"
],
[
"2512.812279089",
"522.276635400"
],
[
"2257.864471973",
"2570.387988536"
],
[
"727.442606768",
"1122.227810926"
],
[
"1178.172684263",
"1678.985230353"
],
[
"903.439560097",
"693.116982016"
],
[
"1460.698925195",
"1009.319400105"
],
[
"238.604576204",
"2182.248919830"
],
[
"1401.691703114",
"2067.043697170"
],
[
"774.127896593",
"181.880735995"
],
[
"1502.566252928",
"837.041881157"
],
[
"1001.931355325",
"944.770293309"
],
[
"2129.067800850",
"212.318242646"
],
[
"1377.624809707",
"28.031812403"
],
[
"1864.620412384",
"2758.203423852"
],
[
"2035.572287259",
"2688.284891004"
],
[
"577.863712454",
"2539.090043052"
],
[
"2439.716466601",
"2391.436941732"
],
[
"1579.069423862",
"38.280307643"
],
[
"2281.519629878",
"2531.430225531"
],
[
"792.443151067",
"2669.639235716"
],
[
"419.320071150",
"2390.507830055"
],
[
"1562.548702505",
"2813.806695542"
],
[
"2603.430045525",
"2171.977599185"
],
[
"226.101084438",
"2128.485407420"
],
[
"1874.696201421",
"111.205284854"
],
[
"460.495266314",
"429.554850424"
],
[
"41.875325685",
"1338.048547798"
],
[
"274.212912197",
"655.872414454"
],
[
"2294.403735518",
"518.628367023"
],
[
"1849.052677239",
"2231.929417677"
],
[
"600.488595984",
"628.733614805"
],
[
"1076.163081973",
"89.344180809"
],
[
"731.876210341",
"232.695253573"
],
[
"2569.953626496",
"1656.234048105"
],
[
"120.870104814",
"981.905822064"
],
[
"1888.191026511",
"2560.511960229"
],
[
"2060.977015898",
"1176.407816321"
],
[
"2281.179077675",
"2131.783359038"
],
[
"1829.558694081",
"958.249820136"
],
[
"1179.841692993",
"2438.560628210"
],
[
"939.922034439",
"1158.143898088"
],
[
"1488.164932804",
"1802.664242328"
],
[
"2799.346394647",
"1434.023417003"
],
[
"2792.842062084",
"1549.687185222"
],
[
"404.407750048",
"1297.678922221"
],
[
"1171.198764392",
"347.679664187"
],
[
"1683.904469942",
"1603.625628317"
],
[
"1181.266229723",
"1258.999952245"
],
[
"2626.314774541",
"772.508925335"
],
[
"1872.514286886",
"1374.633710682"
],
[
"2514.053523706",
"825.172115146"
],
[
"1450.346897670",
"1427.854542036"
],
[
"1640.912943133",
"1176.911143606"
],
[
"2055.324093713",
"1061.433605294"
],
[
"1084.238722002",
"2743.407878805"
],
[
"2388.326329429",
"788.379824926"
],
[
"1950.142389386",
"957.271841161"
],
[
"714.412357688",
"633.843411484"
],
[
"92.255100811",
"1666.364726415"
],
[
"125.833117237",
"1805.366585444"
],
[
"2704.646575729",
"1876.620229179"
],
[
"1934.336564386",
"2681.772954703"
],
[
"2743.704547916",
"1118.568410453"
],
[
"2176.919084464",
"303.073773939"
],
[
"2339.514913101",
"1782.955465864"
],
[
"2413.547779568",
"506.696969310"
],
[
"2298.895480964",
"398.712462227"
],
[
"2201.154838147",
"1857.154452795"
],
[
"1468.744382835",
"250.069255485"
],
[
"902.739789162",
"1774.004911048"
],
[
"833.663257126",
"1923.879856912"
],
[
"1475.156907126",
"85.182890320"
],
[
"817.705358267",
"749.409652372"
],
[
"1645.854254141",
"2057.872653835"
],
[
"326.761815766",
"1419.056234262"
],
[
"1081.393661042",
"2151.825191072"
],
[
"2324.953692575",
"2419.041330470"
],
[
"2316.202359707",
"647.801971079"
],
[
"1567.605283229",
"392.213018434"
],
[
"539.961919049",
"2423.256164809"
],
[
"2754.595050697",
"1292.837163522"
],
[
"1446.479490953",
"2759.844318555"
],
[
"1067.598784872",
"1708.124275453"
],
[
