Gridland Provinces


Problem Statement :


The Kingdom of Gridland contains  provinces. Each province is defined as a  grid where each cell in the grid represents a city. Every cell in the grid contains a single lowercase character denoting the first character of the city name corresponding to that cell.

From a city with the coordinates , it is possible to move to any of the following cells in  unit of time (provided that the destination cell is within the confines of the grid):

A knight wants to visit all the cities in Gridland. He can start his journey in any city and immediately stops his journey after having visited each city at least once. Moreover, he always plans his journey in such a way that the total time required to complete it is minimum.

After completing his tour of each province, the knight forms a string by concatenating the characters of all the cells in his path. How many distinct strings can he form in each province?

Input Format

The first line contains a single integer, , denoting the number of provinces. The  subsequent lines describe each province over the following three lines:
The first line contains an integer, , denoting the number of columns in the province.
Each of the next two lines contains a string, , of length  denoting the characters for the first and second row of the province.

Constraints

1  <=  P  <=  15
1  <=  N  <=  600
Si E { a - z }

Output Format

For each province, print the number of distinct strings the knight can form on a new line.


Solution :



title-img


                            Solution in C :

In   C++  :









#include <bits/stdc++.h>

using namespace std;
#define PB push_back
#define MP make_pair
#define LL long long
#define int LL
#define FOR(i,a,b) for(int i = (a); i <= (b); i++)
#define RE(i,n) FOR(i,1,n)
#define REP(i,n) FOR(i,0,(int)(n)-1)
#define R(i,n) REP(i,n)
#define VI vector<int>
#define PII pair<int,int>
#define LD long double
#define FI first
#define SE second
#define st FI
#define nd SE
#define ALL(x) (x).begin(), (x).end()
#define SZ(x) ((int)(x).size())

template<class C> void mini(C& _a4, C _b4) { _a4 = min(_a4, _b4); }
template<class C> void maxi(C& _a4, C _b4) { _a4 = max(_a4, _b4); }

template<class TH> void _dbg(const char *sdbg, TH h){ cerr<<sdbg<<'='<<h<<endl; }
template<class TH, class... TA> void _dbg(const char *sdbg, TH h, TA... a) {
  while(*sdbg!=',')cerr<<*sdbg++;cerr<<'='<<h<<','; _dbg(sdbg+1, a...);
}

template<class T> ostream& operator<<(ostream& os, vector<T> V) {
  os << "["; for (auto& vv : V) os << vv << ","; os << "]";
  return os;
}

#ifdef LOCAL
#define debug(...) _dbg(#__VA_ARGS__, __VA_ARGS__)
#else
#define debug(...) (__VA_ARGS__)
#define cerr if(0)cout
#endif

const int MAX = 611;
int n;
string t[2];
int P = 1e9 + 9;
int pp[MAX*2];
int hp[2][MAX];
int hs[2][MAX];
int MOD = (1ll<<61) - 1;
int razy(__int128 a, __int128 b){
  return (a * b) % MOD;
}
struct hh{
  vector<int> h;
  hh(){h.PB(0);}
  void add(char z){
    h.PB(h.back() + razy(z, pp[SZ(h)-1]));
    h.back() %= MOD;
  }
  int daj(int i,int j){
    return (h[2*j] - h[2*i+2] + MOD) % MOD;
  }
};
void test(){
  cin >> n >> t[0] >> t[1];
  unordered_set<int> secik;
  R(_,2){
    hh z[2];
    R(i,n){
      int pom = i&1;
      z[0].add(t[pom][i]);
      z[0].add(t[!pom][i]);
      z[1].add(t[!pom][i]);
      z[1].add(t[pom][i]);
    }
    R(i,n){
      R(j,2){
        hp[j][i+1] = razy(hp[j][i], P) + t[j][i] + razy(t[!j][i], pp[2*i+1]);
        hp[j][i+1] %= MOD;
      }
      R(j,2){
        hs[j][i+1] = razy(hs[j][i], P) + t[j][n-i-1] + razy(t[!j][n-i-1], pp[2*i+1]);
        hs[j][i+1] %= MOD;
      }
    }
    R(i,n)R(k,2){
      hs[k][n-i] = razy(hs[k][n-i], pp[2 * i]);
    }
    R(k,2)
      secik.insert(hs[k][n]);
    R(j,n)R(i,j)R(k,2){
      int hash = hp[k][i+1];
      hash += z[k ^ (i&1)].daj(i,j);
      hash += hs[k ^ (i&1) ^ (j&1)][n-j];
      secik.insert(hash%MOD);
    }
    reverse(ALL(t[0]));
    reverse(ALL(t[1]));
  }
  cout << SZ(secik) << "\n";
}

