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 Python (3.4.3) (judge verdict: Accepted)

import numpy as np
N, K = map(int, input().split())
ub = K * 2
b = [[0 for j in range(ub + 1)] for i in range(ub + 1)]
ADD = {"B": 0, "W": K}


def create_bound(p, K, ub):
    if p - K + 1 >= 0:
        return [(p - K + 1, p + 1)]
    else:
        return [(0, p + 1), ((p - K + 1) % ub, ub)]


for i in range(N):
    x, y, c = input().split()
    nx = int(x) % ub
    ny = (int(y) + ADD[c]) % ub
    for add in [0, K]:
        for xlb, xub in create_bound((nx + add) % ub, K, ub):
            for ylb, yub in create_bound((ny + add) % ub, K, ub):
                b[xlb][ylb] += 1
                b[xlb][yub] -= 1
                b[xub][ylb] -= 1
                b[xub][yub] += 1

b_arr = np.array(b)
for i in range(ub):
    b_arr[i + 1, :] += b_arr[i, :]
for j in range(ub):
    b_arr[:, j + 1] += b_arr[:, j]
print(b_arr.max())