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 | [1638.25518692315, 1638.25518692325] | 见来源署名 | packomania.com ↗ |
| 本站纪录 | 1638.255186943 | 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。
以此前 193 个圆心为基准，在 1e-9 整数格点上将每个坐标限制在 ±32 单位窗口，求解公共半径最小化整数 MILP；约束包括大圆边界和 193*192/2=18528 个成对分离条件。HiGHS 得到半径 1638.255186943 的可行解，并通过独立整数验证器、几何审计和装箱验证。后续可扩大局部窗口或以此解作为新基准继续搜索。
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": [
[
"507.937050850",
"2811.970103986"
],
[
"2080.634804832",
"404.009531980"
],
[
"433.385124192",
"791.758013541"
],
[
"1203.833232675",
"441.167063314"
],
[
"315.240621088",
"680.628436101"
],
[
"2731.976516940",
"430.398609850"
],
[
"2114.583680818",
"1726.073690017"
],
[
"2815.411670957",
"2260.919171127"
],
[
"2786.821147315",
"681.283001867"
],
[
"2693.239822096",
"426.655235353"
],
[
"1963.768831649",
"2183.350139421"
],
[
"891.122699323",
"2765.526933919"
],
[
"1082.228899765",
"2632.233935027"
],
[
"2300.752599556",
"861.011372253"
],
[
"2349.782879032",
"2247.399957092"
],
[
"1997.137127476",
"3208.042642174"
],
[
"640.473306496",
"2891.939405991"
],
[
"283.670596689",
"1140.348354260"
],
[
"3193.511115096",
"2062.844584384"
],
[
"1212.601400364",
"883.178678901"
],
[
"394.737392007",
"2667.515643353"
],
[
"581.988123443",
"1067.002011417"
],
[
"2167.850699196",
"114.576206824"
],
[
"3049.228094740",
"855.787276702"
],
[
"487.626541676",
"510.588311832"
],
[
"1507.597422770",
"916.383441917"
],
[
"1983.039985749",
"2932.359607487"
],
[
"35.535122247",
"1482.981683979"
],
[
"130.957081640",
"1075.095253009"
],
[
"1665.484562851",
"30.241599779"
],
[
"1311.310301641",
"3210.738890621"
],
[
"657.124901503",
"366.514211916"
],
[
"2671.569125373",
"2866.709356377"
],
[
"863.120622752",
"3042.732421473"
],
[
"123.011053694",
"2162.077289444"
],
[
"221.138341833",
"2385.807147419"
],
[
"1867.127136094",
"53.463893821"
],
[
"3222.647631268",
"1413.670378651"
],
[
"2454.456226544",
"265.973856438"
],
[
"2907.190576122",
"2015.622713763"
],
[
"3230.020840196",
"1742.763780063"
],
[
"1378.605953158",
"1623.543951351"
],
[
"2422.274743318",
"3027.522415847"
],
[
"1032.763614037",
"1573.052179847"
],
[
"943.537031799",
"204.473420037"
],
[
"55.848101314",
"1814.469773689"
],
[
"2055.894061556",
"3171.205468417"
],
[
"1671.013389813",
"3228.103189346"
],
[
"2093.295060699",
"1436.218983704"
],
[
"1305.017264216",
"85.385592812"
],
[
"1304.611613951",
"647.114876539"
],
[
"2471.331431421",
"623.743229226"
],
[
"1951.690575945",
"84.431126284"
],
[
"2408.469924889",
"2211.059872900"
],
[
"671.146920336",
"2364.455666993"
],
[
"2931.444602087",
"2549.627481555"
],
[
"1659.280270371",
"2371.813882424"
],
[
"988.555337485",
"2161.305917305"
],
[
"864.850140048",
"1764.360390650"
],
[
"2148.797720693",
"2731.564297146"
],
[
"1185.708587731",
"1227.644537416"
],
[
"2848.624578326",
"2648.000873127"
],
[
"714.834141128",
"2705.361009538"
],
[
"601.149734944",
"2648.547799678"
],
[
"2148.882675860",
"2386.124625233"
],
[
"482.238301358",
"2703.905848797"
],
[
"1265.532475819",
"1629.600633359"
],
[
"2727.378952741",
"507.100566704"
],
[
"2821.102133005",
"607.025365837"
],
[
"3126.441027491",
