1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
typedef unsigned int ui;
typedef pair<int, int> pii;
typedef vector<int> vi;
typedef vector<long long> vll;
const ll LINF = 1e18;
const int INF = 1e9;

const int N = 8;
char board[N][N];
char target_board[N][N];

bool is_valid(int x, int n) { return x >= 0 && x < n; }

int count_moves_for_square(int x, int y, int n, int m) {
    static const array<pii, 4> offsets = {{{1, 0}, {0, 1}, {-1, 0}, {0, -1}}};

    int cnt = 0;
    for (size_t i = 0; i < offsets.size(); ++i) {
        int new_x = x + offsets[i].first;
        int new_y = y + offsets[i].second;
        if (is_valid(new_x, n) && is_valid(new_y, m)) {
            ++cnt;
        }
    }
    return cnt;
}

ll count_all_transitions(int n, int m, int n_pawns) {
    ll binom_coeff[n * m + 1][n_pawns + 1];
    ll n_moves[n_pawns + 1];
    memset(n_moves, 0, sizeof(n_moves));
    memset(binom_coeff, 0, sizeof(binom_coeff));

    binom_coeff[0][0] = 1;
    for (int i = 1; i <= n * m; ++i) {
        binom_coeff[i][0] = 1;
        for (int k = 1; k <= min(n_pawns, i); ++k) {
            binom_coeff[i][k] = binom_coeff[i - 1][k] + binom_coeff[i - 1][k - 1];
        }
    }

    for (int i = 0; i < n; ++i) {
        for (int j = 0; j < m; ++j) {
            int idx = i * m + j;
            int cnt_moves = count_moves_for_square(i, j, n, m);

            for (int k = n_pawns; k >= 1; --k) {
                ll new_states = binom_coeff[idx][k - 1];
                n_moves[k] += cnt_moves * new_states + n_moves[k - 1];

                if (k <= 1) {
                    // cerr << i << ' ' << j << ' ' << k << ' ' << n_moves[k] << '\n';
                    continue;
                }

                // Remove moves for occupied squares
                if (j >= 1) {
                    n_moves[k] -= 2 * binom_coeff[idx - 1][k - 2];
                }

                if (i >= 1) {
                    n_moves[k] -= 2 * binom_coeff[idx - 1][k - 2];
                }

                // cerr << i << ' ' << j << ' ' << k << ' ' << n_moves[k] << '\n';
            }
        }
    }

    return n_moves[n_pawns];
}

ll count_moves(char board[N][N], int n, int m) {
    static const array<pii, 4> offsets = {{{1, 0}, {0, 1}, {-1, 0}, {0, -1}}};
    int cnt = 0;

    for (int i = 0; i < n; ++i) {
        for (int j = 0; j < m; ++j) {
            if (board[i][j] != 'O') {
                continue;
            }

            for (const pii &offset : offsets) {
                int new_x = i + offset.first;
                int new_y = j + offset.second;
                if (is_valid(new_x, n) && is_valid(new_y, m) && board[new_x][new_y] != 'O') {
                    ++cnt;
                }
            }
        }
    }

    return cnt;
}

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    int n, m;
    cin >> n >> m;

    int n_pawns = 0;
    int start_parity = 0, target_parity = 0;
    for (int i = 0; i < n; ++i) {
        for (int j = 0; j < m; ++j) {
            cin >> board[i][j];
            if (board[i][j] == 'O') {
                ++n_pawns;
                start_parity += i + j;
                start_parity %= 2;
            }
        }
    }

    for (int i = 0; i < n; ++i) {
        for (int j = 0; j < m; ++j) {
            cin >> target_board[i][j];
            if (target_board[i][j] == 'O') {
                target_parity += i + j;
                target_parity %= 2;
            }
        }
    }

    // cerr << "Parity: " << start_parity << ' ' << target_parity << endl;
    // check if parity of start and target are equal
    if (start_parity != target_parity) {
        cout << 0 << endl;
        return 0;
    }

    ll all_transitions = count_all_transitions(n, m, n_pawns);
    // cerr << "All transitions: " << all_transitions << endl;

    int moves_from_target = count_moves(target_board, n, m);
    // cerr << "Moves from target: " << moves_from_target << endl;
    cout << fixed << setprecision(15) << static_cast<double>(2 * moves_from_target) / all_transitions << endl;

    return 0;
}