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: Wrong Answer)

import numpy as np

n, k = map(int, input().split())
xyc = [list(input().split()) for _ in range(n)]

b = np.zeros((2 * k, k), np.int64)

for x, y, c in xyc:
    x = int(x)
    y = int(y)

    q, r = divmod(x, k)
    if q % 2:
        y += k
    x = r

    if c == "W":
        y += k

    y %= 2 * k

    b[y][x] += 1

tmp = np.vstack([b, b])
bb = np.hstack([tmp, np.roll(tmp, k)])

bb = np.pad(bb, (1, 0), "constant")
bb = np.cumsum(bb, axis=0)
bb = np.cumsum(bb, axis=1)

cnt = bb[k:, k:] - bb[k:, :-k] - bb[:-k, k:] + bb[:-k, :-k]
ans = cnt.max()
print(ans)