"2132.954323150"
],
[
"1152.152611076",
"1550.930395842"
],
[
"3105.736546422",
"1090.848783795"
],
[
"2069.436242252",
"133.560301661"
],
[
"3046.554065126",
"957.381806950"
],
[
"1758.232348287",
"79.761342687"
],
[
"2905.406705588",
"2270.094440744"
],
[
"854.612658433",
"2680.618552698"
],
[
"2839.745492427",
"1913.543638396"
],
[
"97.935994188",
"1395.987726399"
],
[
"3063.167671876",
"2268.956096968"
],
[
"1768.578003087",
"1474.487164238"
],
[
"327.984921674",
"798.611728993"
],
[
"599.536012042",
"2795.787550288"
],
[
"2972.702086996",
"2123.991166592"
],
[
"2543.377851453",
"2244.538991022"
],
[
"1273.662197872",
"1943.831399487"
],
[
"2981.313422587",
"2414.520109945"
],
[
"1255.330536702",
"783.942037936"
],
[
"1618.166269286",
"1395.263996317"
],
[
"266.934637087",
"1395.310452091"
],
[
"1297.532848456",
"2119.214386236"
],
[
"1677.243818204",
"2223.773294008"
],
[
"1075.539482866",
"1120.039851951"
],
[
"2520.140384954",
"2637.345618230"
],
[
"1217.764369423",
"3123.106212086"
],
[
"1943.299004815",
"1446.136870449"
],
[
"408.266813493",
"2164.136811472"
],
[
"612.997365187",
"488.854741354"
],
[
"1560.651456604",
"1215.020530086"
],
[
"1773.366690884",
"1838.036515403"
],
[
"227.921287989",
"2248.759767415"
],
[
"1353.754949384",
"1773.745682384"
],
[
"1459.172541622",
"2809.602975229"
],
[
"886.718019815",
"1599.456267251"
],
[
"405.892870844",
"1762.439954999"
],
[
"2702.664615903",
"2731.185543408"
],
[
"1108.125412542",
"2047.584468858"
],
[
"1663.662865214",
"2861.609778222"
],
[
"2088.813256136",
"537.364614837"
],
[
"2136.900055475",
"2561.490308680"
],
[
"1864.982053305",
"1640.530107785"
],
[
"2313.692231240",
"578.473234232"
],
[
"172.765751578",
"1215.473801204"
],
[
"2319.303711510",
"2696.869814938"
],
[
"2087.445390193",
"1600.532446566"
],
[
"737.549985955",
"2527.173674280"
],
[
"1618.616481533",
"485.690375336"
],
[
"620.619786413",
"2198.867678353"
],
[
"2418.242379611",
"788.571310181"
],
[
"476.179246766",
"661.241186997"
],
[
"2478.715427405",
"2048.532244296"
],
[
"2067.473363690",
"1266.876900012"
],
[
"1033.232304031",
"2770.590894043"
],
[
"985.662721070",
"2528.214874649"
],
[
"267.517144015",
"997.147958410"
],
[
"1022.219020455",
"3019.347215742"
],
[
"2640.339363713",
"682.793573143"
],
[
"1247.301794744",
"2901.645284074"
],
[
"1368.073312797",
"475.717879677"
],
[
"3105.001958745",
"1292.847448101"
],
[
"2639.308591694",
"1854.331740733"
],
[
"525.697113827",
"1967.537918887"
],
[
"447.627244821",
"1525.997794728"
],
[
"2856.654098717",
"1701.230507446"
],
[
"798.428773929",
"2885.038148438"
],
[
"2502.418194120",
"2867.072888526"
],
[
"854.263967193",
"2302.703527971"
],
[
"2011.423529797",
"1841.915047548"
],
[
"519.010714437",
"916.854845122"
],
[
"1848.246957713",
"2066.987050101"
],
[
"1050.997830767",
"887.330360712"
],
[
"2157.425169690",
"778.804887616"
],
[
"2528.318958776",
"437.128295013"
],
[
"1801.891557124",
"1251.967036788"
],
[
"2623.508142381",
"954.589281001"
],
[
"2879.824305530",
"813.850670029"
],
[
"2859.034506862",
"1420.240590085"
],
[
"707.819631761",
"1419.880304250"
],
[
"2739.796599509",
"2117.355362310"
],
[
"1141.263966526",
