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就不是质数\n",[936,1083,1085],{"class":938,"line":1084},7,[936,1086,1087],{"class":946},"    }\n",[936,1089,1091,1094,1097],{"class":938,"line":1090},8,[936,1092,1093],{"class":942},"    return",[936,1095,1096],{"class":946}," true;",[936,1098,1099],{"class":1068},"          \u002F\u002F 所有比它小的都除不尽，就是质数\n",[936,1101,1103],{"class":938,"line":1102},9,[936,1104,1105],{"class":946},"}\n",[936,1107,1109,1111,1114,1117],{"class":938,"line":1108},10,[936,1110,988],{"class":964},[936,1112,1113],{"class":981}," main",[936,1115,1116],{"class":946},"()",[936,1118,998],{"class":946},[936,1120,1122,1125,1128],{"class":938,"line":1121},11,[936,1123,1124],{"class":964},"    int",[936,1126,1127],{"class":1012}," a",[936,1129,972],{"class":946},[936,1131,1133,1136,1139,1141],{"class":938,"line":1132},12,[936,1134,1135],{"class":1012},"    cin ",[936,1137,1138],{"class":946},">>",[936,1140,1127],{"class":1012},[936,1142,972],{"class":946},[936,1144,1146,1149,1151,1154,1156,1159],{"class":938,"line":1145},13,[936,1147,1148],{"class":942},"    if",[936,1150,1007],{"class":946},[936,1152,1153],{"class":981},"isPrime",[936,1155,985],{"class":946},[936,1157,1158],{"class":1012},"a",[936,1160,1161],{"class":946},"))\n",[936,1163,1165,1168,1171,1174,1177,1180,1183,1186],{"class":938,"line":1164},14,[936,1166,1167],{"class":1012},"        cout ",[936,1169,1170],{"class":946},"\u003C\u003C",[936,1172,1173],{"class":946}," \"",[936,1175,1176],{"class":950},"Prime",[936,1178,1179],{"class":946},"\"",[936,1181,1182],{"class":946}," \u003C\u003C",[936,1184,1185],{"class":1012}," endl",[936,1187,972],{"class":946},[936,1189,1191],{"class":938,"line":1190},15,[936,1192,1193],{"class":942},"    else\n",[936,1195,1197,1199,1201,1203,1206,1208,1210,1212],{"class":938,"line":1196},16,[936,1198,1167],{"class":1012},[936,1200,1170],{"class":946},[936,1202,1173],{"class":946},[936,1204,1205],{"class":950},"Not prime",[936,1207,1179],{"class":946},[936,1209,1182],{"class":946},[936,1211,1185],{"class":1012},[936,1213,972],{"class":946},[936,1215,1217],{"class":938,"line":1216},17,[936,1218,1105],{"class":946},[922,1220,1221,1222,1224,1225,1228,1229,1232,1233,1236],{},"然后你会注意到在函数 ",[933,1223,1153],{}," 中，如果形参 ",[933,1226,1227],{},"x"," 有约数，那么必有一个小于等于它平方根的约数。因此我们可以充分利用定义在头文件 ",[933,1230,1231],{},"\u003Ccmath>"," 里的 ",[933,1234,1235],{},"std::sqrt"," 函数计算平方根。",[926,1238,1240],{"className":928,"code":1239,"language":930,"meta":931,"style":931},"#include \u003Ciostream>\n#include \u003Ccmath>         \u002F\u002F 这里增加一个头文件引入\nusing namespace std;\nbool isPrime(int x) {\n    int r;\n    r = sqrt(x);                  \u002F\u002F 计算平方根\n    for (int i{2}; i \u003C= r; i++) { \u002F\u002F r 为上界\n        if (x % i == 0)   \u002F\u002F 只要存在一个能除尽的约数\n            return false; \u002F\u002F 就不是质数\n    }\n    return true;          \u002F\u002F 所有比它小的都除不尽，就是质数\n}\nint main() {\n    int a;\n    cin >> a;\n    if (isPrime(a))\n        cout \u003C\u003C \"Prime\" \u003C\u003C endl;\n    else\n        cout \u003C\u003C \"Not prime\" \u003C\u003C