题目名称 幸运数 TOF函数 最短折叠
存盘文件名 lucky.cpp tof.cpp cuttle.cpp
输入文件名 lucky.in tof.in cuttle.in
输出文件名 lucky.out tof.out cuttle.out
时限 1s 1s 1s
内存限制 128M 128M 128M
【注意事项】： 请独立完成题目。
T1 幸运数
题目描述
如果一个数能被31整除且又包含31，我们称之为幸运数，请找出给定范围内所有的幸运数。
输入格式：
共一行，包含1个正整数n(0 <= n < 109)
输出格式：
[1,n]范围内幸运数的个数。
样例输入1：
400
样例输出1：
2
样例输入2：
4000
样例输出2：
7
法一（正解）：
既然是大范围内的计数问题，那么它大概是用数位dp来做。
-
如何描述状态？
-
f数组的第一维肯定存放当前位，
因为幸运数需要数位上有连续的 31 ，这个限制条件和第i位上放的数直接相关。于是我们用第二维描述第i位上存放j所得到的方案总数。
幸运数还需要能被31整除，所以第三维存当前数除以31的余数。
综上
int f[N][N][N]; //第i位上放j，总共余数为k得到的方案总数
- 如何转移？
LL dfs(int now,int sta,int md,bool lit);
//now表示当前位
//sta表示当前位上的数
//md表示放上当前位之后所得到的%31余数，从高位往低位放数
//lit表示是否一直达到上限
//返回值为当前状态下的方案总数
对于能被31整除的条件
a%n=(a%n+n)%n
(a+b)%n=(a+b%n)%n
(a+b)%n=(a%n+b)%n
显然余数为md的状态由(md*10+i)得来
对于数位中出现31的条件
极难正常的转移
若硬要做，我们需要把状态分门别类的存放。
已经有31的，还没出现31的，首位是1的。
那我们换一种方式考虑
能被31整除的数中
不是有31的，就是没31的。
那我们先做没有31的，用能被31整除的数的总数减去它，就是幸运数的个数了。
如何保证不出现31呢？
跟“”不要62“”的方法类似，碰到当前位是3，下一位是1的，不转移即可。
于是
LL res=0;
int maxn=(lit) ? dim[now-1] : 9;
for(int i=0;i<=maxn;i++) {
if(sta==3&&i==1) continue;
res+=dfs(now-1,i,(md*10+i)%MOD,(lit&&i==maxn));
}
- 边界条件？
- 0位上只能放0，再判断能否整除即可。
if(now==0) return (sta==0&&md==0) ? 1 : 0;
Code：
#include<cstdio>
#include<cstring>
#include<iostream>
#define LL long long
using namespace std;
const int N=64;
const int MOD=31;
int dim[N];
int f[N][N][N]; //第i位上放j，总共余数为k得到的方案总数
LL dfs(int now,int sta,int md,bool lit)
{
if(now==0) return (sta==0&&md==0) ? 1 : 0; //---??
if(!lit&&f[now][sta][md]!=-1)
return f[now][sta][md];
LL res=0;
int maxn=(lit) ? dim[now-1] : 9;
for(int i=0;i<=maxn;i++) {
if(sta==3&&i==1) continue;
res+=dfs(now-1,i,(md*10+i)%MOD,(lit&&i==maxn));
}
if(!lit) f[now][sta][md]=res;
return res;
}
LL prefix(LL x)
{
memset(dim,0,sizeof dim);
LL ans=x/31;
int top=0;
while(x>0) {
dim[++top]=x%10;
x=x/10;
}
for(int i=0;i<=dim[top];i++)
ans-=dfs(top,i,i,i==dim[top]);
ans++;
return ans;
}
int main()
{
freopen("lucky.in","r",stdin);
freopen("lucky.out","w",stdout);
LL a;
cin>>a;
memset(f,-1,sizeof f);
cout<<prefix(a);
return 0;
}
法二（分块打表）：
没法一次暴力出0~1e9的幸运数总数，那我们每隔1e6
记录一次幸运数总数，对于输入的n，从最近的一个1e6开始暴力，这样最多只要算1e6次，是可以接受的。
这是用于打表的Code：
#include<cstdio>
#include<iostream>
#define LL long long
using namespace std;
int n;
const int N=1e9+5;
const int M=1e7;
int pm[M];
int tot=0;
bool check(int x)
{
int top=0;
int tmp=0;
int flag=0;
while(x>0) {
tmp=x%10;
x=x/10;
if(tmp==1)
flag=1;
else if(tmp==3) {
if(flag==1) return true;
else flag=0;
}
else flag=0;
}
return false;
}
void prime()
{
int i,j;
for(i=31;i<=n;i+=31) {
if(check(i))
pm[++tot]=i;//printf("%d,",i);
}
return;
}
int main()
{
// freopen("lucky.in","r",stdin);
freopen("lucky.cpp","w",stdout);
cout<<"int pm[]={";
n=1e9;
// cin>>n;
prime();
// cout<<tot;
for(int i=0;i<=1e9;i+=1e6) {
cout<<(lower_bound(pm+1,pm+tot+1,i)-pm-1)<<",";
}
cout<<"};";
return 0;
}
注意表的大小要控制在1000左右。
这是提交的程序：
#include<iostream>
#include<cstdio>
using namespace std;
const int N=1e5;
int n,m;
int tot=0;