"2302.683829307"
],
[
"1669.306972486",
"988.440324828"
],
[
"659.245780487",
"1716.897232572"
],
[
"1972.627764051",
"1008.750356956"
],
[
"2599.191027468",
"1572.169409723"
],
[
"3120.263043572",
"1577.438557045"
],
[
"1446.979588726",
"2315.873156250"
],
[
"2878.217044884",
"1116.846407153"
],
[
"1495.927243075",
"732.666070006"
],
[
"850.106961092",
"386.449178736"
],
[
"2594.479811676",
"1258.204754950"
],
[
"1012.083268064",
"1365.985595568"
],
[
"1099.722427548",
"588.273654267"
],
[
"2373.505037668",
"2425.224428003"
],
[
"1087.575214686",
"1776.883399531"
],
[
"1589.072419291",
"1647.592250574"
],
[
"1615.171022868",
"2590.455824097"
],
[
"167.012916594",
"1632.090201593"
],
[
"3085.287486631",
"1898.539336268"
],
[
"2311.308808635",
"1425.701569974"
],
[
"459.762166179",
"1220.904190663"
],
[
"2243.843705436",
"304.200381322"
],
[
"1917.054504488",
"2742.476755318"
],
[
"829.573991960",
"1993.688309699"
],
[
"1275.265251324",
"2598.459977513"
],
[
"1157.788681130",
"256.142870953"
],
[
"214.544931195",
"1971.780807276"
],
[
"1845.740504985",
"3084.704828016"
],
[
"1351.209521673",
"1400.069712100"
],
[
"1489.778958111",
"3089.937145944"
],
[
"2716.117049057",
"2445.502096123"
],
[
"1536.699146246",
"1989.663307498"
],
[
"1340.355013071",
"1040.205258901"
],
[
"773.672914631",
"719.798599869"
],
[
"1899.883855501",
"2386.891085170"
],
[
"2304.910289990",
"1070.656115628"
],
[
"2329.498370425",
"1780.235263496"
],
[
"1838.985754701",
"696.116684205"
],
[
"794.378000783",
"1093.212032701"
],
[
"2195.282810297",
"2976.186733658"
],
[
"1519.383706338",
"194.886676270"
],
[
"440.318122496",
"2450.347771416"
],
[
"1880.521543394",
"319.399577800"
],
[
"2175.329769135",
"2129.483725791"
]
],
"radius": "1638.255186943"
}{
"n": 193
}{
"centers": [
[
"507.937050850",
"2811.970103986"
],
[
"2080.634804832",
"404.009531980"
],
[
"433.385124192",
"791.758013541"
],
[
"1203.833232675",
"441.167063314"
],
[
"315.240621088",
"680.628436101"
],
[
"2731.976516940",
"430.398609850"
],
[
"2114.583680818",
"1726.073690017"
],
[
"2815.411670957",
"2260.919171127"
],
[
"2786.821147315",
"681.283001867"
],
[
"2693.239822096",
"426.655235353"
],
[
"1963.768831649",
"2183.350139421"
],
[
"891.122699323",
"2765.526933919"
],
[
"1082.228899765",
"2632.233935027"
],
[
"2300.752599556",
"861.011372253"
],
[
"2349.782879032",
"2247.399957092"
],
[
"1997.137127476",
"3208.042642174"
],
[
"640.473306496",
"2891.939405991"
],
[
"283.670596689",
"1140.348354260"
],
[
"3193.511115096",
"2062.844584384"
],
[
"1212.601400364",
"883.178678901"
],
[
"394.737392007",
"2667.515643353"
],
[
"581.988123443",
"1067.002011417"
],
[
"2167.850699196",
"114.576206824"
],
[
"3049.228094740",
"855.787276702"
],
[
"487.626541676",
"510.588311832"
],
[
"1507.597422770",
"916.383441917"
],
[
"1983.039985749",
"2932.359607487"
],
[
"35.535122247",
"1482.981683979"
],
[
"130.957081640",
"1075.095253009"
],
[
"1665.484562851",
"30.241599779"
],
[
"1311.310301641",
"3210.738890621"
],
[
"657.124901503",
"366.514211916"
],
[
"2671.569125373",
"2866.709356377"
],
[
"863.120622752",
"3042.732421473"
],
[
"123.011053694",
"2162.077289444"
],
[