endl;\n}\n",[933,1241,1242,1252,1267,1277,1293,1302,1323,1358,1378,1386,1390,1398,1402,1412,1420,1430,1444,1462,1467,1486],{"__ignoreMap":931},[936,1243,1244,1246,1248,1250],{"class":938,"line":939},[936,1245,943],{"class":942},[936,1247,947],{"class":946},[936,1249,951],{"class":950},[936,1251,954],{"class":946},[936,1253,1254,1256,1258,1261,1264],{"class":938,"line":957},[936,1255,943],{"class":942},[936,1257,947],{"class":946},[936,1259,1260],{"class":950},"cmath",[936,1262,1263],{"class":946},">",[936,1265,1266],{"class":1068},"         \u002F\u002F 这里增加一个头文件引入\n",[936,1268,1269,1271,1273,1275],{"class":938,"line":975},[936,1270,961],{"class":960},[936,1272,965],{"class":964},[936,1274,969],{"class":968},[936,1276,972],{"class":946},[936,1278,1279,1281,1283,1285,1287,1289,1291],{"class":938,"line":1001},[936,1280,978],{"class":964},[936,1282,982],{"class":981},[936,1284,985],{"class":946},[936,1286,988],{"class":964},[936,1288,992],{"class":991},[936,1290,995],{"class":946},[936,1292,998],{"class":946},[936,1294,1295,1297,1300],{"class":938,"line":1043},[936,1296,1124],{"class":964},[936,1298,1299],{"class":1012}," r",[936,1301,972],{"class":946},[936,1303,1304,1307,1310,1313,1315,1317,1320],{"class":938,"line":1072},[936,1305,1306],{"class":1012},"    r ",[936,1308,1309],{"class":946},"=",[936,1311,1312],{"class":981}," sqrt",[936,1314,985],{"class":946},[936,1316,1227],{"class":1012},[936,1318,1319],{"class":946},");",[936,1321,1322],{"class":1068},"                  \u002F\u002F 计算平方根\n",[936,1324,1325,1327,1329,1331,1333,1335,1337,1339,1341,1344,1346,1348,1350,1352,1355],{"class":938,"line":1084},[936,1326,1004],{"class":942},[936,1328,1007],{"class":946},[936,1330,988],{"class":964},[936,1332,1013],{"class":1012},[936,1334,1016],{"class":960},[936,1336,1019],{"class":1012},[936,1338,1022],{"class":946},[936,1340,1025],{"class":1012},[936,1342,1343],{"class":946},"\u003C=",[936,1345,1299],{"class":1012},[936,1347,1022],{"class":946},[936,1349,1035],{"class":1012},[936,1351,1038],{"class":946},[936,1353,1354],{"class":946}," {",[936,1356,1357],{"class":1068}," \u002F\u002F r 为上界\n",[936,1359,1360,1362,1364,1366,1368,1370,1372,1374,1376],{"class":938,"line":1090},[936,1361,1046],{"class":942},[936,1363,1007],{"class":946},[936,1365,1052],{"class":1051},[936,1367,1055],{"class":946},[936,1369,1025],{"class":1051},[936,1371,1060],{"class":946},[936,1373,1063],{"class":960},[936,1375,995],{"class":946},[936,1377,1069],{"class":1068},[936,1379,1380,1382,1384],{"class":938,"line":1102},[936,1381,1075],{"class":942},[936,1383,1078],{"class":946},[936,1385,1081],{"class":1068},[936,1387,1388],{"class":938,"line":1108},[936,1389,1087],{"class":946},[936,1391,1392,1394,1396],{"class":938,"line":1121},[936,1393,1093],{"class":942},[936,1395,1096],{"class":946},[936,1397,1099],{"class":1068},[936,1399,1400],{"class":938,"line":1132},[936,1401,1105],{"class":946},[936,1403,1404,1406,1408,1410],{"class":938,"line":1145},[936,1405,988],{"class":964},[936,1407,1113],{"class":981},[936,1409,1116],{"class":946},[936,1411,998],{"class":946},[936,1413,1414,1416,1418],{"class":938,"line":1164},[936,1415,1124],{"class":964},[936,1417,1127],{"class":1012},[936,1419,972],{"class":946},[936,1421,1422,1424,1426,1428],{"class":938,"line":1190},[936,1423,1135],{"class":1012},[936,1425,1138],{"class":946},[936,1427,1127],{"class":1012},[936,1429,972],{"class":946},[936,1431,