int32_t main() {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout << fixed << setprecision(11);
  cerr << fixed << setprecision(6);
  pp[0] = 1;
  R(i,MAX*2-1)pp[i+1] = razy(pp[i], P);
  int t;cin >> t;while(t--)test();
}
  








In   Java :









import java.util.HashSet;
import java.util.Scanner;
import java.util.Set;

public class Solution2 {
private final static Scanner scanner = 
new Scanner(System.in);
private final static long mod1 = 2147483607,
 f1 = 107, f2 = 101;
private final static long[] arr1 = new long[10000], 
arr2 = new long[10000]; 
private final static Set<Long> result = new HashSet<>();
public static void main(String[] args) {
for (int i = 0; i < arr1.length; ++i) {
arr1[i] = i > 0 ? arr1[i - 1] * f1 % mod1 : 1;
arr2[i] = i > 0 ? arr2[i - 1] * f2 % mod1 : 1;
}

for (int t = scanner.nextInt(); t > 0; --t) {
result.clear();
scanner.nextInt();
char[] c1 = scanner.next().toCharArray();
char[] c2 = scanner.next().toCharArray();

for (int i = 0; i < c1.length; ++i) {
process(c1, c2, i, false);
process(c2, c1, i, false); 
process(c1, c2, i, true);
process(c2, c1, i, true);
}
reverse(c1);
reverse(c2);
for (int i = 0; i < c1.length; ++i) {
process(c1, c2, i, false);
process(c2, c1, i, false); 
process(c1, c2, i, true);
process(c2, c1, i, true);
}
System.out.println(result.size());
}
}

static void process(char[] s1, char[] s2, int k, boolean b) {
long p1 = 0, p2 = 0, p3 = 0, p4 = 0;
for (int i = 0; i < k; ++i) {
p1 = (p1 + s1[i] * arr1[k - 1 - i]) % mod1;
p1 = (p1 + s2[i] * arr1[k + i]) % mod1;
p3 = (p3 + s1[i] * arr2[k - 1 - i]) % mod1;
p3 = (p3 + s2[i] * arr2[k + i]) % mod1;
}
if (b) {
p1 = (p1 + s2[k] * arr1[k * 2]) % mod1;
p1 = (p1 + s1[k] * arr1[k * 2 + 1]) % mod1;
p3 = (p3 + s2[k] * arr2[k * 2]) % mod1;
p3 = (p3 + s1[k] * arr2[k * 2 + 1]) % mod1;
char[] s = s1; s1 = s2; s2 = s;
++k;
}
for (int i = k; i < s1.length; ++i) {
p2 = (p2 + s1[i] * arr1[s1.length * 2 + k - 1 - i]) % mod1;
p2 = (p2 + s2[i] * arr1[i + k]) % mod1;
p4 = (p4 + s1[i] * arr2[s1.length * 2 + k - 1 - i]) % mod1;
p4 = (p4 + s2[i] * arr2[i + k]) % mod1;
}
result.add(((p1 + p2) % mod1) * mod1 + (p3 + p4) % mod1);