"1619.792459453",
"2399.812025032"
],
[
"1710.078842581",
"2229.232515647"
],
[
"1132.552135467",
"2582.888142536"
],
[
"1278.986651237",
"647.602088257"
],
[
"2421.573988554",
"1626.126552277"
],
[
"1252.314704745",
"2741.815569370"
],
[
"2507.910919455",
"660.259673081"
],
[
"465.818818508",
"1925.608592649"
],
[
"2204.775388723",
"2266.959990764"
],
[
"1019.750914041",
"1297.017043797"
],
[
"1995.052208567",
"2256.180950664"
],
[
"1476.866582937",
"1964.640862123"
],
[
"2718.686516887",
"1696.165580981"
],
[
"2414.948613621",
"965.750267066"
],
[
"1229.430549789",
"2283.313135655"
],
[
"933.758165252",
"2647.401376890"
],
[
"2594.490803595",
"2025.249653810"
],
[
"2529.214994874",
"1809.926480891"
],
[
"1113.944659953",
"509.870809216"
],
[
"881.183634841",
"2116.092795349"
],
[
"1831.686943039",
"2406.499911608"
],
[
"1264.473769158",
"2058.075511779"
],
[
"887.130731916",
"558.650334131"
],
[
"2645.038508327",
"951.532448355"
],
[
"661.606860058",
"2032.360842210"
],
[
"121.937300949",
"1476.672958022"
],
[
"2363.114181984",
"1980.551200912"
],
[
"793.899617256",
"1256.530638734"
],
[
"729.495002025",
"2526.892944550"
],
[
"241.250197341",
"1965.015881158"
],
[
"702.809196172",
"2278.305930929"
],
[
"1987.685341165",
"1833.693475588"
],
[
"2122.919504534",
"2049.880249139"
],
[
"431.591299898",
"594.284728221"
],
[
"1495.417540181",
"1617.542427928"
],
[
"1862.848298561",
"2059.480350307"
],
[
"1048.410026102",
"1934.311789559"
],
[
"1346.095374222",
"425.520077929"
],
[
"1972.583185496",
"254.629575166"
],
[
"2341.935913368",
"1405.032054424"
],
[
"2441.325130746",
"2226.411013598"
],
[
"603.800303456",
"808.811904250"
],
[
"2072.179821189",
"1363.071945594"
],
[
"2246.828690079",
"1148.067823296"
],
[
"2608.098518803",
"1474.187440885"
],
[
"1790.686916039",
"460.878810712"
],
[
"1471.591621350",
"2213.584981920"
],
[
"1113.795949196",
"1067.540397127"
],
[
"1288.963855266",
"1798.227025754"
],
[
"1886.662527139",
"1581.149635730"
],
[
"1338.137764767",
"885.354366415"
],
[
"2174.243946115",
"1631.755242740"
],
[
"1255.254139076",
"159.608682916"
],
[
"2533.172687783",
"1195.068971687"
],
[
"1661.144944228",
"1390.973875448"
],
[
"831.620601904",
"984.999053658"
],
[
"1626.568696311",
"960.264964682"
],
[
"1859.642920777",
"1160.019340392"
],
[
"1693.831316051",
"182.232093954"
],
[
"653.628108523",
"417.209445842"
],
[
"1057.458191140",
"773.895957340"
],
[
"1855.129284447",
"751.827571072"
],
[
"1700.236531337",
"1820.010155890"
],
[
"662.633793821",
"1752.669685827"
],
[
"888.100223346",
"1527.002033829"
],
[
"2090.228541686",
"2505.924464764"
],
[
"973.394707130",
"2377.294093484"
],
[
"576.390381638",
"1442.349954215"
],
[
"1384.066087937",
"2509.507411318"
],
[
"2088.729485695",
"530.208650436"
],
[
"423.970190697",
"2191.498560480"
],
[
"2162.638351068",
"853.877431172"
],
[
"1546.380748544",
"650.341899282"
],
[
"302.624302348",
"862.932292054"
],
[
"1691.711646436",
"2655.905036541"
],
[
"535.251461371",
"1110.893185933"
],
[
"1404.639473512",
"1196.323050908"
],
[
"1221.249068342",
"1488.543788182"
],
[
"337.548929801",
"1680.097216838"
],
[
"197.574621535",
"1190.499940887"
],
[
"950.312113390",
"270.445685807"
]
],
"radius": "1427.198483682"
}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。