1432,1434,1436,1438,1440,1442],{"class":938,"line":1196},[936,1433,1148],{"class":942},[936,1435,1007],{"class":946},[936,1437,1153],{"class":981},[936,1439,985],{"class":946},[936,1441,1158],{"class":1012},[936,1443,1161],{"class":946},[936,1445,1446,1448,1450,1452,1454,1456,1458,1460],{"class":938,"line":1216},[936,1447,1167],{"class":1012},[936,1449,1170],{"class":946},[936,1451,1173],{"class":946},[936,1453,1176],{"class":950},[936,1455,1179],{"class":946},[936,1457,1182],{"class":946},[936,1459,1185],{"class":1012},[936,1461,972],{"class":946},[936,1463,1465],{"class":938,"line":1464},18,[936,1466,1193],{"class":942},[936,1468,1470,1472,1474,1476,1478,1480,1482,1484],{"class":938,"line":1469},19,[936,1471,1167],{"class":1012},[936,1473,1170],{"class":946},[936,1475,1173],{"class":946},[936,1477,1205],{"class":950},[936,1479,1179],{"class":946},[936,1481,1182],{"class":946},[936,1483,1185],{"class":1012},[936,1485,972],{"class":946},[936,1487,1489],{"class":938,"line":1488},20,[936,1490,1105],{"class":946},[1492,1493,1494],"blockquote",{},[922,1495,1496,1497,1500,1501,1504,1505,1507],{},"一般地，",[933,1498,1499],{},"sqrt"," 函数的声明为 ",[933,1502,1503],{},"double sqrt(double arg);","。因此在上述代码第六行调用过程中，发生了整型与浮点类型之间的隐式转换。（这里忽略了许多细节，实际上 ",[933,1506,1499],{}," 接收整型实参时可能是一个模板函数。）",[922,1509,1510],{},"这样速度会快不少。如果你把整个程序的结构用之前的图示标示出来的话，就长成这样：",[1512,1513,1516],"div",{"className":1514},[1515],"fig",[1512,1517],{"id":1518,"className":1519},"fig1",[1520],"raphael",[922,1522,1523],{},"所以函数可以像这样一层一层嵌套起来调用。那么问题就来了，一个函数能否“调用它自己”？",[918,1525,1526],{"id":1526},"认识递归",[922,1528,1529],{},"这个“调用自己”听上去特别玄乎。还是用一个例子来引入好了：求一个正整数的阶乘 $n! =\\displaystyle \\prod_^ni=1\\times2\\times\\cdots\\times n$。你能否不适用循环语句来实现吗？",[922,1531,1532,1533,1536,1537,1540,1541,1544],{},"可以这样考虑，将 $n!$ 定义为：\n$$1! = 1$$\n$$n! =(n-1)!\\times n$$\n这样就可以\n$$\\begin{aligned}\n2!&= 1!\\times 2=&2\\\n3!&= 2!\\times 3=&6\\\n4!&= 3!\\times 4=&24\\\n&\\qquad\\cdots&\n\\end{aligned}$$\n如此一路推算下去。如果用函数 ",[933,1534,1535],{},"int fact(int);"," 来表示阶乘的话，则 ",[933,1538,1539],{},"fact(n)"," 的值与 ",[933,1542,1543],{},"n * fact(n - 1)"," 的值相等。可以把这一思想这样表示出来：",[926,1546,1548],{"className":928,"code":1547,"language":930,"meta":931,"style":931},"#include \u003Ciostream>\nusing namespace std;\nint fact(int n) {\n    if (n == 1) {\n        return 1;\n    } else {\n        return n * fact(n - 1);\n    }\n}\nint main() {\n    cout \u003C\u003C fact(4) \u003C\u003C endl;\n}\n",[933,1549,1550,1560,1570,1588,1606,1615,1625,1649,1653,1657,1667,1689],{"__ignoreMap":931},[936,1551,1552,1554,1556,1558],{"class":938,"line":939},[936,1553,943],{"class":942},[936,1555,947],{"class":946},[936,1557,951],{"class":950},[936,1559,954],{"class":946},[936,1561,1562,1564,1566,1568],{"class":938,"line":957},[936,1563,961],{"class":960},[936,1565,965],{"class":964},[936,1567,969],{"class":968},[936,1569,972],{"class":946},[936,1571,1572,1574,1577,1579,1581,1584,1586],{"class":938,"line":975},[936,1573,988],{"class":964},[936,1575,1576],{"class":981}," fact",[936,1578,985],{"class":946},[936,1580,988],{"class":964},[936,1582,1583],{"class":991}," n",[936,1585,995],{"class":946},[936,1587,998],{"class":946},[936,1589,1590,1592,1594,1597,1599,1602,1604],{"class":938,"line":1001},[936,1591,1148],{"class":942},[936,1593,1007],{"class":946},[936,1595,1596],{"class":1012},"n ",[936,1598,1060],{"class":946},[936,1600,1601],{"class":960}," 1",[936,1603,995],{"class":946},[936,1605,998],{"class":946},[936,1607,1608,1611,1613],{"class":938,"line":1043},[936,1609,1610],{"class":942},"        return",[936,1612,1601],{"class":960},[936,1614,972],{"class":946},[936,1616,1617,1620,1623],{"class":938,"line":1072},[936,1618,1619],{"class":946},"    }",[936,1621,1622],{"class":942}," else",[936,1624,998],{"class":946},[936,1626,1627,1629,1632,1635,1637,1639,1641,1644,1646],{"class":938,"line":1084},[936,1628,1610],{"class":942},[936,1630,1631],{"class":1051}," n ",[936,1633,1634],{"class":946},"*",[936,1636,1576],{"class":981},[936,1638,985],{"class":946},[936,1640,1596],{"class":1051},[936,1642,1643],{"class":946},"-",[936,1645,1601],{"class":960},[936,1647,1648],{"class":946},");\n",[936,1650,1651],{"class":938,"line":1090},[936,1652,1087],{"class":946},[936,1654,1655],{"class":938,"line":1102},[936,1656,1105],{"class":946},[936,1658,1659,1661,1663,1665],{"class":938,"line":1108},[936,1660,988],{"class":964},[936,1662,1113],{"class":981},[936,1664,1116],{"class":946},[936,1666,998],{"class":946},[936,1668,1669,1672,1674,1676,1678,1681,1683,1685,1687],{"class":938,"line":1121},[936,1670,1671],{"class":1012},"    cout ",[936,1673,1170],{"class":946},[936,1675,1576],{"class":981},[936,1677,985],{"class":946},[936,1679,1680],{"class":960},"4",[936,1682,995],{"class":946},[936,1684,1182],{"class":946},[936,1686,1185],{"class":1012},[936,1688,972],{"class":946},[936,1690,1691],{"class":938,"line":1132},[936,1692,1105],{"class":946},[922,1694,1695],{},"理解起来有难度？其实不用想太多，这种写法和原来的普通调用没有任何区别。请看下面的图示：",[1512,1697,1699],{"className":1698},[1515],[1512,1700],{"id":1701,"className":1702},"fig2",[1520],[922,1704,1705,1706,1709],{},"我们称这种在函数体内调用函数自身的编程方法为",[1707,1708,256],"strong",{},"（Recursion）。从刚才的图中也可看出，递归式的调用与普通的函数嵌套调用是一样的，都是“一层一层”地执行，然后从内到外“一层一层”地返回到调用方。",[922,1711,1712,1713,1716,1717,1720,1721,1724],{},"递归唯一比较特殊的地方在于它需要有一个递归终止的条件。试想如果刚才的 ",[933,1714,1715],{},"fact"," 函数中缺少 ",[933,1718,1719],{},"if (n == 1) return 1;"," 这个判断的话，则调用将持续不断、永不停止地进行下去——直到计算机的内存放不下那么多函数的空间为止（这种现象俗称“爆栈”）。所以在进行递归调用前需要",[1707,1722,1723],{},"谨慎地考虑它终止的条件","。",[918,1726,1727],{"id":1727},"经典例题",[922,1729,1730],{},"说到递归，跑不了的就是这个著名的“汉诺塔问题”：",[922,1732,1733],{},"相传在古印度圣庙中，有一种被称为汉诺塔的游戏。该游戏是在一块铜板装置上，有三根杆（编号 A、B、C），在 A 杆自下而上、由大到小按顺序放置 $n$ 个金盘。游戏的目标：把 A 杆上的金盘全部移到 C 杆上，并仍保持原有顺序叠好。操作规则：每次只能移动一个盘子，并且在移动过程中三根杆上都始终保持大盘在下、小盘在上，操作过程中盘子可以置于 A、B、C 任一杆上。下面的图示为 $n=6$ 的情形：",[922,1735,1736],{},[1737,1738],"img",{"alt":1739,"src":1740},"Hanoi","https:\u002F\u002Fs1.ax1x.com\u002F2020\u002F07\u002F30\u002Fauhym6.png",[922,1742,1743],{},"::: tip\n汉诺塔问题经常用作递归的入门例题，但是它理解起来的难度仍然比较大。不过我敢确定的是，如果你能理解这道题，那么你就掌握了递归的核心要点。这也是为什么这里会选它作为例题。我会尽量清楚地去解释这个过程，但是也不能够保证完全讲明白。如果你看了很久还是没有理解的话，姑且跳过去吧，等有了更多的编程经验之后再回过头来看也不晚。\n:::",[922,1745,1746],{},"我们想编写一个程序来做这件事情。输入一个整数 n，输出移动盘子的步骤。