for (int i = k; i < s1.length - 1; i += 2) {
p1 = (p1 + s2[i] * arr1[i * 2]) % mod1;
p1 = (p1 + s1[i] * arr1[i * 2 + 1]) % mod1;
p1 = (p1 + s1[i + 1] * arr1[i * 2 + 2]) % mod1;
p1 = (p1 + s2[i + 1] * arr1[i * 2 + 3]) % mod1;
p2 = (p2 + s2[i] * (mod1 - arr1[i * 2])) % mod1;
p2 = (p2 + s2[i+1] * (mod1 - arr1[i * 2 + 1])) % mod1;
p2 = (p2 + s1[i] * (mod1 - arr1[s1.length * 2 - 1])) % mod1;
p2 = (p2 + s1[i+1] * (mod1 - arr1[s1.length * 2 - 2])) % mod1;
p2 = (p2 * f1 * f1) % mod1;

p3 = (p3 + s2[i] * arr2[i * 2]) % mod1;
p3 = (p3 + s1[i] * arr2[i * 2 + 1]) % mod1;
p3 = (p3 + s1[i + 1] * arr2[i * 2 + 2]) % mod1;
p3 = (p3 + s2[i + 1] * arr2[i * 2 + 3]) % mod1;
p4 = (p4 + s2[i] * (mod1 - arr2[i * 2])) % mod1;
p4 = (p4 + s2[i+1] * (mod1 - arr2[i * 2 + 1])) % mod1;
p4 = (p4 + s1[i] * (mod1 - arr2[s1.length * 2 - 1])) % mod1;
p4 = (p4 + s1[i+1] * (mod1 - arr2[s1.length * 2 - 2])) % mod1;
p4 = (p4 * f2 * f2) % mod1;

result.add(((p1 + p2) % mod1) * mod1 + (p3 + p4) % mod1);
}
}

private static void reverse(char[] str) {
for (int i = str.length / 2 - 1; i >= 0; --i) {
char t = str[i];
str[i] = str[str.length - 1 - i];
str[str.length - 1 - i] = t;
}
}

}










In   C   :








#include <stdio.h>
#include <stdlib.h>
#define MOD1 1000000007
#define MOD2 1000000009
#define HASH_SIZE 123455
typedef struct _node{
int x;
int y;
struct _node *next;
} node;
void solve(int i,int j);
void insert(int x,int y);
void freehash();
long long modInverse(long long a,long long mod);
char a[2][601];
int hash_count,N;
long long tr1[1200],tl1[1200],br1[1200],
bl1[1200],dr1[1200],dl1[1200],ur1[1200],
ul1[1200],mod1[1201],inmod1[1201];
long long tr2[1200],tl2[1200],br2[1200],
bl2[1200],dr2[1200],dl2[1200],ur2[1200],
ul2[1200],mod2[1201],inmod2[1201];
node *hash[HASH_SIZE]={0};