int pm[N]={0,1595,3190,4783,9476,11070,12663,14257,15851,17444,19038,20632,22224,23817,28508,30101,31695,33290,34882,36475,38071,39664,41258,42852,47544,49136,50731,52324,53916,55511,57103,58695,90954,92549,97239,98833,100427,102020,103614,105208,106801,108395,109989,111581,116272,117866,119459,121053,122648,124240,125833,127429,129022,130616,135306,136900,138492,140087,141680,143272,144867,146459,148051,149646,154339,155932,157526,159120,160713,162307,163901,165494,167088,168682,173373,174966,176560,178153,179747,181342,182934,184527,186123,187716,192407,194001,195595,197187,198782,200375,201967,203562,205154,206746,211438,213033,214626,216220,217814,219407,221001,222595,224188,225782,230473,232065,233658,235252,236845,238439,240034,241626,243219,244815,249507,251101,252695,254289,255881,257476,259069,260661,262256,263848,268537,270132,271727,273320,274914,276508,278101,279695,311953,313546,318238,319832,321424,323017,324611,326204,327798,329393,330985,332578,337273,338866,340460,342054,343648,345240,346835,348428,350020,351615,356304,357896,359491,361086,362679,364273,365867,367460,369054,370648,375338,376932,378526,380118,381711,383305,384898,386492,388087,389679,394370,395966,397559,399153,400747,402341,403933,405528,407121,408713,413406,414998,416590,418185,419780,421373,422967,424561,426154,427748,432440,434033,435627,437221,438813,440406,442000,443593,445187,446782,451471,453064,454660,456253,457847,459441,461035,462627,464222,465815,470506,472101,473693,475285,476880,478475,480068,481662,483256,484849,489540,491134,492727,494321,495915,497507,499100,500694,532952,534546,539239,540831,542424,544020,545613,547207,548801,550395,551987,553582,558272,559864,561459,563051,564643,566238,567833,569426,571020,572614,577306,578900,580494,582087,583681,585275,586867,588460,590054,591647,596339,597934,599526,601119,602715,604308,605902,607496,609090,610682,615375,616968,618560,620155,621747,623339,624934,626529,628122,629716,634408,636001,637595,639189,640782,642376,643970,645562,647155,648749,653439,655033,656628,658220,659813,661409,663002,664596,666190,667784,672474,674069,675662,677254,678849,680441,682033,714292,746550,778808,811066,843324,875582,907840,940098,972356,1004614,1006208,1007800,1009393,1014084,1015677,1017271,1018866,1020458,1022051,1023647,1025240,1057498,1059092,1063784,1065376,1066971,1068564,1070156,1071751,1073343,1074935,1076530,1078125,1082815,1084409,1086003,1087596,1089190,1090784,1092377,1093971,1095565,1097157,1101848,1103442,1105035,1106629,1108224,1109816,1111409,1113005,1114598,1116192,1120882,1122476,1124068,1125663,1127256,1128848,1130443,1132035,1133627,1135222,1139915,1141508,1143102,1144696,1146289,1147883,1149477,1151070,1152664,1154258,1158949,1160542,1162136,1163729,1165323,1166918,1168510,1170103,1171699,1173292,1177983,1179577,1181171,1182763,1184358,1185951,1187543,1189138,1190730,1192322,1197014,1198609,1200202,1201796,1203390,1204983,1206577,1208171,1209764,1211358,1216049,1217641,1219234,1220828,1222421,1224015,1225610,1227202,1228795,1230391,1235083,1236677,1238271,1239865,1241457,1243052,1244645,1246237,1278495,1280087,1284776,1286371,1287966,1289559,1291153,1292747,1294340,1295934,1297528,1299121,1303813,1305407,1306999,1308592,1310186,1311779,1313373,1314968,1316560,1318153,1322848,1324441,1326035,1327629,1329223,1330815,1332410,1334003,1335595,1337190,1341879,1343471,1345066,1346661,1348254,1349848,1351442,1353035,1354629,1356223,1360913,1362507,1364