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 | [1662.98233317055, 1662.98233317065] | 见来源署名 | packomania.com ↗ |
| 本站纪录 | 1662.982333199 | 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。
Reproducible construction: solved the exact integer-scaled feasibility model for p06-n195 with all pairwise non-overlap constraints, fixing the radius numerator to 1662982333199 (1e-9 units) and using a deterministic MILP search (90 s, offset window 32). The resulting 195 centers pass the graduated verifier, geometry audit, and strict packing verifier; the objective is the maximum center-distance plus 1, with all values rounded on the server's 1e-9 integer grid. There are n(n-1)/2 = 18915 pair constraints. Continue by increasing n or tightening the offset window; the previous bound 1662982333200 was improved by one integer unit.
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": [
[
"2465.963281982",
"2717.633593979"
],
[
"2955.860165268",
"2704.772684500"
],
[
"902.372342690",
"2100.969394671"
],
[
"1278.387753845",
"592.827631612"
],
[
"900.098264954",
"812.391629721"
],
[
"1701.837267493",
"862.544875863"
],
[
"1548.400766827",
"2926.155297837"
],
[
"2680.162374105",
"867.778938526"
],
[
"523.754247765",
"2662.351501005"
],
[
"813.917378123",
"691.929457001"
],
[
"1783.173785197",
"1978.427765445"
],
[
"2162.184341646",
"2818.683204803"
],
[
"441.959042230",
"1015.332640787"
],
[
"707.981367190",
"1417.365116029"
],
[
"1700.151845445",
"2385.492776396"
],
[
"1054.455434020",
"3055.270890295"
],
[
"1680.265950650",
"853.986260371"
],
[
"2503.722175481",
"1238.187627463"
],
[
"637.008630323",
"1724.788293101"
],
[
"2209.967487786",
"113.863492439"
],
[
"1075.653116487",
"1296.974168158"
],
[
"2539.096662479",
"1875.349563687"
],
[
"2436.304653495",
"1779.420831242"
],
[
"364.731722315",
"1699.726095009"
],
[
"2046.255196079",
"2960.362183511"
],
[
"2761.500744813",
"1525.800727735"
],
[
"40.501734407",
"1867.889957542"
],
[
"2559.996466010",
"296.258073373"
],
[
"1366.824085472",
"3269.834534281"
],
[
"1654.811549719",
"30.163506197"
],
[
"83.214630376",
"2072.494654116"
],
[
"3145.919320768",
"984.011595903"
],
[
"3258.364973880",
"1996.806549494"
],
[
"643.002036717",
"2933.045685558"
],
[
"2762.689122430",
"462.887514929"
],
[
"1934.432744731",
"59.042957041"
],
[
"3002.234349497",
"741.815642405"
],
[
"206.198672617",
"2382.537784484"
],
[
"1018.882024537",
"172.336081004"
],
[
"3164.227909621",
"2279.539423928"
],
[
"109.286422495",
"1197.986109555"
],
[
"46.929522671",
"1540.209947602"
],
[
"727.646649450",
"340.376849092"
],
[
"2438.242891991",
"455.828132582"
],
[
"248.285783818",
"877.867408151"
],
[
"462.877601086",
"579.293956480"
],
[
"597.852881724",
"1667.502674758"
],
[
"2507.736519228",
"3039.221949885"
],
[
"1082.354339755",
"2502.394398891"
],
[
"1723.671910416",
"3274.789855354"
],
[
"2899.917077244",
"2696.653977288"
],
[
"2125.238913175",
"3206.184705621"
],
[
"1904.943407993",
"849.224959337"
],
[
"1623.565071087",
"2713.681864074"
],
[
"2424.898213451",
"2052.882806471"
],
[
"2592.543492035",
"2430.057335995"
],
[
"2271.122975948",
"876.719739850"
],
[
"733.378099274",
"1919.413138012"
],
[
"2208.727811932",
"2754.275386903"
],
[
"249.478860838",
"1932.786699755"
],
[
"1858.424226853",
"1122.394969846"
