#include <bits/stdc++.h> using namespace std; #define int long long #define fi first #define se second #define all(a) a.begin(), a.end() #define pii pair<int,int> #define el '\n' typedef vector<vector<int>> matrix; const int maxN = 505; const int base = 311; const int MOD = 1000000003; const int base2 = 307; const int MOD2 = 1000000007; const int INF = 1e15; const int NEG_INF = -1e15; const int LOG = 20; int n, q, m, a, b, t, k, C; string s; // int dist[maxN]; int A[maxN], B[maxN]; int match[maxN], parent[maxN], block[maxN]; bool visited[maxN]; int fac[maxN]; int cntT[30], cntS[30]; int dx[] = {-1, -1, -1, 0, 0, 1, 1, 1}; int dy[] = {-1, 0, 1, -1, 1, -1, 0, 1}; int H[205][205], H2[maxN], R[maxN], power[maxN], power2[maxN], pb1[LOG], pb2[LOG]; string SS[maxN]; string S, T; int getHash( int i, int j, const int H[]){ if(i > j) return 0; return (H[j] - H[i-1] * power[j-i+1] + MOD*MOD) % MOD; } int getHash2( int i, int j){ return (H2[j] - H2[i-1] * power2[j-i+1] + MOD2*MOD2) % MOD2; } // int getHashT( int i, int j ){ // return (R[j] - R[i-1] * power[j-i+1] + MOD*MOD) % MOD; // } int add(int a, int b, int mod) { return (a + b) % mod; } int mul(int a, int b, int mod) { return (a * b) % mod; } pii dp[LOG][maxN][maxN]; vector<pii> final_hashes; void solve_dir(int dir){ int dr = dx[dir]; int dc = dy[dir]; for(int i = 1; i <= m; i++){ for(int j = 1; j <= n; j++){ dp[0][i][j] = {SS[i][j], SS[i][j]}; } } int cur_len = 1; // 2^0 for (int p = 1; p < LOG; p++) { for (int i = 1; i <= m; i++) { for (int j = 1; j <= n; j++) { int step_r = (dr * (cur_len % m)) % m; int ni = ((i - 1 + step_r) % m + m) % m + 1; int step_c = (dc * (cur_len % n)) % n; int nj = ((j - 1 + step_c) % n + n) % n + 1; int h1 = add(mul(dp[p-1][i][j].fi, pb1[p-1], MOD), dp[p-1][ni][nj].fi, MOD); int h2 = add(mul(dp[p-1][i][j].se, pb2[p-1], MOD2), dp[p-1][ni][nj].se, MOD2); dp[p][i][j] = {h1, h2}; } } cur_len *= 2; } for (int i = 1; i <= m; i++) { for (int j = 1; j <= n; j++) { int h1 = 0, h2 = 0; int ci = i, cj = j; for (int p = LOG - 1; p >= 0; p--) { if ((k >> p) & 1) { h1 = add(mul(h1, pb1[p], MOD), dp[p][ci][cj].fi, MOD); h2 = add(mul(h2, pb2[p], MOD2), dp[p][ci][cj].se, MOD2); int step = (1LL << p); int step_r = (dr * (step % m)) % m; ci = ((ci - 1 + step_r) % m + m) % m + 1; int step_c = (dc * (step % n)) % n; cj = ((cj - 1 + step_c) % n + n) % n + 1; } } final_hashes.push_back({h1, h2}); } } } void run_test() { cin >> m >> n >> k; // n = S.length(); // m = T.length(); // S = '_' + S; // T = '_' + T; // int HashT = 0; // for( int i = 1; i <= n; i++ ){ // H[i] = (H[i-1] * base + S[i] - 'a' + 1) % MOD; // } // for( int i = 1; i <= m; i++ ){ // HashT = (HashT * base + T[i] - 'a' + 1) % MOD; // } for(int i = 1; i <= m; i++){ cin >> SS[i]; SS[i] = '_' + SS[i]; } pb1[0] = base; pb2[0] = base2; for (int p = 1; p < LOG; p++) { pb1[p] = mul(pb1[p-1], pb1[p-1], MOD); pb2[p] = mul(pb2[p-1], pb2[p-1], MOD2); } for(int i = 0; i < 8; i++){ solve_dir(i); } sort(all(final_hashes)); int numerator = 0; int cur_cnt = 0; for (int i = 0; i < final_hashes.size(); i++) { cur_cnt++; if (i == final_hashes.size() - 1 || final_hashes[i] != final_hashes[i+1]) { numerator += cur_cnt * cur_cnt; cur_cnt = 0; } } int total_choices = m * n * 8; int denominator = total_choices * total_choices; int common = __gcd(numerator, denominator); cout << numerator / common << "/" << denominator / common << el; } signed main(){ ios::sync_with_stdio(false); cin.tie(nullptr); // freopen("lightsout.in", "r", stdin); // freopen("lightsout.out", "w", stdout); int test = 1; // cin >> test; while (test-- > 0) { run_test(); } }