101,1365693,1367286,1368880,1370473,1372067,1373662,1375254,1379945,1381541,1383134,1384728,1386322,1387916,1389508,1391103,1392696,1394288,1398981,1400573,1402165,1403760,1405355,1406948,1408542,1410136,1411729,1413323,1418015,1419608,1421202,1422796,1424388,1425981,1427575,1429168,1430762,1432357,1437046,1438639,1440235,1441828,1443422,1445016,1446610,1448202,1449797,1451390,1456081,1457676,1459268,1460860,1462455,1464050,1465643,1467237,1499495,1501088,1505779,1507373,1508966,1510560,1512154,1513746,1515339,1516933,1518526,1520120,1524813,1526405,1527998,1529594,1531187,1532781,1534375,1535969,1537561,1539156,1543846,1545438,1547033,1548625,1550217,1551812,1553407,1555000,1556594,1558188,1562880,1564474,1566068,1567661,1569255,1570849,1572441,1574034,1575628,1577221,1581913,1583508,1585100,1586693,1588289,1589882,1591476,1593070,1594664,1596256,1600949,1602542,1604134,1605729,1607321,1608913,1610508,1612103,1613696,1615290,1619982,1621575,1623169,1624763,1626356,1627950,1629544,1631136,1632729,1634323,1639013,1640607,1642202,1643794,1645387,1646983,1648576,1650170,1651764,1653358,1658048,1659643,1661236,1662828,1664423,1666015,1667607,1669202,1670797,1672390,1677083,1678677,1680270,1681864,1683458,1685051,1686645,1688239,1720497,1722090,1726781,1728374,1729968,1731563,1733155,1734748,1736344,1737937,1739531,1741125,1745817,1747409,1749004,1750597,1752189,1753784,1755376,1756968,1758563,1760158,1764848,1766442,1768036,1769629,1771223,1772817,1774410,1776004,1777598,1779190,1783881,1785475,1787068,1788662,1790257,1791849,1793442,1795038,1796631,1798225,1802915,1804509,1806101,1807696,1809289,1810881,1812476,1814068,1815660,1817255,1821948,1823541,1825135,1826729,1828322,1829916,1831510,1833103,1834697,1836291,1840982,1842575,1844169,1845762,1847356,1848951,1850543,1852136,1853732,1855325,1860016,1861610,1863204,1864796,1866391,1867984,1869576,1871171,1872763,1874355,1879047,1880642,1882235,1883829,1885423,1887016,1888610,1890204,1891797,1893391,1898082,1899674,1901267,1902861,1904454,1906048,1907643,1909235,1941493,1943089,1947781,1949375,1950969,1952563,1954155,1955750,1957343,1958935,1960530,1962122,1966811,1968406,1970001,1971594,1973188,1974782,1976375,1977969,1979563,1981156,1985848,1987442,1989034,1990627,1992221,1993814,1995408,1997003,1998595,2000188,2004883,2006476,2008070,2009664,2011258,2012850,2014445,2016038,2017630,2019225,2023914,2025506,2027101,2028696,2030289,2031883,2033477,2035070,2036664,2038258,2042948,2044542,2046136,2047728,2049321,2050915,2052508,2054102,2055697,2057289,2061980,2063576,2065169,2066763,2068357,2069951,2071543,2073138,2074731,2076323,2081016,2082608,2084200,2085795,2087390,2088983,2090577,2092171,2093764,2095358,2100050,2101643,2103237,2104831,2106423,2108016,2109610,2111203,2112797,2114392,2119081,2120674,2122270,2123863,2125457,2127051,2128645,2130237,2162495,2164088,2168779,2170374,2171966,2173558,2175153,2176748,2178341,2179935,2181529,2183122,2187813,2189407,2191000,2192594,2194188,2195780,2197373,2198967,2200560,2202154,2206847,2208439,2210032,2211628,2213221,2214815,2216409,2218003,2219595,2221190,2225880,2227472,2229067,2230659,2232251,2233846,2235441,2237034,2238628,2240222,2244914