Problem:
Score : 500 points

Problem Statement

AtCoDeer is thinking of painting an infinite two-dimensional grid in a checked pattern of side K.
Here, a checked pattern of side K is a pattern where each square is painted black or white so that each connected component of each color is a K × K square.
Below is an example of a checked pattern of side 3:

AtCoDeer has N desires.
The i-th desire is represented by x_i, y_i and c_i.
If c_i is B, it means that he wants to paint the square (x_i,y_i) black; if c_i is W, he wants to paint the square (x_i,y_i) white.
At most how many desires can he satisfy at the same time?

Constraints

1 ≤ N ≤ 10^5

1 ≤ K ≤ 1000

0 ≤ x_i ≤ 10^9

0 ≤ y_i ≤ 10^9

If i ≠ j, then (x_i,y_i) ≠ (x_j,y_j).

c_i is B or W.

N, K, x_i and y_i are integers.

Input

Input is given from Standard Input in the following format:

N K
x_1 y_1 c_1
x_2 y_2 c_2
:
x_N y_N c_N

Output

Print the maximum number of desires that can be satisfied at the same time.

Sample Input 1

4 3
0 1 W
1 2 W
5 3 B
5 4 B

Sample Output 1

4

He can satisfy all his desires by painting as shown in the example above.

Sample Input 2

2 1000
0 0 B
0 1 W

Sample Output 2

2

Sample Input 3

6 2
1 2 B
2 1 W
2 2 B
1 0 B
0 6 W
4 5 W

Sample Output 3

4

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
ll white_map[2001][2001];
ll black_map[2001][2001];
ll n, k;
ll k2;

inline ll get_white(int x, int y) {
    if (x < 0 or y < 0 or x >= k2 or y >= k2) return 0LL;
    return white_map[x][y];
}
inline ll get_black(int x, int y) {
    if (x < 0 or y < 0 or x >= k2 or y >= k2) return 0LL;
    return black_map[x][y];
}
inline ll get_white_range(int sx, int sy, int dx, int dy) {
    if (dx < sx or dy < sy) return 0LL;
    return get_white(dx, dy) - get_white(dx, sy-1) - get_white(sx-1, dy) + get_white(sx-1, sy-1);
}
inline ll get_black_range(int sx, int sy, int dx, int dy) {
    if (dx < sx or dy < sy) return 0LL;
    return get_black(dx, dy) - get_black(dx, sy-1) - get_black(sx-1, dy) + get_black(sx-1, sy-1);
}
int main() {
    

    cin >> n >> k;
    k2 = k*2;
    for (int i=0; i<n; i++) {
        ll x, y;
        char color;
        scanf("%lld %lld %c\n", &x, &y, &color);
        x %= k2;
        y %= k2;
        // printf("(%d, %d)\n", x, y);
        if (color == 'W') {
            white_map[x][y]++;
        } else {
            black_map[x][y]++;
        }
    }
    for (ll i=0; i<k2-1; i++) {
        for (ll j=0; j <k2; j++) {
            white_map[i+1][j] += white_map[i][j];
            black_map[i+1][j] += black_map[i][j];
        }
    }
    for (ll i=0; i<k2; i++) {
        for (ll j=0; j <k2-1; j++) {
            white_map[i][j+1] += white_map[i][j];
            black_map[i][j+1] += black_map[i][j];
        }
    }
    
    ll ans = 0;
    for (ll i=0; i < k; i++) {
        for (ll j=0; j < k; j++) {
            ll local_ans = get_white_range(0,0, i-1, j-1);
            local_ans += get_white_range(i, j, i+k-1, j+k-1);
            local_ans += get_white_range(i+k, j+k, k2-1, k2-1);
            local_ans += get_white_range(0, j+k, i-1, k2-1);
            local_ans += get_white_range(i+k, 0, k2-1, j-1);
            local_ans += get_black_range(i, j+k, i+k-1, k2-1);
            local_ans += get_black_range(0, j, i-1, j+k-1);
            local_ans += get_black_range(i+k, j, k2-1, j+k-1);
            local_ans += get_black_range(i, 0, i+k-1, j-1);
            ans = max(ans, local_ans);
        }
    }
    for (ll i=0; i < k; i++) {
        for (ll j=0; j < k; j++) {
            ll local_ans = get_black_range(0,0, i-1, j-1);
            local_ans += get_black_range(i, j, i+k-1, j+k-1);
            local_ans += get_black_range(i+k, j+k, k2-1, k2-1);
            local_ans += get_black_range(0, j+k, i-1, k2-1);
            local_ans += get_black_range(i+k, 0, k2-1, j-1);
            local_ans += get_white_range(i, j+k, i+k-1, k2-1);
            local_ans += get_white_range(0, j, i-1, j+k-1);
            local_ans += get_white_range(i+k, j, k2-1, j+k-1);
            local_ans += get_white_range(i, 0, i+k-1, j-1);
            ans = max(ans, local_ans);
        }
    }
    
    cout << ans << '\n';
    return 0;
    /*
    //dump map
    for (ll i=0; i < k2; i++) {
        for (ll j=0; j < k2; j++) {
            // cout << white_map[i][j] << ' ';
            cout << get_white(i, j) << ' ';
            // cout << get_white_range(i, j, i, j) << ' ';
        }
        cout << '\n';
    }
    cout << "========================" << endl;
    for (ll i=0; i < k2; i++) {
        for (ll j=0; j < k2; j++) {
            // cout << black_map[i][j] << ' ';
            cout << get_black(i, j) << ' ';
            // cout << get_black_range(i, j, i, j) << ' ';
        }
        cout << '\n';
    }
	return 0;
	*/
}