"221.138341833",
"2385.807147419"
],
[
"1867.127136094",
"53.463893821"
],
[
"3222.647631268",
"1413.670378651"
],
[
"2454.456226544",
"265.973856438"
],
[
"2907.190576122",
"2015.622713763"
],
[
"3230.020840196",
"1742.763780063"
],
[
"1378.605953158",
"1623.543951351"
],
[
"2422.274743318",
"3027.522415847"
],
[
"1032.763614037",
"1573.052179847"
],
[
"943.537031799",
"204.473420037"
],
[
"55.848101314",
"1814.469773689"
],
[
"2055.894061556",
"3171.205468417"
],
[
"1671.013389813",
"3228.103189346"
],
[
"2093.295060699",
"1436.218983704"
],
[
"1305.017264216",
"85.385592812"
],
[
"1304.611613951",
"647.114876539"
],
[
"2471.331431421",
"623.743229226"
],
[
"1951.690575945",
"84.431126284"
],
[
"2408.469924889",
"2211.059872900"
],
[
"671.146920336",
"2364.455666993"
],
[
"2931.444602087",
"2549.627481555"
],
[
"1659.280270371",
"2371.813882424"
],
[
"988.555337485",
"2161.305917305"
],
[
"864.850140048",
"1764.360390650"
],
[
"2148.797720693",
"2731.564297146"
],
[
"1185.708587731",
"1227.644537416"
],
[
"2848.624578326",
"2648.000873127"
],
[
"714.834141128",
"2705.361009538"
],
[
"601.149734944",
"2648.547799678"
],
[
"2148.882675860",
"2386.124625233"
],
[
"482.238301358",
"2703.905848797"
],
[
"1265.532475819",
"1629.600633359"
],
[
"2727.378952741",
"507.100566704"
],
[
"2821.102133005",
"607.025365837"
],
[
"3126.441027491",
"2132.954323150"
],
[
"1152.152611076",
"1550.930395842"
],
[
"3105.736546422",
"1090.848783795"
],
[
"2069.436242252",
"133.560301661"
],
[
"3046.554065126",
"957.381806950"
],
[
"1758.232348287",
"79.761342687"
],
[
"2905.406705588",
"2270.094440744"
],
[
"854.612658433",
"2680.618552698"
],
[
"2839.745492427",
"1913.543638396"
],
[
"97.935994188",
"1395.987726399"
],
[
"3063.167671876",
"2268.956096968"
],
[
"1768.578003087",
"1474.487164238"
],
[
"327.984921674",
"798.611728993"
],
[
"599.536012042",
"2795.787550288"
],
[
"2972.702086996",
"2123.991166592"
],
[
"2543.377851453",
"2244.538991022"
],
[
"1273.662197872",
"1943.831399487"
],
[
"2981.313422587",
"2414.520109945"
],
[
"1255.330536702",
"783.942037936"
],
[
"1618.166269286",
"1395.263996317"
],
[
"266.934637087",
"1395.310452091"
],
[
"1297.532848456",
"2119.214386236"
],
[
"1677.243818204",
"2223.773294008"
],
[
"1075.539482866",
"1120.039851951"
],
[
"2520.140384954",
"2637.345618230"
],
[
"1217.764369423",
"3123.106212086"
],
[
"1943.299004815",
"1446.136870449"
],
[
"408.266813493",
"2164.136811472"
],
[
"612.997365187",
"488.854741354"
],
[
"1560.651456604",
"1215.020530086"
],
[
"1773.366690884",
"1838.036515403"
],
[
"227.921287989",
"2248.759767415"
],
[
"1353.754949384",
"1773.745682384"
],
[
"1459.172541622",
"2809.602975229"
],
[
"886.718019815",
"1599.456267251"
],
[
"405.892870844",
"1762.439954999"
],
[
"2702.664615903",
"2731.185543408"
],
[
"1108.125412542",
"2047.584468858"
],
[
"1663.662865214",
"2861.609778222"
],
[
"2088.813256136",
"537.364614837"
],
[
"2136.900055475",
"2561.490308680"
],
[
"1864.982053305",
"1640.530107785"
],
[
"2313.692231240",
"578.473234232"
],
[
"172.765751578",
"1215.473801204"
],
[
"2319.303711510",
"2696.869814938"
],
[
"2087.445390193",
"1600.532446566"
],
[