int main(){
int T,i,j;
long long t1,t2;
scanf("%d",&T);
while(T--){
hash_count=0;
scanf("%d%s%s",&N,&a[0][0],&a[1][0]);
for(i=0,t1=t2=1;i<N;i++,t1=t1*26%MOD1,t2=t2*26%MOD2)
if(i){
tl1[i]=((a[0][i]-'a')*t1+tl1[i-1])%MOD1;
bl1[i]=((a[1][i]-'a')*t1+bl1[i-1])%MOD1;
tl2[i]=((a[0][i]-'a')*t2+tl2[i-1])%MOD2;
bl2[i]=((a[1][i]-'a')*t2+bl2[i-1])%MOD2;
}
else{
tl1[i]=a[0][i]-'a';
bl1[i]=a[1][i]-'a';
tl2[i]=a[0][i]-'a';
bl2[i]=a[1][i]-'a';
}
for(i=N-1;i>=0;i--,t1=t1*26%MOD1,t2=t2*26%MOD2){
tl1[2*N-i-1]=((a[1][i]-'a')*t1+tl1[2*N-i-2])%MOD1;
bl1[2*N-i-1]=((a[0][i]-'a')*t1+bl1[2*N-i-2])%MOD1;
tl2[2*N-i-1]=((a[1][i]-'a')*t2+tl2[2*N-i-2])%MOD2;
bl2[2*N-i-1]=((a[0][i]-'a')*t2+bl2[2*N-i-2])%MOD2;
}
for(i=N-1,t1=t2=1;i>=0;i--,t1=t1*26%MOD1,t2=t2*26%MOD2)
if(i!=N-1){
tr1[N-i-1]=((a[0][i]-'a')*t1+tr1[N-i-2])%MOD1;
br1[N-i-1]=((a[1][i]-'a')*t1+br1[N-i-2])%MOD1;
tr2[N-i-1]=((a[0][i]-'a')*t2+tr2[N-i-2])%MOD2;
br2[N-i-1]=((a[1][i]-'a')*t2+br2[N-i-2])%MOD2;
}
else{
tr1[N-i-1]=a[0][i]-'a';
br1[N-i-1]=a[1][i]-'a';
tr2[N-i-1]=a[0][i]-'a';
br2[N-i-1]=a[1][i]-'a';
}
for(i=0;i<N;i++,t1=t1*26%MOD1,t2=t2*26%MOD2){
tr1[i+N]=((a[1][i]-'a')*t1+tr1[i+N-1])%MOD1;
br1[i+N]=((a[0][i]-'a')*t1+br1[i+N-1])%MOD1;
tr2[i+N]=((a[1][i]-'a')*t2+tr2[i+N-1])%MOD2;
br2[i+N]=((a[0][i]-'a')*t2+br2[i+N-1])%MOD2;
}
for(i=0,t1=t2=1;i<N;i++){
if(i%2){
ul1[2*i]=((a[0][i]-'a')*t1+ul1[2*i-1])%MOD1;
dl1[2*i]=((a[1][i]-'a')*t1+dl1[2*i-1])%MOD1;
ul2[2*i]=((a[0][i]-'a')*t2+ul2[2*i-1])%MOD2;
dl2[2*i]=((a[1][i]-'a')*t2+dl2[2*i-1])%MOD2;
}
else
if(!i){
ul1[2*i]=a[1][i]-'a';
dl1[2*i]=a[0][i]-'a';
ul2[2*i]=a[1][i]-'a';
dl2[2*i]=a[0][i]-'a';
}
else{
ul1[2*i]=((a[1][i]-'a')*t1+ul1[2*i-1])%MOD1;
dl1[2*i]=((a[0][i]-'a')*t1+dl1[2*i-1])%MOD1;
ul2[2*i]=((a[1][i]-'a')*t2+ul2[2*i-1])%MOD2;
dl2[2*i]=((a[0][i]-'a')*t2+dl2[2*i-1])%MOD2;
}
t1=t1*26%MOD1;
t2=t2*26%MOD2;
if(i%2){
ul1[2*i+1]=((a[1][i]-'a')*t1+ul1[2*i])%MOD1;
dl1[2*i+1]=((a[0][i]-'a')*t1+dl1[2*i])%MOD1;
ul2[2*i+1]=((a[1][i]-'a')*t2+ul2[2*i])%MOD2;
dl2[2*i+1]=((a[0][i]-'a')*t2+dl2[2*i])%MOD2;
}
else{
ul1[2*i+1]=((a[0][i]-'a')*t1+ul1[2*i])%MOD1;
dl1[2*i+1]=((a[1][i]-'a')*t1+dl1[2*i])%MOD1;
ul2[2*i+1]=((a[0][i]-'a')*t2+ul2[2*i])%MOD2;
dl2[2*i+1]=((a[1][i]-'a')*t2+dl2[2*i])%MOD2;
}
t1=t1*26%MOD1;
t2=t2*26%MOD2;
}
for(i=N-1,t1=t2=1;i>=0;i--)
if(i==N-1){
ur1[2*(N-1-i)]=a[1][i]-'a';