为了解决这个问题，我现在先写一个函数：",[926,1748,1750],{"className":928,"code":1749,"language":930,"meta":931,"style":931},"void moveDisk(int n, char src, char dest, char trans);\n",[933,1751,1752],{"__ignoreMap":931},[936,1753,1754,1757,1760,1762,1764,1766,1769,1772,1775,1777,1779,1782,1784,1786,1789],{"class":938,"line":939},[936,1755,1756],{"class":964},"void",[936,1758,1759],{"class":981}," moveDisk",[936,1761,985],{"class":946},[936,1763,988],{"class":964},[936,1765,1583],{"class":991},[936,1767,1768],{"class":946},",",[936,1770,1771],{"class":964}," char",[936,1773,1774],{"class":991}," src",[936,1776,1768],{"class":946},[936,1778,1771],{"class":964},[936,1780,1781],{"class":991}," dest",[936,1783,1768],{"class":946},[936,1785,1771],{"class":964},[936,1787,1788],{"class":991}," trans",[936,1790,1648],{"class":946},[922,1792,1793,1794,1797,1798,1801,1802,1805,1806,1808,1809,1811,1812,1814,1815,1818,1819,1822],{},"这个函数是做什么的呢？当执行它的时候，它会输出将 ",[933,1795,1796],{},"n"," 个（前 n 小的）盘子从 ",[933,1799,1800],{},"src"," 这根杆挪到 ",[933,1803,1804],{},"dest"," 这根杆的步骤。其中 ",[933,1807,1800],{}," 就是起始杆， ",[933,1810,1804],{}," 就是目标杆。最后剩下的那根杆被称为“中转杆”。比如我们的目标就是将 ",[933,1813,1796],{}," 个盘子从杆 A 移动到杆 C，那么直接执行 ",[933,1816,1817],{},"moveDisk(n, 'A', 'C', 'B');"," 就可以输出中间的步骤了。为了行文方便，称最小的盘子为 1 号盘，比它大一点的称为 2 号盘……最大的那个盘子称为 n 号盘。那么怎么去实现这个 ",[933,1820,1821],{},"moveDisk"," 函数呢？",[922,1824,1825,1826,1828,1829,1832],{},"先考虑最简单的情况：",[933,1827,1796],{}," 为 ",[933,1830,1831],{},"1"," 的时候。如果只有一个盘子，那么直接将它从起始杆挪到目标杆就可以了。也就是：",[926,1834,1836],{"className":928,"code":1835,"language":930,"meta":931,"style":931},"void moveDisk(int n, char src, char dest, char trans) {\n    if (n == 1) {\n        \u002F\u002F 将 1 号盘（最小的）从 src 杆挪到 dest 杆\n        cout \u003C\u003C \"Move disk 1 from \" \u003C\u003C src \u003C\u003C \" to \" \u003C\u003C dest \u003C\u003C endl;\n        return;\n    }\n    \u002F\u002F [...]\n}\n",[933,1837,1838,1872,1888,1893,1931,1937,1941,1946],{"__ignoreMap":931},[936,1839,1840,1842,1844,1846,1848,1850,1852,1854,1856,1858,1860,1862,1864,1866,1868,1870],{"class":938,"line":939},[936,1841,1756],{"class":964},[936,1843,1759],{"class":981},[936,1845,985],{"class":946},[936,1847,988],{"class":964},[936,1849,1583],{"class":991},[936,1851,1768],{"class":946},[936,1853,1771],{"class":964},[936,1855,1774],{"class":991},[936,1857,1768],{"class":946},[936,1859,1771],{"class":964},[936,1861,1781],{"class":991},[936,1863,1768],{"class":946},[936,1865,1771],{"class":964},[936,1867,1788],{"class":991},[936,1869,995],{"class":946},[936,1871,998],{"class":946},[936,1873,1874,1876,1878,1880,1882,1884,1886],{"class":938,"line":957},[936,1875,1148],{"class":942},[936,1877,1007],{"class":946},[936,1879,1596],{"class":1012},[936,1881,1060],{"class":946},[936,1883,1601],{"class":960},[936,1885,995],{"class":946},[936,1887,998],{"class":946},[936,1889,1890],{"class":938,"line":975},[936,1891,1892],{"class":1068},"        \u002F\u002F 将 1 号盘（最小的）从 src 杆挪到 dest 杆\n",[936,1894,1895,1897,1899,1901,1904,1906,1908,1911,1913,1915,1918,1920,1922,1925,1927,1929],{"class":938,"line":1001},[936,1896,1167],{"class":1051},[936,1898,1170],{"class":946},[936,1900,1173],{"class":946},[936,1902,1903],{"class":950},"Move disk 1 from ",[936,1905,1179],{"class":946},[936,1907,1182],{"class":946},[936,1909,1910],{"class":1051}," src ",[936,1912,1170],{"class":946},[936,1914,1173],{"class":946},[936,1916,1917],{"class":950}," to ",[936,1919,1179],{"class":946},[936,1921,1182],{"class":946},[936,1923,1924],{"class":1051}," dest ",[936,1926,1170],{"class":946},[936,1928,1185],{"class":1051},[936,1930,972],{"class":946},[936,1932,1933,1935],{"class":938,"line":1043},[936,1934,1610],{"class":942},[936,1936,972],{"class":946},[936,1938,1939],{"class":938,"line":1072},[936,1940,1087],{"class":946},[936,1942,1943],{"class":938,"line":1084},[936,1944,1945],{"class":1068},"    \u002F\u002F [...]