"737.549985955",
"2527.173674280"
],
[
"1618.616481533",
"485.690375336"
],
[
"620.619786413",
"2198.867678353"
],
[
"2418.242379611",
"788.571310181"
],
[
"476.179246766",
"661.241186997"
],
[
"2478.715427405",
"2048.532244296"
],
[
"2067.473363690",
"1266.876900012"
],
[
"1033.232304031",
"2770.590894043"
],
[
"985.662721070",
"2528.214874649"
],
[
"267.517144015",
"997.147958410"
],
[
"1022.219020455",
"3019.347215742"
],
[
"2640.339363713",
"682.793573143"
],
[
"1247.301794744",
"2901.645284074"
],
[
"1368.073312797",
"475.717879677"
],
[
"3105.001958745",
"1292.847448101"
],
[
"2639.308591694",
"1854.331740733"
],
[
"525.697113827",
"1967.537918887"
],
[
"447.627244821",
"1525.997794728"
],
[
"2856.654098717",
"1701.230507446"
],
[
"798.428773929",
"2885.038148438"
],
[
"2502.418194120",
"2867.072888526"
],
[
"854.263967193",
"2302.703527971"
],
[
"2011.423529797",
"1841.915047548"
],
[
"519.010714437",
"916.854845122"
],
[
"1848.246957713",
"2066.987050101"
],
[
"1050.997830767",
"887.330360712"
],
[
"2157.425169690",
"778.804887616"
],
[
"2528.318958776",
"437.128295013"
],
[
"1801.891557124",
"1251.967036788"
],
[
"2623.508142381",
"954.589281001"
],
[
"2879.824305530",
"813.850670029"
],
[
"2859.034506862",
"1420.240590085"
],
[
"707.819631761",
"1419.880304250"
],
[
"2739.796599509",
"2117.355362310"
],
[
"1141.263966526",
"2302.683829307"
],
[
"1669.306972486",
"988.440324828"
],
[
"659.245780487",
"1716.897232572"
],
[
"1972.627764051",
"1008.750356956"
],
[
"2599.191027468",
"1572.169409723"
],
[
"3120.263043572",
"1577.438557045"
],
[
"1446.979588726",
"2315.873156250"
],
[
"2878.217044884",
"1116.846407153"
],
[
"1495.927243075",
"732.666070006"
],
[
"850.106961092",
"386.449178736"
],
[
"2594.479811676",
"1258.204754950"
],
[
"1012.083268064",
"1365.985595568"
],
[
"1099.722427548",
"588.273654267"
],
[
"2373.505037668",
"2425.224428003"
],
[
"1087.575214686",
"1776.883399531"
],
[
"1589.072419291",
"1647.592250574"
],
[
"1615.171022868",
"2590.455824097"
],
[
"167.012916594",
"1632.090201593"
],
[
"3085.287486631",
"1898.539336268"
],
[
"2311.308808635",
"1425.701569974"
],
[
"459.762166179",
"1220.904190663"
],
[
"2243.843705436",
"304.200381322"
],
[
"1917.054504488",
"2742.476755318"
],
[
"829.573991960",
"1993.688309699"
],
[
"1275.265251324",
"2598.459977513"
],
[
"1157.788681130",
"256.142870953"
],
[
"214.544931195",
"1971.780807276"
],
[
"1845.740504985",
"3084.704828016"
],
[
"1351.209521673",
"1400.069712100"
],
[
"1489.778958111",
"3089.937145944"
],
[
"2716.117049057",
"2445.502096123"
],
[
"1536.699146246",
"1989.663307498"
],
[
"1340.355013071",
"1040.205258901"
],
[
"773.672914631",
"719.798599869"
],
[
"1899.883855501",
"2386.891085170"
],
[
"2304.910289990",
"1070.656115628"
],
[
"2329.498370425",
"1780.235263496"
],
[
"1838.985754701",
"696.116684205"
],
[
"794.378000783",
"1093.212032701"
],
[
"2195.282810297",
"2976.186733658"
],
[
"1519.383706338",
"194.886676270"
],
[
"440.318122496",
"2450.347771416"
],
[
"1880.521543394",
"319.399577800"
],
[
"2175.329769135",
"2129.483725791"
]
],
"radius": "1638.255186943"
}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。