dr1[2*(N-1-i)]=a[0][i]-'a';
ur2[2*(N-1-i)]=a[1][i]-'a';
dr2[2*(N-1-i)]=a[0][i]-'a';
t1=t1*26%MOD1;
t2=t2*26%MOD2;
ur1[2*(N-1-i)+1]=((a[0][i]-'a')*t1+ur1[2*(N-1-i)])%MOD1;
dr1[2*(N-1-i)+1]=((a[1][i]-'a')*t1+dr1[2*(N-1-i)])%MOD1;
ur2[2*(N-1-i)+1]=((a[0][i]-'a')*t2+ur2[2*(N-1-i)])%MOD2;
dr2[2*(N-1-i)+1]=((a[1][i]-'a')*t2+dr2[2*(N-1-i)])%MOD2;
t1=t1*26%MOD1;
t2=t2*26%MOD2;
}
else{
if((N-i)%2==0){
ur1[2*(N-1-i)]=((a[0][i]-'a')*t1+ur1[2*(N-1-i)-1])%MOD1;
dr1[2*(N-1-i)]=((a[1][i]-'a')*t1+dr1[2*(N-1-i)-1])%MOD1;
ur2[2*(N-1-i)]=((a[0][i]-'a')*t2+ur2[2*(N-1-i)-1])%MOD2;
dr2[2*(N-1-i)]=((a[1][i]-'a')*t2+dr2[2*(N-1-i)-1])%MOD2;
}
else{
ur1[2*(N-1-i)]=((a[1][i]-'a')*t1+ur1[2*(N-1-i)-1])%MOD1;
dr1[2*(N-1-i)]=((a[0][i]-'a')*t1+dr1[2*(N-1-i)-1])%MOD1;
ur2[2*(N-1-i)]=((a[1][i]-'a')*t2+ur2[2*(N-1-i)-1])%MOD2;
dr2[2*(N-1-i)]=((a[0][i]-'a')*t2+dr2[2*(N-1-i)-1])%MOD2;
}
t1=t1*26%MOD1;
t2=t2*26%MOD2;
if((N-i)%2==0){
ur1[2*(N-1-i)+1]=((a[1][i]-'a')*t1+ur1[2*(N-1-i)])%MOD1;
dr1[2*(N-1-i)+1]=((a[0][i]-'a')*t1+dr1[2*(N-1-i)])%MOD1;
ur2[2*(N-1-i)+1]=((a[1][i]-'a')*t2+ur2[2*(N-1-i)])%MOD2;
dr2[2*(N-1-i)+1]=((a[0][i]-'a')*t2+dr2[2*(N-1-i)])%MOD2;
}
else{
ur1[2*(N-1-i)+1]=((a[0][i]-'a')*t1+ur1[2*(N-1-i)])%MOD1;
dr1[2*(N-1-i)+1]=((a[1][i]-'a')*t1+dr1[2*(N-1-i)])%MOD1;
ur2[2*(N-1-i)+1]=((a[0][i]-'a')*t2+ur2[2*(N-1-i)])%MOD2;
dr2[2*(N-1-i)+1]=((a[1][i]-'a')*t2+dr2[2*(N-1-i)])%MOD2;
}
t1=t1*26%MOD1;
t2=t2*26%MOD2;
}
for(i=0;i<=2*N;i++)
if(i){
mod1[i]=mod1[i-1]*26%MOD1;
inmod1[i]=modInverse(mod1[i],MOD1);
mod2[i]=mod2[i-1]*26%MOD2;
inmod2[i]=modInverse(mod2[i],MOD2);
}
else
mod1[i]=inmod1[i]=mod2[i]=inmod2[i]=1;
for(i=0;i<=N;i++)
for(j=i;j<=N;j++)
solve(i,j);
printf("%d\n",hash_count);
freehash();
}
return 0;
}
void solve(int i,int j){
long long t1,t2,t3,t4,t5,t6,t7,t8,t9;
long long tt1,tt2,tt3,tt4,tt5,tt6,tt7,tt8,tt9;
t1=tr1[N+i-1];
t2=br1[N+i-1];
if(i!=N){
t1=(t1-tr1[N-i-1]+MOD1)%MOD1;
t2=(t2-br1[N-i-1]+MOD1)%MOD1;
}
t1=t1*inmod1[N-i]%MOD1;
t2=t2*inmod1[N-i]%MOD1;
t3=tl1[2*N-j-1];
t4=bl1[2*N-j-1];
if(j){
t3=(t3-tl1[j-1]+MOD1)%MOD1;
t4=(t4-bl1[j-1]+MOD1)%MOD1;
}
t3=t3*inmod1[j]%MOD1;
t4=t4*inmod1[j]%MOD1;
if(!j)
t5=t6=0;
else{
t5=ul1[2*j-1];
t6=dl1[2*j-1];
if(i){
t5=(t5-ul1[2*i-1]+MOD1)%MOD1;
t6=(t6-dl1[2*i-1]+MOD1)%MOD1;
}
}
if(i==N)
t7=t8=0;
else{
t7=ur1[2*(N-i)-1];
t8=dr1[2*(N-i)-1];
if(j!=N){