\n",[936,1947,1948],{"class":938,"line":1090},[936,1949,1105],{"class":946},[922,1951,1952,1953,1828,1955,1957],{},"现在考虑 ",[933,1954,1796],{},[933,1956,1016],{}," 的情形。我们想要把两个盘子从 A 杆挪到 C 杆。怎么做呢？稍微尝试以下就可以得出这样的解法：",[1959,1960,1961,1965,1968],"ol",{},[1962,1963,1964],"li",{},"将最小的 1 号盘从 A 杆挪到 B 杆；",[1962,1966,1967],{},"将 2 号盘从 A 杆挪到 C 杆；",[1962,1969,1970],{},"将 1 号盘从 B 杆挪到 C 杆。",[922,1972,1973,1974,1828,1976,1979],{},"这两种情况是比较简单的。现在来看 ",[933,1975,1796],{},[933,1977,1978],{},"3"," 的情形：要把三个盘子从 A 杆挪到 C 杆。同样也分为 3 步：",[1959,1981,1982,1985,1988],{},[1962,1983,1984],{},"将 1 号、2 号两个盘子从 A 杆挪到 B 杆；",[1962,1986,1987],{},"将 3 号盘从 A 杆挪到 C 杆；",[1962,1989,1990],{},"将 1 号、2 号两个盘子从 B 杆挪到 C 杆。",[922,1992,1993],{},"这里，第二步没有问题。第一步和第三步如何同时移动两个盘子呢？没关系，只要稍微考虑一下就可以：",[1959,1995,1996,2010,2012],{},[1962,1997,1998,1999],{},"将 1 号、2 号两个盘子从 A 杆挪到 B 杆，具体做法是：\n",[1959,2000,2001,2004,2007],{},[1962,2002,2003],{},"将 1 号盘从 A 杆挪到 C 杆；",[1962,2005,2006],{},"将 2 号盘从 A 杆挪到 B 杆；",[1962,2008,2009],{},"将 1 号盘从 C 杆挪到 B 杆；",[1962,2011,1987],{},[1962,2013,2014,2015],{},"将 1 号、2 号两个盘子从 B 杆挪到 C 杆。\n",[1959,2016,2017,2020,2023],{},[1962,2018,2019],{},"将 1 号盘从 B 杆挪到 A 杆；",[1962,2021,2022],{},"将 2 号盘从 B 杆挪到 C 杆；",[1962,2024,2003],{},[922,2026,2027],{},"稍微停顿一下。这里你会发现两个拆开的步骤非常相似。它们和之前挪两个盘子的步骤是一致的，都是：",[2029,2030,2031,2034,2037],"ul",{},[1962,2032,2033],{},"1 号：起始杆 → 中转杆",[1962,2035,2036],{},"2 号：起始杆 → 目标杆",[1962,2038,2039],{},"1 号：中转杆 → 目标杆",[922,2041,2042],{},"事实上，将 3 个盘子整个从 A 杆挪到 C 杆也是这样的过程！",[2029,2044,2045,2048,2051],{},[1962,2046,2047],{},"1,2 号：起始杆 → 中转杆",[1962,2049,2050],{},"3   号：起始杆 → 目标杆",[1962,2052,2053],{},"1,2 号：中转杆 → 目标杆",[922,2055,2056,2057,2059,2060,2062,2063,2066,2067,2070,2071,2073,2074,2076,2077,2079,2080,2082,2083,2085],{},"所以就发现了规律。如果你要将 ",[933,2058,1796],{}," 号盘子从 ",[933,2061,1800],{}," 杆挪走，你首先需要把上面碍事的 ",[933,2064,2065],{},"n - 1"," 个盘子放到一边去（放在 ",[933,2068,2069],{},"trans"," 杆上）。等把这 ",[933,2072,2065],{}," 个盘子处理妥当了，就可以安安稳稳地将 ",[933,2075,1796],{}," 号盘子挪到 ",[933,2078,1804],{}," 杆。最后，只需要再把刚才撂在一边的 ",[933,2081,2065],{}," 个盘子放回 ",[933,2084,1804],{}," 杆上就行了。这个过程就是这样的：",[926,2087,2089],{"className":928,"code":2088,"language":930,"meta":931,"style":931},"\u002F\u002F 将上面 n - 1 个盘子放在中转杆上\nmoveDisk(n - 1, src, trans, dest);\n\u002F\u002F 将 n 号盘子放在目标杆上\ncout \u003C\u003C \"Move disk \" \u003C\u003C n \u003C\u003C \" from \" \u003C\u003C src \u003C\u003C \" to \" \u003C\u003C dest \u003C\u003C endl;\n\u002F\u002F 将 n - 1 个盘子放到目标杆上\nmoveDisk(n - 1, trans, dest, src);\n",[933,2090,2091,2096,2122,2127,2176,2181],{"__ignoreMap":931},[936,2092,2093],{"class":938,"line":939},[936,2094,2095],{"class":1068},"\u002F\u002F 将上面 n - 1 个盘子放在中转杆上\n",[936,2097,2098,2100,2102,2104,2106,2108,2110,2112,2114,2116,2118,2120],{"class":938,"line":957},[936,2099,1821],{"class":981},[936,2101,985],{"class":946},[936,2103,1596],{"class":1012},[936,2105,1643],{"class":946},[936,2107,1601],{"class":960},[936,2109,1768],{"class":946},[936,2111,1774],{"class":1012},[936,2113,1768],{"class":946},[936,2115,1788],{"class":1012},[936,2117,1768],{"class":946},[936,2119,1781],{"class":1012},[936,2121,1648],{"class":946},[936,2123,2124],{"class":938,"line":975},[936,2125,2126],{"class":1068},"\u002F\u002F 