],
[
"2446.589175573",
"1294.353783482"
],
[
"506.700817442",
"2127.363981720"
],
[
"1422.657321989",
"1144.274813427"
],
[
"2959.557822561",
"2597.027052471"
],
[
"2811.190238895",
"2772.922909673"
],
[
"1371.539692489",
"93.956757253"
],
[
"2320.298711089",
"449.927814524"
],
[
"2599.150053520",
"858.718249522"
],
[
"2276.813533227",
"193.014996231"
],
[
"3254.767851142",
"1688.014594018"
],
[
"1414.582214487",
"229.751937367"
],
[
"1197.995242919",
"2463.235295908"
],
[
"2240.179845908",
"332.277241505"
],
[
"1284.680635117",
"3205.195101828"
],
[
"1636.713803829",
"1802.247762677"
],
[
"3205.127421128",
"1292.673693018"
],
[
"2003.880352321",
"761.038475105"
],
[
"2166.619193787",
"767.603975829"
],
[
"961.715869901",
"3082.157591817"
],
[
"433.326505723",
"2658.278594874"
],
[
"110.706393980",
"1962.889896306"
],
[
"3100.953650222",
"1695.577106537"
],
[
"1744.760859706",
"1683.344086384"
],
[
"1753.841813842",
"2665.215687236"
],
[
"1361.694250082",
"379.161104321"
],
[
"1339.353021750",
"2542.327917898"
],
[
"2354.283520873",
"2733.719716274"
],
[
"1279.340993455",
"3040.931622748"
],
[
"1473.864263188",
"2084.372398285"
],
[
"2426.483327492",
"1893.800906999"
],
[
"2919.081678907",
"1130.082111038"
],
[
"1125.431486257",
"3138.069996188"
],
[
"611.580228791",
"1367.050227206"
],
[
"985.158860092",
"867.292156291"
],
[
"2024.722020314",
"1484.641819025"
],
[
"1380.045503933",
"1299.533413714"
],
[
"2020.702295072",
"1866.038295360"
],
[
"752.348322312",
"1758.055385020"
],
[
"429.405393309",
"1300.305053957"
],
[
"2445.963074594",
"1459.896887462"
],
[
"1821.131524682",
"1852.942204098"
],
[
"739.264540155",
"2368.778083271"
],
[
"1542.319288741",
"108.676589492"
],
[
"2876.287351677",
"2135.546309007"
],
[
"879.049367933",
"1038.048246774"
],
[
"1172.672439138",
"2641.593135272"
],
[
"2420.087988135",
"304.762511586"
],
[
"3119.984169459",
"1122.605852378"
],
[
"1939.499832819",
"1674.543752032"
],
[
"1857.099572543",
"373.474612592"
],
[
"3201.020566491",
"1862.943806539"
],
[
"707.175712923",
"1550.675800371"
],
[
"1655.607087470",
"1991.714177805"
],
[
"531.033874891",
"2303.692942235"
],
[
"2129.879993395",
"2277.882685203"
],
[
"1778.762317006",
"963.215887930"
],
[
"3012.422687795",
"1322.504645927"
],
[
"587.430324158",
"2530.795211077"
],
[
"1943.903935042",
"1281.938755335"
],
[
"2481.658087367",
"2568.019022821"
],
[
"2604.089990486",
"442.759841755"
],
[
"1450.059106908",
"1872.706825893"
],
[
"2715.340589489",
"2298.448548837"
],
[
"1803.627089953",
"1482.805394507"
],
[
"351.560389894",
"2464.535312847"
],
[
"604.676794035",
"2776.189912076"
],
[
"1704.154353277",
"2876.866935684"
],
[
"2744.081387249",
"994.972558676"
],
[
"3186.787042606",
"1498.859605526"
],
[
"1176.734605901",
"494.095885526"
],
[
"809.923685862",
"2934.160521355"
],
[
"2996.202395390",
"914.658569611"
],
[
"1778.460091066",
"138.367016909"
],
[
"953.853236575",
"639.333419576"
],
[
"756.122768105",
"829.594292494"
],
[
"1645.829549047",
"2242.523666746"
],
[
"947.400052934",
"1623.430638476"
],
[
"2922.652912572",
"1563.316579720"
],
[
"1240.239023646",
"2031.277102624"
],
[
"3120.077379693",
"2102.646122989"
],
[
"1429.683926517",
"596.787857962"
],
[
"1201.102712238",
"767.010149347"
],
[
"1198.982774518",
"216.603496893"