t7=(t7-ur1[2*(N-j)-1]+MOD1)%MOD1;
t8=(t8-dr1[2*(N-j)-1]+MOD1)%MOD1;
}
}
tt1=tr2[N+i-1];
tt2=br2[N+i-1];
if(i!=N){
tt1=(tt1-tr2[N-i-1]+MOD2)%MOD2;
tt2=(tt2-br2[N-i-1]+MOD2)%MOD2;
}
tt1=tt1*inmod2[N-i]%MOD2;
tt2=tt2*inmod2[N-i]%MOD2;
tt3=tl2[2*N-j-1];
tt4=bl2[2*N-j-1];
if(j){
tt3=(tt3-tl2[j-1]+MOD2)%MOD2;
tt4=(tt4-bl2[j-1]+MOD2)%MOD2;
}
tt3=tt3*inmod2[j]%MOD2;
tt4=tt4*inmod2[j]%MOD2;
if(!j)
tt5=tt6=0;
else{
tt5=ul2[2*j-1];
tt6=dl2[2*j-1];
if(i){
tt5=(tt5-ul2[2*i-1]+MOD2)%MOD2;
tt6=(tt6-dl2[2*i-1]+MOD2)%MOD2;
}
}
if(i==N)
tt7=tt8=0;
else{
tt7=ur2[2*(N-i)-1];
tt8=dr2[2*(N-i)-1];
if(j!=N){
tt7=(tt7-ur2[2*(N-j)-1]+MOD2)%MOD2;
tt8=(tt8-dr2[2*(N-j)-1]+MOD2)%MOD2;
}
}
t9=t1;
if(i%2)
t9+=t6;
else
t9+=t5;
if((j-i)%2)
t9+=t3*mod1[j*2]%MOD1;
else
t9+=t4*mod1[j*2]%MOD1;
t9%=MOD1;
tt9=tt1;
if(i%2)
tt9+=tt6;
else
tt9+=tt5;
if((j-i)%2)
tt9+=tt3*mod2[j*2]%MOD2;
else
tt9+=tt4*mod2[j*2]%MOD2;
tt9%=MOD2;
insert(t9,tt9);
t9=t2;
if(i%2)
t9+=t5;
else
t9+=t6;
if((j-i)%2)
t9+=t4*mod1[j*2]%MOD1;
else
t9+=t3*mod1[j*2]%MOD1;
t9%=MOD1;
tt9=tt2;
if(i%2)
tt9+=tt5;
else
tt9+=tt6;
if((j-i)%2)
tt9+=tt4*mod2[j*2]%MOD2;
else
tt9+=tt3*mod2[j*2]%MOD2;
tt9%=MOD2;
insert(t9,tt9);
t9=t3;
if((N-j)%2)
t9+=t8;
else
t9+=t7;
if((j-i)%2)
t9+=t1*mod1[(N-i)*2]%MOD1;
else
t9+=t2*mod1[(N-i)*2]%MOD1;
t9%=MOD1;
tt9=tt3;
if((N-j)%2)
tt9+=tt8;
else
tt9+=tt7;
if((j-i)%2)
tt9+=tt1*mod2[(N-i)*2]%MOD2;
else
tt9+=tt2*mod2[(N-i)*2]%MOD2;
tt9%=MOD2;
insert(t9,tt9);
t9=t4;
if((N-j)%2)
t9+=t7;
else
t9+=t8;
if((j-i)%2)
t9+=t2*mod1[(N-i)*2]%MOD1;
else
t9+=t1*mod1[(N-i)*2]%MOD1;
t9%=MOD1;
tt9=tt4;
if((N-j)%2)
tt9+=tt7;
else
tt9+=tt8;
if((j-i)%2)
tt9+=tt2*mod2[(N-i)*2]%MOD2;
else
tt9+=tt1*mod2[(N-i)*2]%MOD2;
tt9%=MOD2;
insert(t9,tt9);
return;
}
void insert(int x,int y){
int bucket=(x+y)%HASH_SIZE;
node *t=hash[bucket];
while(t){
if(t->x==x && t->y==y)
return;
t=t->next;
}
t=(node*)malloc(sizeof(node));
t->x=x;
t->y=y;
t->next=hash[bucket];
hash[bucket]=t;
hash_count++;
return;
}
void freehash(){
int i;
node *t,*p;
for(i=0;i<HASH_SIZE;i++){
t=hash[i];
while(t){
p=t->next;
free(t);
t=p;
}
hash[i]=NULL;
}
return;
}
long long modInverse(long long a,long long mod){
long long b0 = mod, t, q;
long long x0 = 0, x1 = 1;
while (a > 1) {
q = a / mod;
t = mod; mod = a % mod; a = t;
t = x0; x0 = x1 - q * x0; x1 = t;
}
if (x1 < 0) x1 += b0;
return x1;
}