将 n 号盘子放在目标杆上\n",[936,2128,2129,2132,2134,2136,2139,2141,2143,2145,2147,2149,2152,2154,2156,2158,2160,2162,2164,2166,2168,2170,2172,2174],{"class":938,"line":1001},[936,2130,2131],{"class":1012},"cout ",[936,2133,1170],{"class":946},[936,2135,1173],{"class":946},[936,2137,2138],{"class":950},"Move disk ",[936,2140,1179],{"class":946},[936,2142,1182],{"class":946},[936,2144,1631],{"class":1012},[936,2146,1170],{"class":946},[936,2148,1173],{"class":946},[936,2150,2151],{"class":950}," from ",[936,2153,1179],{"class":946},[936,2155,1182],{"class":946},[936,2157,1910],{"class":1012},[936,2159,1170],{"class":946},[936,2161,1173],{"class":946},[936,2163,1917],{"class":950},[936,2165,1179],{"class":946},[936,2167,1182],{"class":946},[936,2169,1924],{"class":1012},[936,2171,1170],{"class":946},[936,2173,1185],{"class":1012},[936,2175,972],{"class":946},[936,2177,2178],{"class":938,"line":1043},[936,2179,2180],{"class":1068},"\u002F\u002F 将 n - 1 个盘子放到目标杆上\n",[936,2182,2183,2185,2187,2189,2191,2193,2195,2197,2199,2201,2203,2205],{"class":938,"line":1072},[936,2184,1821],{"class":981},[936,2186,985],{"class":946},[936,2188,1596],{"class":1012},[936,2190,1643],{"class":946},[936,2192,1601],{"class":960},[936,2194,1768],{"class":946},[936,2196,1788],{"class":1012},[936,2198,1768],{"class":946},[936,2200,1781],{"class":1012},[936,2202,1768],{"class":946},[936,2204,1774],{"class":1012},[936,2206,1648],{"class":946},[922,2208,2209],{},"所以整段代码就是这样的：",[926,2211,2213],{"className":928,"code":2212,"language":930,"meta":931,"style":931},"#include \u003Ciostream>\nusing namespace std;\nvoid moveDisk(int n, char src, char dest, char trans) {\n    if (n == 1) {\n        cout \u003C\u003C \"Move disk 1 from \" \u003C\u003C src \u003C\u003C \" to \" \u003C\u003C dest \u003C\u003C endl;\n        return;\n    } \u002F\u002F 因为这里 return 了，所以不需要写 else\n    moveDisk(n - 1, src, trans, dest);\n    cout \u003C\u003C \"Move disk \" \u003C\u003C n \u003C\u003C \" from \" \u003C\u003C src \u003C\u003C \" to \" \u003C\u003C dest \u003C\u003C endl;\n    moveDisk(n - 1, trans, dest, src);\n}\nint main() {\n    int n;\n    cin >> n;\n    moveDisk(n, 'A', 'C', 'B');\n}\n",[933,2214,2215,2225,2235,2269,2285,2319,2325,2332,2359,2405,2431,2435,2445,2453,2463,2502],{"__ignoreMap":931},[936,2216,2217,2219,2221,2223],{"class":938,"line":939},[936,2218,943],{"class":942},[936,2220,947],{"class":946},[936,2222,951],{"class":950},[936,2224,954],{"class":946},[936,2226,2227,2229,2231,2233],{"class":938,"line":957},[936,2228,961],{"class":960},[936,2230,965],{"class":964},[936,2232,969],{"class":968},[936,2234,972],{"class":946},[936,2236,2237,2239,2241,2243,2245,2247,2249,2251,2253,2255,2257,2259,2261,2263,2265,2267],{"class":938,"line":975},[936,2238,1756],{"class":964},[936,2240,1759],{"class":981},[936,2242,985],{"class":946},[936,2244,988],{"class":964},[936,2246,1583],{"class":991},[936,2248,1768],{"class":946},[936,2250,1771],{"class":964},[936,2252,1774],{"class":991},[936,2254,1768],{"class":946},[936,2256,1771