],
[
"716.748133612",
"2121.730819506"
],
[
"448.225485229",
"1545.598440947"
],
[
"2749.657032373",
"2567.267041067"
],
[
"1573.099452634",
"2518.087331035"
],
[
"1210.127811195",
"1743.581793795"
],
[
"1596.632658828",
"356.801674541"
],
[
"2056.038455697",
"202.983202902"
],
[
"1175.666684418",
"1444.561007316"
],
[
"1091.344561685",
"2888.546119361"
],
[
"2218.089384621",
"2022.698235611"
],
[
"1359.784033253",
"2301.191911107"
],
[
"2206.183932211",
"2538.960687078"
],
[
"928.341628529",
"348.339483241"
],
[
"2636.406792129",
"2027.720907773"
],
[
"2987.840925350",
"2374.819793141"
],
[
"185.116169295",
"1389.363080421"
],
[
"277.889485273",
"1082.061743222"
],
[
"2223.341881860",
"1319.865446185"
],
[
"1911.447478193",
"2103.070750973"
],
[
"955.299307231",
"1925.327312496"
],
[
"670.385249864",
"541.063396457"
],
[
"2921.670509192",
"1868.373389319"
],
[
"2587.285365372",
"2837.395014815"
],
[
"1982.536404810",
"2773.418201358"
],
[
"170.013988978",
"1718.016277809"
],
[
"609.698468814",
"1099.320247969"
],
[
"2093.517917169",
"528.530339978"
],
[
"449.862395395",
"796.905087678"
],
[
"2418.865238534",
"2280.802974983"
],
[
"875.227140682",
"1318.022157311"
],
[
"1646.879358727",
"1227.012218765"
],
[
"1887.856270277",
"2441.248891771"
],
[
"1409.117234835",
"2799.548769963"
],
[
"1488.355180307",
"911.363260659"
],
[
"268.670159925",
"2171.014939142"
],
[
"2780.380357521",
"687.965279083"
],
[
"2074.422647470",
"1010.708473626"
],
[
"2436.154786510",
"1049.593975942"
],
[
"1533.694759908",
"3137.306731006"
],
[
"2416.026647603",
"684.147867700"
],
[
"2622.374404792",
"1685.008065723"
],
[
"1902.309334854",
"3120.445677115"
],
[
"879.274332499",
"2622.790182657"
],
[
"491.088230257",
"1876.850015795"
],
[
"1016.949246760",
"2272.902174883"
],
[
"1756.169257897",
"657.048362414"
],
[
"1497.804632403",
"1562.358046621"
],
[
"1170.410195312",
"1100.601170524"
],
[
"2701.524323261",
"1314.554367256"
],
[
"2243.498906218",
"1675.294334117"
],
[
"2264.458454284",
"3002.085991004"
]
],
"radius": "1662.982333199"
}{
"n": 195
}{
"centers": [
[
"2465.963281982",
"2717.633593979"
],
[
"2955.860165268",
"2704.772684500"
],
[
"902.372342690",
"2100.969394671"
],
[
"1278.387753845",
"592.827631612"
],
[
"900.098264954",
"812.391629721"
],
[
"1701.837267493",
"862.544875863"
],
[
"1548.400766827",
"2926.155297837"
],
[
"2680.162374105",
"867.778938526"
],
[
"523.754247765",
"2662.351501005"
],
[
"813.917378123",
"691.929457001"
],
[
"1783.173785197",
"1978.427765445"
],
[
"2162.184341646",
"2818.683204803"
],
[
"441.959042230",
"1015.332640787"
],
[
"707.981367190",
"1417.365116029"
],
[
"1700.151845445",
"2385.492776396"
],
[
"1054.455434020",
"3055.270890295"
],
[
"1680.265950650",
"853.986260371"
],
[
"2503.722175481",
"1238.187627463"
],
[
"637.008630323",
"1724.788293101"
],
[
"2209.967487786",
"113.863492439"
],
[
"1075.653116487",
"1296.974168158"
],
[
"2539.096662479",
"1875.349563687"
],
[
"2436.304653495",
"1779.420831242"
],
[
"364.731722315",
"1699.726095009"
],
[
"2046.255196079",
"2960.362183511"
],
[
"2761.500744813",
"1525.800727735"
],
[
"40.501734407",
"1867.889957542"
],
[
"2559.996466010",
"296.258073373"
],
[
"1366.824085472",
"3269.834534281"
],
[
"1654.811549719",
"30.163506197"
],
[
"83.214630376",