In   Python3  :








class Solution:
    def __init__(self):
        self.size = int(input())
        self.array = []
        self.array.append(input().strip())
        self.array.append(input().strip())
        self.hash = set()

    def add_to_hash(self, results):
        for set_str in results:
            self.hash.add(hash(set_str))
            self.hash.add(hash(set_str[::-1]))

    def calculate(self):
        if len(set(self.array[0]+self.array[1]))==1:
            return 1
        results = self.get_circles(self.array[0]+self.array[1][::-1])
        full_string1 = self.array[1][::-1]+self.array[0]
        full_string2 = self.array[0][::-1]+self.array[1]
        full_zigzags=self.get_zigzag('',1,0,+1)
        self.add_to_hash(results)
        results=set(full_zigzags.keys())
        for i in range(1,self.size):
            for zig,diverge in full_zigzags.items():
                if diverge < i:
                    continue
                j = self.size -i
                if i%2 == 0:
                    new_str = full_string2[j:-j]+zig[i*2:]
                else:
                    new_str = full_string1[j:-j]+zig[i*2:]
                results.add(new_str)

        self.add_to_hash(results)
        full_zigzags=self.get_zigzag('',0,0,+1)
        results=set(full_zigzags.keys())
        for i in range(1,self.size):
            for zig,diverge in full_zigzags.items():
                if diverge < i:
                    continue
                j = self.size -i
                if i%2 == 0:
                    new_str = full_string1[j:-j]+zig[i*2:]
                else:
                    new_str = full_string2[j:-j]+zig[i*2:]
                results.add(new_str)
        self.add_to_hash(results)
        return len(self.hash)

    def get_circles(self,loop):
        circles = set()
        circles.add(loop)
        for i in range(self.size*2-1):
            loop = loop[1:]+loop[0]
            circles.add(loop)
        return circles

    def get_zigzag(self,seed,row,col,right):
        output = list(seed)
        steps = 0
        zigzags = {}
        while col >=0 and col <self.size:
            output.append(self.array[row][col])
            steps+=1
            row = 0 if row == 1 else 1
            if steps < self.size*2:
                zigzags[self.add_circle_diversion(row,col,right,''.join(output))]=steps//2
            output.append(self.array[row][col])
            steps+=1
            col+=right
        zigzags[''.join(output)]=self.size
        return zigzags

    def add_circle_diversion(self, row, col, right, built_str):
        built_str+=self.array[row][col::right]
        remaining = 2*self.size-len(built_str)
        if remaining == 0:
            return built_str
        row = 0 if row == 1 else 1
        built_str+=self.array[row][::-1*right][:remaining]
        return built_str

def main():
    cases = int(input())
    for i in range(cases):
        my_object = Solution()
        print(my_object.calculate())


def get_int_list(in_str):
    return [int(i) for i in in_str.strip().split()]


if __name__ == "__main__":
    main()
                        




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