],{"class":964},[936,2258,1781],{"class":991},[936,2260,1768],{"class":946},[936,2262,1771],{"class":964},[936,2264,1788],{"class":991},[936,2266,995],{"class":946},[936,2268,998],{"class":946},[936,2270,2271,2273,2275,2277,2279,2281,2283],{"class":938,"line":1001},[936,2272,1148],{"class":942},[936,2274,1007],{"class":946},[936,2276,1596],{"class":1012},[936,2278,1060],{"class":946},[936,2280,1601],{"class":960},[936,2282,995],{"class":946},[936,2284,998],{"class":946},[936,2286,2287,2289,2291,2293,2295,2297,2299,2301,2303,2305,2307,2309,2311,2313,2315,2317],{"class":938,"line":1043},[936,2288,1167],{"class":1051},[936,2290,1170],{"class":946},[936,2292,1173],{"class":946},[936,2294,1903],{"class":950},[936,2296,1179],{"class":946},[936,2298,1182],{"class":946},[936,2300,1910],{"class":1051},[936,2302,1170],{"class":946},[936,2304,1173],{"class":946},[936,2306,1917],{"class":950},[936,2308,1179],{"class":946},[936,2310,1182],{"class":946},[936,2312,1924],{"class":1051},[936,2314,1170],{"class":946},[936,2316,1185],{"class":1051},[936,2318,972],{"class":946},[936,2320,2321,2323],{"class":938,"line":1072},[936,2322,1610],{"class":942},[936,2324,972],{"class":946},[936,2326,2327,2329],{"class":938,"line":1084},[936,2328,1619],{"class":946},[936,2330,2331],{"class":1068}," \u002F\u002F 因为这里 return 了，所以不需要写 else\n",[936,2333,2334,2337,2339,2341,2343,2345,2347,2349,2351,2353,2355,2357],{"class":938,"line":1090},[936,2335,2336],{"class":981},"    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'",[936,2477,2478],{"class":950},"A",[936,2480,2481],{"class":946},"'",[936,2483,1768],{"class":946},[936,2485,2475],{"class":946},[936,2487,2488],{"class":950},"C",[936,2490,2481],{"class":946},[936,2492,1768],{"class":946},[936,2494,2475],{"class":946},[936,2496,2497],{"class":950},"B",[936,2499,2481],{"class":946},[936,2501,1648],{"class":946},[936,2503,2504],{"class":938,"line":1196},[936,2505,1105],{"class":946},[922,2507,2508],{},"下面这张图片展示了这个递归的思想。每一个步骤都分为三个小部分，直到最简单的情形（点击可展开\u002F收起下一层过程）：",[1512,2510,2512],{"style":2511},"display:flex;height: 500px;justify-content:center;",[2513,2514],"iframe",{"src":2515,"height":2516,"style":2517},"\u002Fassets\u002Fhanoi_recursion.svg",500,"left:0; right: 0; border: 0;",[922,2519,2520,2521,2524],{},"有人说，递归思想就是",[1707,2522,2523],{},"将一个大规模的问题转移到较小规模的同类问题","。所以在解决递归问题时，只需考虑问题的转移方法和终止条件就够了。",[1492,2526,2527],{},[922,2528,2529,2530],{},"可参考更形象的描述：",[1158,2531,2535],{"href":2532,"rel":2533},"https:\u002F\u002Fwww.zhihu.com\u002Fquestion\u002F24385418\u002Fanswer\u002F257751077",[2534],"nofollow","如何理解汉诺塔的递归？ - Fireman A的回答 - 知乎",[918,2537,2538],{"id":2538},"更多",[922,2540,2541],{},"递归是一个非常重要的概念，它在各个领域都有诸多的用途。不过由于本书的侧重点不在算法，所以不会再展开讲。有一种极为有用的算法被称为“深度优先搜索”（DFS），它的核心就是递归思想；感兴趣的读者可以查阅相关资料，这里不再赘述。",[2543,2544,2545],"script",{"setup":931},"\nimport { onMounted } from \"vue\";\nimport { fig1, fig2 } from \"@src\u002Fch03\u002Frecursion\";\n\nonMounted(() => {\n    const [fig1e, fig2e, fig2t] = [\n        \"#fig1\",\n        \"#fig2\",\n        \"#fig2Text\"\n    ].map((s) => document.querySelector(s));\n    fig1(fig1e);\n    fig2(fig2e, fig2t);\n})\n",[2547,2548,2549],"style",{},"html pre.shiki code .s7zQu, html code.shiki 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