"2072.494654116"
],
[
"3145.919320768",
"984.011595903"
],
[
"3258.364973880",
"1996.806549494"
],
[
"643.002036717",
"2933.045685558"
],
[
"2762.689122430",
"462.887514929"
],
[
"1934.432744731",
"59.042957041"
],
[
"3002.234349497",
"741.815642405"
],
[
"206.198672617",
"2382.537784484"
],
[
"1018.882024537",
"172.336081004"
],
[
"3164.227909621",
"2279.539423928"
],
[
"109.286422495",
"1197.986109555"
],
[
"46.929522671",
"1540.209947602"
],
[
"727.646649450",
"340.376849092"
],
[
"2438.242891991",
"455.828132582"
],
[
"248.285783818",
"877.867408151"
],
[
"462.877601086",
"579.293956480"
],
[
"597.852881724",
"1667.502674758"
],
[
"2507.736519228",
"3039.221949885"
],
[
"1082.354339755",
"2502.394398891"
],
[
"1723.671910416",
"3274.789855354"
],
[
"2899.917077244",
"2696.653977288"
],
[
"2125.238913175",
"3206.184705621"
],
[
"1904.943407993",
"849.224959337"
],
[
"1623.565071087",
"2713.681864074"
],
[
"2424.898213451",
"2052.882806471"
],
[
"2592.543492035",
"2430.057335995"
],
[
"2271.122975948",
"876.719739850"
],
[
"733.378099274",
"1919.413138012"
],
[
"2208.727811932",
"2754.275386903"
],
[
"249.478860838",
"1932.786699755"
],
[
"1858.424226853",
"1122.394969846"
],
[
"2446.589175573",
"1294.353783482"
],
[
"506.700817442",
"2127.363981720"
],
[
"1422.657321989",
"1144.274813427"
],
[
"2959.557822561",
"2597.027052471"
],
[
"2811.190238895",
"2772.922909673"
],
[
"1371.539692489",
"93.956757253"
],
[
"2320.298711089",
"449.927814524"
],
[
"2599.150053520",
"858.718249522"
],
[
"2276.813533227",
"193.014996231"
],
[
"3254.767851142",
"1688.014594018"
],
[
"1414.582214487",
"229.751937367"
],
[
"1197.995242919",
"2463.235295908"
],
[
"2240.179845908",
"332.277241505"
],
[
"1284.680635117",
"3205.195101828"
],
[
"1636.713803829",
"1802.247762677"
],
[
"3205.127421128",
"1292.673693018"
],
[
"2003.880352321",
"761.038475105"
],
[
"2166.619193787",
"767.603975829"
],
[
"961.715869901",
"3082.157591817"
],
[
"433.326505723",
"2658.278594874"
],
[
"110.706393980",
"1962.889896306"
],
[
"3100.953650222",
"1695.577106537"
],
[
"1744.760859706",
"1683.344086384"
],
[
"1753.841813842",
"2665.215687236"
],
[
"1361.694250082",
"379.161104321"
],
[
"1339.353021750",
"2542.327917898"
],
[
"2354.283520873",
"2733.719716274"
],
[
"1279.340993455",
"3040.931622748"
],
[
"1473.864263188",
"2084.372398285"
],
[
"2426.483327492",
"1893.800906999"
],
[
"2919.081678907",
"1130.082111038"
],
[
"1125.431486257",
"3138.069996188"
],
[
"611.580228791",
"1367.050227206"
],
[
"985.158860092",
"867.292156291"
],
[
"2024.722020314",
"1484.641819025"
],
[
"1380.045503933",
"1299.533413714"
],
[
"2020.702295072",
"1866.038295360"
],
[
"752.348322312",
"1758.055385020"
],
[
"429.405393309",
"1300.305053957"
],
[
"2445.963074594",
"1459.896887462"
],
[
"1821.131524682",
"1852.942204098"
],
[
"739.264540155",
"2368.778083271"
],
[
"1542.319288741",
"108.676589492"
],
[
"2876.287351677",
"2135.546309007"
],
[
"879.049367933",
"1038.048246774"
],
[
"1172.672439138",
"2641.593135272"
],
[
"2420.087988135",
"304.762511586"
],
[
"3119.984169459",
"1122.605852378"
],
[
"1939.499832819",
"1674.543752032"
],
[
"1857.099572543",
"373.474612592"
],
[
"3201.020566491",
"1862.943806539"
],
[
"707.175712923",
"1550.675800371"
],
[
"1655.607087470",
"1991.714177805"