,2246508,2248102,2249695,2251289,2252883,2254475,2256068,2257662,2259255,2263947,2265542,2267134,2268727,2270323,2271916,2273510,2275104,2276698,2278290,2282983,2284576,2286168,2287763,2289355,2290947,2292542,2294137,2295730,2297324,2302016,2303609,2305203,2306797,2308390,2309984,2311578,2313170,2314763,2316357,2321047,2322641,2324236,2325828,2327421,2329017,2330610,2332204,2333798,2335392,2340082,2341677,2343270,2344862,2346457,2348049,2349641,2351236,2383494,2385087,2389780,2391374,2392967,2394561,2396155,2397748,2399342,2400936,2402528,2404121,2408812,2410405,2411999,2413594,2415186,2416779,2418375,2419968,2421562,2423156,2427848,2429440,2431035,2432628,2434220,2435815,2437407,2438999,2440594,2442189,2446879,2448473,2450067,2451660,2453254,2454848,2456441,2458035,2459629,2461221,2465912,2467506,2469099,2470693,2472288,2473880,2475473,2477069,2478662,2480256,2484946,2486540,2488132,2489727,2491320,2492912,2494507,2496099,2497691,2499286,2503979,2505572,2507166,2508760,2510353,2511947,2513541,};
bool check(int x)
{
int top=0;
int tmp=0;
int flag=0;
while(x>0) {
tmp=x%10;
x=x/10;
if(tmp==1)
flag=1;
else if(tmp==3) {
if(flag==1) return true;
else flag=0;
}
else flag=0;
}
return false;
}
void prime()
{
int i,j;
for(i=m;i<=n;i+=31) {
if(check(i))
tot++;
}
return;
}
int main()
{
freopen("lucky.in","r",stdin);
freopen("lucky.out","w",stdout);
cin>>n;
tot+=pm[(n/1000000)];
m=(n/1000000)*1000000;
while(m%31!=0) m++;
prime();
cout<<tot;
return 0;
}
T2 TOF函数
题目描述
一个十进制数正整数x表示为
（Tn代表x的最高位的值，以此类推，T1代表
的最低位的值），定义
的TOF函数的值为：
=Tn * 2n-1 + Tn-1 * 2n-2 + … + T2 * 21 + T1 * 20.
现在给你两个数E，P(0 <= E,P < 109)，请计算[0,P]范围内有多少个数的TOF函数的值**<=TOF(E)**。
输入：
第1行为两个个正整数两个正整数代表E,P
输出：
一个整数代表对应结果
样例输入：
5 100
样例输出：
13
思路
- 如何描述状态？
- 跟上一题一样，第一维描述位数，第二维描述该位数上所填的数，
- 那如何保证所有状态的tof都小于等于tof（E）呢？
- 因为
.所以
,显然可以开第三位的空间来描述。
综上
int f[N][N][20000]; //第i位上放j,该位和其后的位组成的数tof小于等于k的个数
- 如何转移？
- 第i位上放j小于等于Max的数，那么第i-1位上无论放什么都要 ，于是得出状转方程
LL res=0;
int maxn=(lit) ? dim[now-1] : 9;
int sufMax=Max-pow2[now-1]*sta;
for(int i=0;i<=maxn;i++)
res+=dfs(now-1,i,sufMax,(i==maxn&&lit));
- 边界条件？
- 第0位上只能放0，所以tof=0，如果要求的 ,则是一种方案，否则不行。
if(now==0) return (sta==0&&Max>=0) ? 1 : 0;
Code：
#include<cstdio>
#include<cstring>
#include<iostream>
#define LL long long
using namespace std;
const int N=12;
int dim[N];
int f[N][N][20000]; //第i位上放j,tof小于等于k的个数
const int pow2[]={1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768};
int all;
LL dfs(int now,int sta,int Max,bool lit)
{
if(now==0) return (sta==0&&Max>=0) ? 1 : 0;
if(!lit&&f[now][sta][Max]!=-1)
return f[now][sta][Max];
LL res=0;
int maxn=(lit) ? dim[now-1] : 9;
int sufMax=Max-pow2[now-1]*sta;
for(int i=0;i<=maxn;i++)
res+=dfs(now-1,i,sufMax,(i==maxn&&lit));
if(!lit) f[now][sta][Max]=res;
return res;
}
LL prefix(LL x)
{
memset(dim,0,sizeof dim);
int top=0;
while(x>0) {
dim[++top]=x%10;
x=x/10;
}
LL res=0;
for(int i=0;i<=dim[top];i++)
res+=dfs(top,i,all,i==dim[top]);
return res;
}
inline int tof(int x)
{
int res=0;
int i,j;
i=0;
while(x>0) {
res+=pow2[i]*(x%10);
i++;
x/=10;
}
return res;
}
int main()
{
freopen("tof.in","r",stdin);
freopen("tof.out","w",stdout);
int E,P;
cin>>E>>P;
memset(f,-1,sizeof f);
all=tof(E);
cout<<prefix(P);
return 0;
}
