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

def getBoundary(p, k, k2):
    if p - k + 1 >= 0:
        return [(p - k + 1, p + 1)]
    else:
        return [(0, p + 1), ((p - k + 1) % k2, k2)]

n, k = map(int, input().strip().split())
k2 = k * 2
b = [[0 for i in range(k2 + 1)] for j in range(k2 + 1)]

for i in range(n):
    x_str, y_str, c_str = input().strip().split()
    x = int(x_str) % k2
    y = (int(y_str) + (c_str == 'B') * k) % k2
    for xb, xe in getBoundary(x % k2, k, k2):
        for yb, ye in getBoundary(y % k2, k, k2):
            b[xb][yb] += 1
            b[xb][ye] -= 1
            b[xe][yb] -= 1
            b[xe][ye] += 1
    for xb, xe in getBoundary((x + k) % k2, k, k2):
        for yb, ye in getBoundary((y + k) % k2, k, k2):
            b[xb][yb] += 1
            b[xb][ye] -= 1
            b[xe][yb] -= 1
            b[xe][ye] += 1

arr_b = np.array(b)
for i in range(k2):
    arr_b[i + 1] += arr_b[i]
for j in range(k2):
    arr_b[:, j + 1] += arr_b[:, j]
print(arr_b.max())