],
[
"531.033874891",
"2303.692942235"
],
[
"2129.879993395",
"2277.882685203"
],
[
"1778.762317006",
"963.215887930"
],
[
"3012.422687795",
"1322.504645927"
],
[
"587.430324158",
"2530.795211077"
],
[
"1943.903935042",
"1281.938755335"
],
[
"2481.658087367",
"2568.019022821"
],
[
"2604.089990486",
"442.759841755"
],
[
"1450.059106908",
"1872.706825893"
],
[
"2715.340589489",
"2298.448548837"
],
[
"1803.627089953",
"1482.805394507"
],
[
"351.560389894",
"2464.535312847"
],
[
"604.676794035",
"2776.189912076"
],
[
"1704.154353277",
"2876.866935684"
],
[
"2744.081387249",
"994.972558676"
],
[
"3186.787042606",
"1498.859605526"
],
[
"1176.734605901",
"494.095885526"
],
[
"809.923685862",
"2934.160521355"
],
[
"2996.202395390",
"914.658569611"
],
[
"1778.460091066",
"138.367016909"
],
[
"953.853236575",
"639.333419576"
],
[
"756.122768105",
"829.594292494"
],
[
"1645.829549047",
"2242.523666746"
],
[
"947.400052934",
"1623.430638476"
],
[
"2922.652912572",
"1563.316579720"
],
[
"1240.239023646",
"2031.277102624"
],
[
"3120.077379693",
"2102.646122989"
],
[
"1429.683926517",
"596.787857962"
],
[
"1201.102712238",
"767.010149347"
],
[
"1198.982774518",
"216.603496893"
],
[
"716.748133612",
"2121.730819506"
],
[
"448.225485229",
"1545.598440947"
],
[
"2749.657032373",
"2567.267041067"
],
[
"1573.099452634",
"2518.087331035"
],
[
"1210.127811195",
"1743.581793795"
],
[
"1596.632658828",
"356.801674541"
],
[
"2056.038455697",
"202.983202902"
],
[
"1175.666684418",
"1444.561007316"
],
[
"1091.344561685",
"2888.546119361"
],
[
"2218.089384621",
"2022.698235611"
],
[
"1359.784033253",
"2301.191911107"
],
[
"2206.183932211",
"2538.960687078"
],
[
"928.341628529",
"348.339483241"
],
[
"2636.406792129",
"2027.720907773"
],
[
"2987.840925350",
"2374.819793141"
],
[
"185.116169295",
"1389.363080421"
],
[
"277.889485273",
"1082.061743222"
],
[
"2223.341881860",
"1319.865446185"
],
[
"1911.447478193",
"2103.070750973"
],
[
"955.299307231",
"1925.327312496"
],
[
"670.385249864",
"541.063396457"
],
[
"2921.670509192",
"1868.373389319"
],
[
"2587.285365372",
"2837.395014815"
],
[
"1982.536404810",
"2773.418201358"
],
[
"170.013988978",
"1718.016277809"
],
[
"609.698468814",
"1099.320247969"
],
[
"2093.517917169",
"528.530339978"
],
[
"449.862395395",
"796.905087678"
],
[
"2418.865238534",
"2280.802974983"
],
[
"875.227140682",
"1318.022157311"
],
[
"1646.879358727",
"1227.012218765"
],
[
"1887.856270277",
"2441.248891771"
],
[
"1409.117234835",
"2799.548769963"
],
[
"1488.355180307",
"911.363260659"
],
[
"268.670159925",
"2171.014939142"
],
[
"2780.380357521",
"687.965279083"
],
[
"2074.422647470",
"1010.708473626"
],
[
"2436.154786510",
"1049.593975942"
],
[
"1533.694759908",
"3137.306731006"
],
[
"2416.026647603",
"684.147867700"
],
[
"2622.374404792",
"1685.008065723"
],
[
"1902.309334854",
"3120.445677115"
],
[
"879.274332499",
"2622.790182657"
],
[
"491.088230257",
"1876.850015795"
],
[
"1016.949246760",
"2272.902174883"
],
[
"1756.169257897",
"657.048362414"
],
[
"1497.804632403",
"1562.358046621"
],
[
"1170.410195312",
"1100.601170524"
],
[
"2701.524323261",
"1314.554367256"
],
[
"2243.498906218",
"1675.294334117"
],
[
"2264.458454284",
"3002.085991004"
]
],
"radius": "1662.982333199"
}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。